text:
Q: Why suddenly spring asks me for aspectj? I'm currently using spring security for my project.
After adding this bean (user for remember-me feature):
    <b:bean id="jdbcTokenRepository" class="org.springframework.security.web.authentication.rememberme.JdbcTokenRepositoryImpl">
        <b:property name="createTableOnStartup" value="false"/>
        <b:property name="dataSource" ref="dataSource"/>
    </b:bean>

User token still wasn't saved. All because 'dataSource' was configured to autoCommit=false.
So first thing that I did was (really ugly because I had to create one more class, but it was working):
 import org.springframework.security.web.authentication.rememberme.JdbcTokenRepositoryImpl;
 import org.springframework.transaction.annotation.Transactional;

 @Transactional
 public class TransactionalJdbcTokenRepositoryImpl extends JdbcTokenRepositoryImpl {}

So much better solution for me was the following:
    <b:bean id="jdbcTokenRepository"
                class="org.springframework.security.web.authentication.rememberme.JdbcTokenRepositoryImpl">
        <b:property name="createTableOnStartup" value="false"/>
        <b:property name="dataSource" ref="dataSource"/>
    </b:bean>

    <tx:advice id="tokenRepoAdvice" transaction-manager="transactionManager">
        <tx:attributes>
            <tx:method name="*"/>
        </tx:attributes>
    </tx:advice>

    <aop:config proxy-target-class="false" expose-proxy="false">
        <aop:advisor id="tokenRepoAdvisor"
                             advice-ref="tokenRepoAdvice"
                             pointcut="bean(jdbcTokenRepository)"/>
    </aop:config>

Pure XML no unnecessary classes:
But for some reason spring requires aspectj now!
Caused by: java.lang.ClassNotFoundException: org.aspectj.weaver.reflect.ReflectionWorld$ReflectionWorldException
at org.apache.catalina.loader.WebappClassLoader.loadClass(WebappClassLoader.java:1702)
at org.apache.catalina.loader.WebappClassLoader.loadClass(WebappClassLoader.java:1547)
... 68 more

Why? Without this config my dao layer was working just fine wiht @Transactional annotation. How to avoid adding aspectj library but still preserve xml config?

A: After rereading spring docs, I have found a solution:
    <tx:advice id="tokenRepoAdvice" transaction-manager="transactionManager">
        <tx:attributes>
            <tx:method name="*"/>
        </tx:attributes>
    </tx:advice>

    <bean id="jdbcTokenRepository"
            class="org.springframework.aop.framework.ProxyFactoryBean">
        <property name="proxyInterfaces" alue="org.springframework.security.web.authentication.rememberme.PersistentTokenRepository"/>
        <property name="target">
            <bean class="org.springframework.security.web.authentication.rememberme.JdbcTokenRepositoryImpl">
                <property name="createTableOnStartup" value="false"/>
                <property name="dataSource" ref="dataSource"/>
            </bean>
        </property>
        <property name="interceptorNames">
            <list>
                <value>tokenRepoAdvice</value>
            </list>
        </property>
</bean>

And the initial problem was a parent class of JdbcTokenRepositoryImpl (it has final methods - that's why aspectj was required)

redpajama_set_name: RedPajamaStackExchange


This course prepares librarians on how to work with first responders, including public safety, fire/rescue, public health department, local health care facilities, and emergency planners.
Libraries have a critical role to play in disaster preparation and recovery, but they need a seat at the table with the right people, and they need to have something to offer. During and after a disaster are not the times for libraries to learn how to work with local, state, and federal responders. The time to build and develop these relationships is before a disaster strikes. It is critical to have a seat at the table before, during, and after a disaster.
This course will cover the basics of initiating and developing relationships with all relevant local, state, and federal officials and agencies, as well as with emergency responders.
This self-paced course fulfills one of the requirements of the Medical Library Association (MLA) Disaster Information Specialization, Advanced Level. Students will earn four MLA Continuing Education (CE) credits.
If you are taking this course for Medical Library Association credits, please register and complete the class through our free online registration system.
Note: After reviewing the information on this page, please be sure to complete the pre-course evaluation on the next page before continuing with the course.
This course is divided into the following sections. Use the Next and Back buttons at the bottom of the page to move through the course.
Each section is a single page with content and activities that help reinforce the topics covered. For more information on additional ways to navigate the course, see the "Getting Around" section below.
Each section is summarized with key points, which are the most critical take-home messages of the material.
Throughout this course, the National Library of Medicine will be referred to using the acronym NLM and the Disaster Information Management Research Center will be called NLM Disaster Health.
We use "community" in this course to refer to the geographic neighborhood community (e.g., public library community), as well as work/organizational community (e.g., hospital or academic community).
Identify and explain the roles played by governmental and nongovernmental organizations in different types of disasters.
Recognize library roles that are valuable in community disaster planning.
Develop relationships that will help secure a seat at the table.
Click the Back and Next buttons at the bottom of each page to move through the material.
Use the Menu button in the upper right corner of each page to access the main sections or contact support.
Press the LEFT and RIGHT arrow keys on your keyboard to move through the material.
Use a combination of the above methods to explore the course contents.
At the top of each page, links will indicate where the current page falls within the overall course. You can also click these links to return to this homepage or to the section that contains the page.
Throughout the course, you will encounter links. Any link denoted with an External Link icon ( ) points to a resource outside of this course and will typically open in a new tab or window. All other links are to other areas within the course.
Additionally, clicking on an image will open a larger view of the image in a pop-up box within the page.
If you need assistance using this course or have questions or comments about any of the content presented here, please select the About / Getting Help option from the upper right menu.
This is an example of a pop-up within the training. These are included throughout the course to provide additional information or resources related to a particular topic.


text:
How do you calculate after tax return on a bond?
The effective after-tax yield can be found by multiplying the percentage of yield after taxes by the pre-tax rate of return. If the investment in this example returns 8 percent, that number would be multiplied by 0.70 to get an after-tax yield of 5.6 percent.
How do you calculate after tax return?
Subtract your percentage tax rate on the security's income from 1. Multiply your result by the pretax return to calculate the after-tax return on the income. In this example, assume you pay a 15 percent tax rate on the income. Subtract 15 percent, or 0.15, from 1 to get 0.85.
How do you calculate after tax yield on a municipal bond?
Calculating Tax Equivalent Yield
Find the reciprocal of your tax rate (1 – your tax rate). If you pay 25% tax, your reciprocal would be (1 – . 25) = . 75, or 75%.
Divide this amount into the yield on the tax-free bond to find out the TEY. For example, if the bond in question yields 3%, use (3.0 / . 75) = 4%.
How do I calculate after tax equity?
Calculation of Return on Equity
ROE = Net Income/Average Shareholder's Equity.
Net Income is the amount of income, net of expenses and taxes that a company generates for a given period.
Average shareholders' equity is calculated by adding equity at the beginning of the period.
How do you calculate IRR after tax?
It is calculated by taking the difference between the current or expected future value and the original beginning value, divided by the original value and multiplied by 100.
How do you calculate tax free bond yield?
The interest payment received will be on the face value i.e. Rs 70 for every bond of Rs 1,000. So, Current yield = 70 / 1250 = 5.6 per cent.
How do you calculate tax free equivalent yield?
To calculate the tax free equivalent yield, multiply the corporate bond yield by the total tax rate. This is the tax free equivalent yield. In other words, this is the return needed from a tax free investment to equal the return of a corporate bond.
What is post tax return?
An after-tax return is the profit realized on an investment after deducting taxes due. After-tax returns help investors determine their true earnings. When calculating the after-tax return, it is important to only include income received and costs paid during the reporting period.
Is return on assets before or after tax?
The return on total assets ratio indicates how well a company's investments generate value, making it an important measure of productivity for a business. It is calculated by dividing the company's earnings after taxes (EAT) by its total assets, and multiplying the result by 100%.
Is IRR pre or post tax?
The method of calculating a rate of return (IRR) of a net cash flow is independent of the tax status of the cash flows (pre-tax or after-tax). If the net cash flows used to calculate the IRR are after-tax net cash flows, then the resulting IRR is the IRR of the net cash flow after taxes.
How do I calculate IRR using Excel?
Excel's IRR function calculates the internal rate of return for a series of cash flows, assuming equal-size payment periods. Using the example data shown above, the IRR formula would be =IRR(D2:D14,. 1)*12, which yields an internal rate of return of 12.22%.
How are bonds taxed?
Most bonds are taxable. Generally, only bonds issued by local and state governments (i.e., municipal bonds) are tax-exempt and even then special rules may apply. You must pay tax on both interest payments and on capital gains if you redeem the bond before its maturity date.

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text:
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Abraham and Jacob tithed, didn't they?
Tithing did occur before the Mosaic Law was given. There are actually three kinds of tithing in the Bible…the way Abraham did it, the way Jacob did it, and the way that the Jews did it.
Abraham (Genesis 14):
Abraham tithed to Melchizadek…he gave a tenth of the spoils of war. If we're to tithe as Abraham did, we should give a tenth of the spoils of war to the king. Since that's all the information we have, that's all we should presume about the passage.
Jacob (Genesis 28):
Jacob tithed directly to God out of gratitude…if you want to tithe out of gratitude, that's between you and God. Since that's all the information we have, that's all we should presume about the passage.
The Jews:
As I've written elsewhere, the tithe was a debt owed by Jews to God with specific instructions on who could collect it, how it was to be used, etc. If we're to tithe as the Jews did, we need to read and understand and follow the instructions given to the Jews. If your pastor has done his homework, he can tell you that this isn't happening. Other than the examples of Abraham and Jacob, everything we read in the Bible about tithing is directed at Jews. Unless your pastor is willing to put himself under the entire Mosaic Law, there's no Scriptural reason for him to say that you should tithe. Abraham tithed to the king and Jacob tithed to God Himself. Where do we find instructions that Christians should tithe to the church? There aren't any.
I do believe in giving.
I believe that Christians should be very generous, and that the New Testament example is to give cheerfully and generously to those in need as the Holy Spirit leads us. Anyone who tells you that the Bible tells Christians to tithe is either ignorant or lying, because it simply isn't there. Hey, if someone wants to give ten percent of their income to the church, that's their business…but if someone says "God wants Christians to tithe to the church" I want to know which book and which verse they say tells them that. What could be more reasonable than that?
Here's what I'd suggest:
Get a pen and paper and then ask your pastor for a list of Bible verses that teach Christians to tithe. If he doesn't have a list for you, ask him if he will make one for you. Whatever he tells you, write it down and look it up. See for yourself what the Bible says, and compare what you think with what the pastor says. If (after doing your homework) you disagree with your pastor, go to him respectfully and ask to talk about it. His response will go a long way toward understanding what sort of spiritual leader he is.
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Soil organic matter is the largest carbon (C) pool in the terrestrial C cycle, and soil CO₂ emissions surpass anthropogenic emissions from fossil fuel combustion by a factor of nine. Therefore, mechanisms controlling C stabilisation in soils and its feedback to climate change are widely debated. During decomposition, microbial substrate-use efficiency is an important property because it determines the allocation of substrate C to biosynthesis and respiratory losses. High efficiency values indicate that C primarily remains in soils while low efficiency implies that C is primarily lost into the atmosphere. Despite empirical evidence that efficiency is temperature sensitive, traditional Earth system models treat this property as a constant.
The aim of this thesis was to improve our mechanistic understanding of drivers regulating substrate-use efficiency with special consideration to climate change. It investigated the impacts of (i) temperature, (ii) microbial community composition and (iii) substrate quality on substrate-use efficiency. Within the thesis, a microbial energetics approach was applied and further developed using isothermal calorimetry. Further, the thesis compared common approaches for measuring microbial substrate-use efficiency, and the implications of the resultant empirical data for projected C stocks were tested using a modelling approach.
Substrate-use efficiency was generally temperature sensitive and decreased with increasing temperature. The observed temperature responses were non-linear and varied across land use management systems. The changes in substrate-use efficiency with temperature were driven rather by changes in microbial physiology than by shifts in active microbial communities. Nevertheless, fungi and Gram-negative bacteria tended towards relatively higher efficiencies. Efficiencies varied among utilised substrates, but substrate quality per se was a poor proxy for efficiency. Projected losses from soil C stocks varied across land use management systems and were up to 39 % and 15 % for grassland and forest systems, respectively. Results from the modelling approach confirmed that substrate-use efficiency is one of the factors to which soil C stocks react most sensitively. Findings from this thesis emphasise the importance of furthering our understanding of substrate-use efficiency for reliable climate projections.
I Bölscher T, Wadsö L, Börjesson G, Herrmann AM (2016). Differences in substrate use efficiency: impacts of microbial community composition, land use management, and substrate complexity. Biology and Fertility of Soils 52, 547-559.
IV Herrmann AM, Bölscher T (2015). Simultaneous screening of microbial energetics and CO₂ respiration in soil samples from different ecosystems. Soil Biology and Biochemistry 83, 88-92.


Description: The Council's development plan documents and planning policies are under-pinned by evidence, so as to ensure that the policies are robust and relevant.
The Council's planning policies are underpinned by evidence, to ensure that they are robust and justified. The documents below are part of the evidence base which has been used to support Council planning documents, in particular, the Core Strategy. As studies informing a long-term plan, many remain highly relevant. Additional evidence is also available in support of the recent Development Policies Local Plan.
They are organised below around the topic Matters discussed in the examination of the Core Strategy. To see the studies, please scroll down to find the topic matter of interest. SA and HRA (environmental) information is provided at the bottom of the page.


When photographer Lesley McIntyre's daughter Molly was born in 1984, it was revealed she was suffering from a muscular abnormality and the doctors thought it was highly unlikely she would survive more than a few weeks or months, most likely never leaving the hospital. In spite of this, McIntyre did take her home and Molly lived until her fourteenth birthday. However, in all of this time, her condition was never properly diagnosed.
Before the birth, McIntyre assumed she'd be able to combine her career as a photographer with being a parent. She did achieve this, but not in the manner she anticipated. Grounded in domestic life and unable to pursue the commissions she might have done had her child been more robust, McIntyre began recording the details of the day to day and found observing childhood fascinating. The first pictures were taken shortly after Molly's birth and the last a few days before she died at home in South London. Because of the precarious nature of Molly's life, the photographs have an extraordinary poignancy.
Molly's is a story that crosses all barriers of race, class and gender. Any family, at any time or anywhere in the world can find themselves confronted by the reality of disability. Having spent years fighting so that her child would not be socially and educationally marginalized by her physical disability, McIntyre is fully aware of how resistant many people are to engaging with such subject matter. This is not confined to disability alone – but to death, and in particular child death.
This series of black and white photographs reveal a life full of vitality and the ordinariness of growing up, despite the challenges both Molly and McIntyre faced on a daily basis. An extremely bright and well loved child, mature beyond her young years, McIntyre hopes that Molly's legacy is that she gave so many people the capacity to engage with life and to feel.
The Time Of Her Life, published in hard back by Jonathan Cape and available throughout the exhibition, was Highly Commended by the John Kobal Book Award.


Born in 1961 in St. Paul, Minnesota, the grandson of the Chief of the Winnebago Native American tribe.
Enlisted in the United States Marine Corps in 1980 and entered the Marine Corps Security Battalion Guard school, a rigorous, elite training program in which he was instructed in espionage and counterespionage techniques. Was given TOP SECRET security clearance and was assigned to the U.S. Embassy in Moscow in 1984.
As part of his assignment, he signed a non-fraternization agreement, thereby promising not to engage in friendships with Soviet citizens and to report any such contacts. Lonetree apparently received undo attention because most citizens of Moscow had never seen a Native American before. Despite such attention he was described as lonely and somber and was believed to have begun drinking excessively.
Attended the annual Marine ball in November 1985 and was introduced to Violetta Seina, a 25 year old Russian woman who was an embassy employee. The two began dating soon thereafter. Was part of the the Marine unit assigned to provide security for the 1985 summit meeting between Soviet premier Mikhail Gorbachev and U.S. President Ronald Reagan.
Was introduced by Violetta to her "Uncle Sasha", who in reality was a KGB intelligence officer (named Aleksiy Yefimov). Sasha recruited Lonetree to become a "friend of the Soviet Union." Lonetree eventually provided Sasha with information about the embassy and the U.S. Ambassador. Sasha also asked him to plant a "bug" in the Embassy but Lonetree allegedly refused to do, but instead provided plans for the building.
Was assigned to the U.S. Embassy in Vienna, Austria in March 1986. While in Vienna, Lonetree met with Sasha and provided him with information about embassy employees and with floor plans to the building. In return he was given $2,500 US. Over time, Lonetree provided TOP SECRET documents as well as a burn bag containing more than 100 documents related to U.S. arms reduction.
On December 12, 1986, Sasha turned Lonetree over to another handler, who promised to aid Lonetree in re-uniting with Violetta. Two days later, however, Lonetree reported his actions to the CIA station chief. He was immediately turned over to the Navy Intelligence Service (NIS) and placed under arrest, charged with espionage.
NIS launched a nine month investigation, during which all of the Marine guards in the Moscow embassy were replaced. NIS also arrested Corporal Arnold Bracy, who signed a confession indicating that he acted as a lookout for Lonetree, while Lonetree secreted KGB operatives into the embassy to plant bugs. Also arrested was Marine staff Sergeant Robert Stufflebeam, who was charged wit failing to report fraternization with Soviet women. The charges against Bracy and Stufflebeam were eventually dropped.
NIS was highly criticized for its over-exuberance with its investigation, accused of blowing things way out of proportion in order to show off having disrupted a spy operation. Lonetree was convicted on multiple counts of turning over classified information, was court-martialed in 1987 and sentenced to 30 years in prison. He was the first U.S. Marine Corps member ever convicted of espionage. Because of his cooperation with authorities, his sentence was reduced to 25 years of which he served nine before being released in February 1996.


The 'Changing the Guard' ceremony at Horse Guards is quite similar to the one that takes place at Buckingham Palace, but it doesn't have any marching bands or music (unless you include the guy blasting out a few notes on his bugle). But it is also a lot-less crowded, and unlike the ceremony at Buckingham Palace this one takes place every day of the year -- in all weathers. So you might want to check the schedule for Buckingham Palace first, and if that one isn't running on the day you're going then this one makes a decent alternative.
The ceremony begins at 10.28 AM when the replacement guards start riding towards Horse Guards from Hyde Park Barracks. If you've got plenty of energy (and you will definitely need a lot of energy to keep up with the horses!) then you can watch them pass Apsley House and Wellington Arch before riding all the way down Constitution Hill to Buckingham Palace. They will then ride down the Mall and assemble on the parade ground.
A series of changeovers will then take place on the parade ground, with the horses riding in and out of a door in the courtyard. They will also switch-over the horses that are standing in the horseboxes on Whitehall. Unfortunately it isn't possible to see the entire parade in one day because it takes place in two different locations, so what you have to do is pick a spot beforehand and accept that you will miss half of the show. Luckily Craig has been twice and seen the whole thing from both places, so read his review of the ceremony beforehand, because it goes into detail about the timings and the best place to stand to see as much as possible. He's also included a lot more photographs. Feel free to ask him a question about the ceremony, or write your own review on the forum.
Even if you can't make the ceremony itself, then it's still definitely worth visiting Horse Guards just to see the two mounted sentries standing in the horse boxes (between 10 AM and 4 PM). This is one of the most popular photos for tourists. The guards are not allowed to smile or interact with the crowd, so you will frequently see some students trying to make them laugh. Craig has written a big review about Horse Guards as well.
At 4 PM the horses are replaced by Foot Guards during the Dismounting Ceremony (also known as the 4 O'Clock Parade).


We strongly encourage anyone who joins the HSHRC to become a licensed Ham radio operator so you can use our communication services and enjoy the wonderful world of ham radio. Some folks might consider ham radio to be old timey or something that is not necessary anymore. Nothing could be further from the truth. In emergency situations, like recent hurricanes and even 9/11, it is ham radio operators that provide critical emergency communication operations.
In addition, with the advent of the Internet, there are many new and exciting technologies emerging that bring ham radio technology into the world of high tech. From voice synthesizers, to controlling remote repeaters through the Internet, to connecting geographically separated repeaters through the Internet, the possibilities are endless.
Everything that you need to know about getting your ham radio license is located The FCC Amateur Radio Service.
It is now easier than ever to become a ham radio operator. All you need is your Technician license in order to communicate on the two meter band which is where we communicate locally in Nashville and Middle Tennessee. It's a great first step, and the exam is really pretty easy to prepare for and to take.
ARRL.org Store - The American Radio Relay League has everything you will ever need.
CWCom Software - Program for practicing code on the Internet. We tune it to 4644.
Web site designed and maintained by Smithware - CIO Services in Middle Tennessee.


text:
\section{Introduction}
\label{sec:intro}
A corner stone in gauge theory is the Hitchin--Kobayashi correspondence (\cite{skobayashi82,lubke83,uhlenbeckyau86,donaldson87}). This celebrated generalisation of the Narasimhan and Seshadri theorem asserts that a holomorphic vector bundle over a K\"ahler manifold carries an Hermite--Einstein metric if and only if it is polystable in the sense of Mumford and Takemoto (\cite{Mum62,Tak72}). The interplay between the differential geometric side, hermitian Yang--Mills connections (HYM for short) that originated from physics, and the algebro-geometric side, the stability notion motivated by moduli constructions, has had many applications and became a very fertile source of inspiration. Given that HYM connections are canonically attached to polystable vector bundles, it is natural to investigate their relations to natural maps between vector bundles, such as pullbacks. In this paper, we address the problem of pulling back HYM connections along blow-ups. While the similar problem for extremal K\"ahler metrics has seen many developments in the past ten years \cite{ArPac06,ArPac09,ArPacSing,sze-blowupsII,SeySze,dervan-sektnan-blowups}, relatively little seems to be known about the behaviour of HYM connections under blow-ups \cite{Buchdahl,dervansektnan19}. In this paper, under some mild asumptions, we solve the problem for pullback of {\it semi-stable} vector bundles on blow-ups along smooth centers.

Let $\pi: X' \to X$ be the blow-up of a polarised K\"ahler manifold $(X,[\omega])$ along a submanifold $Z \subset X$, and $E'=\pi^*E$ the pullback of a holomorphic vector bundle $E\to X$. For $0<\varepsilon\ll 1$, $L_\varepsilon:=\pi^*[\omega] -\varepsilon c_1(Z')$ defines a polarisation on $X'$, where we set $Z'=\pi^{-1}(Z)$ the exceptional divisor. There are obstructions for $E'$ to admit HYM connections with respect to $\omega_\varepsilon\in c_1(L_\varepsilon)$, with $0<\varepsilon\ll 1$. In particular, $E$ should be {\it simple} and {\it semi-stable} with respect to $[\omega]$ (see Section \ref{sec:blowup}). In the latter case, $E$ admits a Jordan--Holder filtration by semi-stable sheaves with polystable graded object $\mathrm{Gr}(E)$ (see Section \ref{sec:slopestability} for definitions). A further obstruction comes then from subsheaves of $E$ arising from $\mathrm{Gr}(E)$. While those sheaves have the same slope as $E$, their pullbacks to $X'$ could destabilise $E'$. Our main result asserts that those are actually the only obstructions for $E'$ to carry a HYM connection, under some mild asumptions on $\mathrm{Gr}(E)$.

Recall that a semi-stable holomorphic vector bundle $E\to (X,[\omega])$ is said to be {\it sufficiently smooth} if its graded object $\mathrm{Gr}(E)$ is locally free. Let $\mathfrak{E}_{[\omega]}$ denote the set of all subbundles of $E$ arising in a Jordan--Holder filtration for $E$, or equivalently of same slope as $E$ with respect to $[\omega]$. For $\mathcal{F}\in \mathfrak{E}_{[\omega]}$, denote by $\mu_{L_\varepsilon}(\mathcal{F})=\frac{c_1(\pi^*\mathcal{F})\cdot L_\varepsilon^{n-1}}{\mathrm{rank}(\mathcal{F})}$ the slope of $\pi^*\mathcal{F}$ on $(X', L_\varepsilon)$.
\begin{theorem}
 \label{theo:intro}
 Let $E\to X$ be a simple sufficiently smooth semi-stable holomorphic vector bundle on $(X,[\omega])$.  Assume that the stable components of $\mathrm{Gr}(E)$ are pairwise non-isomorphic.
  Then, there exists $\varepsilon_0>0$ and a sequence of HYM connections $(A_\varepsilon)_{\varepsilon\in (0,\varepsilon_0)}$ on $\pi^*E$ with respect to $(\omega_\varepsilon)_{\varepsilon\in (0,\varepsilon_0)}$ if and only if
 \begin{equation}
 \label{eq:necessarynumericalcondition}
 \forall\; \mathcal{F}\in \mathfrak{E}_{[\omega]},\: \mu_{L_\varepsilon}(\mathcal{F})\underset{\varepsilon\to 0}{<} \mu_{L_\varepsilon}(E).
\end{equation}
 In that case, if $A$ denotes a HYM connection on $\mathrm{Gr}(E)$ with respect to $\omega$, then $(A_\varepsilon)_{\varepsilon\in(0,\varepsilon_0)}$ can be chosen so that $A_\varepsilon \underset{\varepsilon\to 0}{\longrightarrow} \pi^*A$ in any Sobolev norm.
\end{theorem}
 In the statement, the expression $\mu_{L_\varepsilon}(\mathcal{F})\underset{\varepsilon\to 0}{<} \mu_{L_\varepsilon}(E)$ means that the first non-zero term in the $\varepsilon$-expansion for $\mu_{L_\varepsilon}(E)-\mu_{L_\varepsilon}(\mathcal{F})$ is strictly positive.
 \begin{remark}
  Simplicity, semi-stability and condition (\ref{eq:necessarynumericalcondition}) are necessary to produce the connections $(A_\varepsilon)$ from Theorem \ref{theo:intro}. The other two asumptions on $\mathrm{Gr}(E)$ are technical. Assuming $\mathrm{Gr}(E)$ to be locally free enables to see $E$ as a smooth complex deformation of $\mathrm{Gr}(E)$ and to work with the various connections on the same underlying complex vector bundle. We should warn the reader though that if one drops this asumption, Condition (\ref{eq:necessarynumericalcondition}) might not be enough to ensure semi-stability of $\pi^*E$ on $(X',L_\varepsilon)$ (see the extra conditions in \cite[Theorem 1.10]{NapTip}). On the other hand, the asumption on $\mathrm{Gr}(E)$ having no pairwise isomorphic components is purely technical, and ensures that its automorphism group, that will provide obstructions in the perturbative theory, is abelian.
 \end{remark}
We now list some corollaries of Theorem \ref{theo:intro}. First, the stable case :
\begin{corollary}
 \label{cor:stablecase}
 Let $E\to X$ be a stable holomorphic vector bundle on $(X,[\omega])$ and let $A$ be a HYM connection on $E$. Then, there exists $\varepsilon_0>0$ and a sequence of HYM connections $(A_\varepsilon)_{\varepsilon\in (0,\varepsilon_0)}$ on $\pi^*E$ with respect to $(\omega_\varepsilon)_{\varepsilon\in (0,\varepsilon_0)}$ such that $A_\varepsilon \underset{\varepsilon\to 0}{\to} \pi^*A$ in any Sobolev norm.
\end{corollary}
For the semi-stable case, Condition (\ref{eq:necessarynumericalcondition}) reduces to a finite number of intersection product computations. One interesting feature comes from the second term in the expansion of $\mu_{L_\varepsilon}(E)$. It is the opposite of the slope of the restriction of $E$ to $Z$. The following formula is proved in \cite[Section 4.1]{NapTip},  where $m=\dim(Z)$ :
\begin{equation}
 \label{eq:formulaAchim}
\mu_{L_\varepsilon}(E)= \mu_{L}(E) - \dbinom{n-1}{m-1} \mu_{L_{\vert Z}}(E_{|Z}) \varepsilon^{n-m} + O(\varepsilon^{n-m+1}).
\end{equation}
We then have :
\begin{corollary}
 \label{cor:restrictionnumericalcriterion}
 Let $E\to X$ be a simple sufficiently smooth semi-stable holomorphic vector bundle on $(X,[\omega])$. Assume that the stable components of $\mathrm{Gr}(E)$ are pairwise non-isomorphic. Denote by $A$ an HYM connection on $E$. If
 \begin{equation}
 \label{eq:necessarynumericalconditionbis}
 \forall\; \mathcal{F}\in \mathfrak{E}_{[\omega]},\: \mu_{L_{\vert Z}}(E_{\vert Z})<\mu_{L_{\vert Z}}(\mathcal{F}_{\vert Z}),
\end{equation}
  then, there exists $\varepsilon_0>0$ and a sequence of HYM connections $(A_\varepsilon)_{\varepsilon\in (0,\varepsilon_0)}$ on $\pi^*E$ with respect to $(\omega_\varepsilon)_{\varepsilon\in (0,\varepsilon_0)}$ converging to $\pi^*A$ in any Sobolev norm.
\end{corollary}
Condition (\ref{eq:necessarynumericalconditionbis}) was checked on explicit examples in \cite[Section 4.5]{NapTip} to produce stable perturbations of tangent sheaves by blow-ups, and our result provides information on the associated connections and their asymptotic behaviour. Note that by Mehta--Ramanathan theorem \cite{MeRa}, if $[\omega]=c_1(L)$ is integral, and if $Z$ is a generic intersection of divisors in linear systems $\vert L^k \vert$, then $E_{\vert Z}$ is semi-stable as soon as $E$ is. In that case, Condition (\ref{eq:necessarynumericalconditionbis}) cannot be satisfied, and it seems unlikely that Condition (\ref{eq:necessarynumericalcondition}) will hold true. Hence, blowing-up such subvarieties tend to destabilise a semi-stable bundle.

In general, we expect that it should not be too hard to obtain stability of sufficiently smooth pulled back bundles under condition (\ref{eq:necessarynumericalcondition}) with purely algebraic methods. However, we emphasize that the Hitchin--Kobayashi correspondence doesn't provide any information on the asymptotic behaviour of the associated HYM connections, which is then the main content of Theorem \ref{theo:intro}. Nevertheless, we state the following corollary, that extends \cite[Theorem 1.10]{NapTip} to a non-equivariant situation:

\begin{corollary}
 \label{cor:semistablecase}
 Let $E\to X$ be a simple sufficiently smooth semi-stable holomorphic vector bundle on $(X,[\omega])$.  Assume that the stable components of $\mathrm{Gr}(E)$ are pairwise non-isomorphic.
  Then, there exists $\varepsilon_0>0$ such that $\pi^*E\to (X',L_\varepsilon)$ is
  \begin{enumerate}
   \item[(i)] stable if and only if for all $\mathcal{F}\in \mathfrak{E}_{[\omega]},\: \mu_{L_\varepsilon}(\mathcal{F})\underset{\varepsilon\to 0}{<} \mu_{L_\varepsilon}(E),$
   \item[(ii)] semi-stable if and only if for all $\mathcal{F}\in \mathfrak{E}_{[\omega]},\: \mu_{L_\varepsilon}(\mathcal{F})\underset{\varepsilon\to 0}{\leq} \mu_{L_\varepsilon}(E),$
   \item[(iii)] unstable otherwise.
  \end{enumerate}
\end{corollary}

Finally, we comment on previous related works. Theorem \ref{theo:intro} extends results from \cite{Buchdahl,dervansektnan19} where blow-ups of HYM connections along points are considered. In the present paper, we consider blow-ups along any smooth subvariety, and also cover the semi-stable situation, which is technically more involved due to the presence of automorphisms of the graded object that obstruct the linear theory. While  \cite{dervansektnan19} is a gluing construction as in the similar problem of producing extremal K\"ahler metrics on blow-ups \cite{ArPac09,ArPacSing,sze-blowupsII,SeySze,dervan-sektnan-blowups}, one of the key feature in our approach is to apply directly the implicit function theorem to reduce to (an $\varepsilon$ dependent family of) finite dimensional GIT problems on a Kuranishi space parametrising small deformations of $\mathrm{Gr}(E)$, as in \cite{gabor-deformations,dervan-sektnan-blowups}. We then use the new technology developed in \cite{SekTip22} to control the perturbations of the associated moment maps when $\omega_\varepsilon$ varries. This is where our hypothesis on $\Aut(\mathrm{Gr}(E))$ being abelian is used. The main new technical input comes from the fact that the underlying smooth manifold $X$ is fixed in \cite{SekTip22}, while it varries with the blow-up, which requires a carefull analysis of the operator introduced to apply the implicit function theorem.

\subsection*{Outline:} In Section \ref{sec:setup}, we recall basic material about HYM connections and stability. We then perform in Section \ref{sec:blowup} the analysis of the linear theory on the blow-up. Relying on this, in Section \ref{sec:perturbation} we explain how to reduce the problem to finding zeros of finite dimensional moment maps. Then, we conclude the proof of Theorem \ref{theo:intro} and its corollaries in Section \ref{sec:proofmain}.

\subsection*{Acknowledgments:} The authors benefited from visits to LMBA and Gotheborg University; they would like to thank these welcoming institutions for providing stimulating work environments. The idea of this project emerged from discussions with Lars Martin Sektnan, whom we thank for sharing his ideas and insight. CT is partially supported by the grants MARGE ANR-21-CE40-0011 and BRIDGES ANR--FAPESP ANR-21-CE40-0017.

\section{Preliminaries}
\label{sec:setup}
In Sections \ref{sec:HYM} and \ref{sec:slopestability} we introduce the notions of HYM connections and slope stability, together with some general results, and refer the reader to \cite{skobayashi87} and \cite{HuLe}. From Section \ref{sec:blowup} we start to specialise the discussion to blow-ups. In particular, in Section \ref{sec:decomposeLaplace}, we provide various asymptotic expressions for the linearisation of the HYM equation on the blow-up. Those results will be used in Section \ref{sec:perturbation}.

\subsection{The hermitian Yang--Mills equation}
\label{sec:HYM}

Let $E \to X$ be a holomorphic vector bundle over a compact K\"ahler manifold $X$. A hermitian metric on $E$ is \textit{Hermite--Einstein} with respect to a K\"ahler metric with K\"ahler form $\omega$ if the curvature $F_h \in \Omega^2 \left(X, \End E \right)$ of the corresponding Chern connection satisfies
\begin{align}\label{HEeqn} \Lambda_{\omega} \left( iF_h \right) = c \Id_E
\end{align}
for some real constant $c$. Equivalently, if $h$ is some hermitian metric on the smooth complex vector bundle underlying $E$, a hermitian connection $A$ on $(E,h)$ is said to be \textit{hermitian Yang--Mills} if it satisfies
\begin{equation*}
\left\{
\begin{array}{ccc}
F_A^{0,2} & = & 0, \\
\Lambda_{\omega} \left( i F_A \right) & = & c \Id_E.
\end{array}
\right.
\end{equation*}
The first equation of this system implies that the $(0,1)$-part of $A$ determines a holomorphic structure on $E$, while the second that $h$ is Hermite--Einstein for this holomorphic structure. We will try to find hermitian Yang--Mills connections within the complex gauge group orbit, which we now define. The (hermitian) \textit{complex gauge group} is
$$
\mathscr{G}^{\mathbb{C}}(E,h) = \Gamma \left( \GL \left(E, \mathbb{C} \right) \right)\cap \Gamma \left( \End_H (E, h) \right),
$$
where $\End_H (E,h)$ stands for the hermitian endomorphisms of $(E,h)$.
Note that if $\bar\partial$ is the Dolbeault operator defining the holomorphic structure on $E$, then $ f \circ \bar\partial \circ f^{-1}$ defines a biholomorphic complex structure on $E$. Let $d_A = \partial_A + \bar\partial_A$ be the Chern connection of $(E,h)$ with respect to the original complex structure (that is $\bar\partial_A = \bar\partial$). Then the Chern connection $A^f$ of $h$ with respect to $f \circ \bar\partial \circ f^{-1}$ is
$$
d_{A^f} = (f^*)^{-1} \circ \partial_A \circ (f^*) + f \circ \bar\partial \circ f^{-1}.
$$
Solving the hermitian Yang--Mills equation is equivalent to solving $$\Psi (s) = c \Id_E$$
where
$$
\begin{array}{cccc}
\Psi :  & \mathrm{Lie}(\mathscr{G}^{\mathbb{C}}(E,h)) &  \longrightarrow  & \mathrm{Lie}(\mathscr{G}^{\mathbb{C}}(E,h))\\
 & s & \longmapsto & i\Lambda_\omega(F_{A^{\exp(s)}}),
\end{array}
$$
and where $\mathrm{Lie}(\mathscr{G}^{\mathbb{C}}(E,h)):=i\Gamma(\End_H(E,h))$ is the tangent space to $\mathscr{G}^{\mathbb{C}}(E,h)$ at the identity. For a connection $A$ on $E$, the Laplace operator $\Delta_{A}$ is
\begin{align}\label{laplaceop}  \Delta_A =  i \Lambda_{\omega} \left( \bar \partial_{A}  \partial_{A} -  \partial_{A} \bar \partial_{A} \right).
\end{align}
If $A_{\End E}$ denote the connection induced by $A$ on $\End E$, then :
\begin{lemma}\label{lem:linop}
If $A$ is the Chern connection of $(E,\overline{\partial},h)$, the differential of $\Psi$  at identity is $$ d \Psi_{\Id_E} = \Delta_{A_{\End E}}.$$
 If moreover $A$ is assumed to be hermitian Yang--Mills, then the kernel of $ \Delta_{A_{\End E}}$ acting on $\Gamma(\End(E))$ is given by the Lie algebra $\mathfrak{aut}(E)$ of the space of automorphisms $\Aut(E)$ of $(E,\overline{\partial})$.
\end{lemma}
The last statement about the kernel follows from the K\"ahler identities and the Akizuki-Nakano identity that imply $\Delta_{A_{\End E}}=\partial^*_{A}\partial_{A}+\bar\partial_{A}^*\bar\partial_{A}$, the two terms of which are equal if $A$ is Hermitian Yang-Mills. The operator $\Delta_{A_{\End E}}$ being elliptic and self-adjoint, $\mathfrak{aut}(E)$ will then appear as a cokernel in the linear theory for perturbations of hermitian Yang--Mills connections.

\subsection{Slope stability}
\label{sec:slopestability}
We recall some basic facts about slope stability, as introduced by \cite{Mum62,Tak72}, and refer the interested reader to \cite{HuLe} for a detailed treatment. We denote here $L:=[\omega]$ the polarisation of the $n$-dimensional K\"ahler manifold $X$.
\begin{definition}
\label{def:stability}
For $\mathcal{E}$ a torsion-free coherent sheaf on $X$, the slope $\mu_L(\mathcal{E})\in\mathbb{Q}$ (with respect to $L$) is given by the intersection formula
 \begin{equation}
  \label{eq:slope}
  \mu_L(\mathcal{E})=\frac{\deg_L(\mathcal{E})}{\mathrm{rank}(\mathcal{E})},
 \end{equation}
 where $\mathrm{rank}(\mathcal{E})$ denotes the rank of $\mathcal{E}$ while $\deg_L(\mathcal{E})=c_1(\mathcal{E})\cdot L^{n-1}$ stands for its degree. Then, $\mathcal{E}$ is said to be \emph{slope semi-stable} (resp. \emph{slope stable}) with respect to $L$ if for any coherent subsheaf $\mathcal{F}$ of $\mathcal{E}$ with $0<\mathrm{rank}( \mathcal{F})<\mathrm{rank}(\mathcal{E})$, one has
\begin{eqnarray*}
\mu_L(\mathcal{F})\leq \mu_L(\mathcal{E}) \:\textrm{( resp. } \mu_L(\mathcal{F})< \mu_L(\mathcal{E}) ).
\end{eqnarray*}
A direct sum of slope stable sheaves of the same slope
 is said to be \emph{slope polystable}.
\end{definition}
In this paper, we will often omit ``slope'' and simply refer to stability of a sheaf, the polarisation being implicit. We will make the standard identification of a holomorphic vector bundle $E$ with its sheaf of sections, and thus talk about slope stability notions for vector bundles as well. In that case slope stability relates nicely to differential geometry via the Hitchin--Kobayashi correspondence :
\begin{theorem}[\cite{skobayashi82,lubke83,uhlenbeckyau86,donaldson87}]
\label{thm:HKcorrespondence}
There exists a Hermite--Einstein metric on $E$ with respect to $\omega$ if and only if $E$ is polystable with respect to $L$
\end{theorem}
We will be mostly interested in semi-stable vector bundles. A \emph{Jordan--H\"older filtration} for a torsion-free sheaf $\mathcal{E}$ is a filtration by coherent subsheaves:
 \begin{align}
\label{eq:JHfiltration}
0 = \mathcal{F}_0 \subset \mathcal{F}_1 \subset \hdots \subset \mathcal{F}_\ell = \mathcal{E},
\end{align}
such that the corresponding quotients,
\begin{align}
\label{eq:JHquots}
\mathcal{G}_i = \frac{\mathcal{F}_i}{\mathcal{F}_{i-1}},
\end{align}
 for $i=1, \hdots, \ell$, are stable with slope $\mu_L(\mathcal{G}_i)=\mu_L(\mathcal{E})$.
 In particular, the graded object of this filtration
 \begin{equation}
  \label{eq:JHgraded}
  \mathrm{Gr}(\mathcal{E}):=\bigoplus_{i=1}^l \mathcal{G}_i
 \end{equation}
is polystable. From \cite[Section 1]{HuLe}, we have the standard existence and uniqueness result:
\begin{proposition}
 \label{prop:JHfiltration}
 Any semi-stable coherent torsion-free sheaf $\mathcal{E}$ on $(X,L)$ admits a Jordan--H\"older filtration, and the graded object $\mathrm{Gr}(\mathcal{E})$ of such filtrations is unique up to isomorphism.
\end{proposition}
When $E$ is locally-free and semi-stable, we say that it is {\it sufficiently smooth} if $\mathrm{Gr}(E)$ is locally-free. In that case, we denote $\mathfrak{E}_{[\omega]}$ the set of holomorphic subbundles of $E$ built out of successive extensions of some of the stable components of $\mathrm{Gr}(E)$. Equivalently, $\mathfrak{E}_{[\omega]}$ is the set of holomorphic subbundles of $E$ arising in a Jordan-Holder filtration for $E$. Finally, we recall that a necessary condition for $E$ to be stable is simplicity, that is $\Aut(E)=\mathbb{C}^*\cdot\Id_E$.
\subsection{Geometry of the blow-up}
\label{sec:blowup}
We consider now $Z \subset X$ a $m$-dimensional complex submanifold of codimension $r=n-m\geq 2$ and the blow-up map
$$
\pi : \mathrm{Bl}_Z(X) \to X.
$$
We will denote by $X'=\mathrm{Bl}_Z(X)$ the blown-up manifold and by $Z'=\pi^{-1}(Z)$ the exceptional divisor.  We denote by
$$
L_\varepsilon:= \pi^* L - \varepsilon [Z']
$$
a polarisation on $X'$, for $0< \varepsilon \ll 1$. Let $E\to X$ be a holomorphic vector bundle, and denote by $E'=\pi^*E$ the pulled back bundle. For any holomorphic subbundle $F\subset E$, the intersection numbers $\mu_{L_\varepsilon}(\pi^*E)-\mu_{L_\varepsilon}(\pi^*F)$ admit expansions in $\varepsilon$, with first term given by $\mu_L(E)-\mu_L(F)$. For that reason, given the Hitchin--Kobayashi correspondence in Theorem \ref{thm:HKcorrespondence}, semi-stability of $E$ on $(X,L)$ is a necessary condition for its pullback $E'$ to admit an HYM connection with respect to a K\"ahler metric in $L_\varepsilon$, for all $0< \varepsilon \ll 1$. Another necessary condition is simplicity of $E'$, which, by Hartogs' theorem, is equivalent to simplicity of $E$. Then, natural candidates to test for stability of $E'$ are given by the pullbacks of elements in $\mathfrak{E}_{[\omega]}$, and Condition (\ref{eq:necessarynumericalcondition}) clearly is necessary for $E'$ to be stable in the polarisations we consider, and thus to admit an HYM connection. Hence, we will assume $E$ to be simple, semi-stable, and to satisfy (\ref{eq:necessarynumericalcondition}). We now turn back to the differential geometry of the blow-up.

\subsubsection{Decomposition on spaces of sections}
\label{sec:decompositions}
We have a commutative diagramm:
$$
\begin{array}{ccc}
 Z' & \stackrel{\iota}{\longrightarrow} & X' \\
 \downarrow &  & \downarrow \\
 Z & \stackrel{\iota_0}{\longrightarrow} & X
\end{array}
$$
where $\iota_0$ and $\iota$ denote the inclusions, while the vertical arrows are given by the projection map $\pi$. We then have a pullback map on sections
$$
\pi^* : \Gamma(X, \End(E)) \longrightarrow \Gamma(X',\End(\pi^*E))
$$
as well as a restriction map :
$$
\iota^* : \Gamma(X',\End(\pi^*E)) \longrightarrow \Gamma(Z',\End(\iota^*\pi^*E)).
$$
Our goal now is to fit those maps in a short exact sequence, that will in the end split the space $\Gamma(X',\End(\pi^*E))$.
  If $N_Z=TX_{\vert Z}/ TZ$ denotes the normal bundle of $Z$ in $X$, then $Z'\simeq \mathbb{P}(N_Z)$, and we can fix a $(1,1)$-form $\lambda \in c_1(\mathcal{O}_{\mathbb{P}(N_Z)}(1))$ that restricts to K\"ahler metrics on the fibers of $\mathbb{P}(N_Z)\to Z$. We also fix a K\"ahler form $\omega\in c_1(L)$ on $X$, and consider its restriction to $Z$. We then have a K\"ahler $\mathbb{C}\mathbb{P}^{r-1}$-fibration :
$$
\pi : (Z', \lambda) \longrightarrow (Z, \omega).
$$
By averaging along fibers as described in \cite[Section 2.3]{SekTip}, we obtain a splitting
\begin{equation}
 \label{eq:firstsplitting}
\Gamma(Z',\End(\iota^*\pi^*E))=\pi^*(\Gamma(Z,\End(\iota_0^*E)))\oplus\Gamma_0(Z',\End(\iota^*\pi^*E)).
\end{equation}
 We will omit the $\iota^*$ and $\pi^*$ to simplify notation. Using the projection on the second factor
 $$
 p_0: \Gamma(Z',\End(E))\to \Gamma_0(Z',\End(E))
 $$
 in (\ref{eq:firstsplitting}), we deduce a short exact sequence :
$$
0 \longrightarrow \Gamma(X, \End(E)) \stackrel{\pi^*}{\longrightarrow} \Gamma(X',\End(E)) \stackrel{p_0\circ \iota^*}{\longrightarrow} \Gamma_0(Z',\End(E)) \longrightarrow 0.
$$
We can actually split this sequence by mean of a linear extension operator
$$
\iota_* : \Gamma_0(Z',\End(E)) \longrightarrow \Gamma(X',\End(E))
$$
such that
$$
p_0\circ \iota^*\circ \iota_* = \mathrm{Id}.
$$
This can be done using bump functions and a standard partition of unity argument. The outcome is an isomorphism :
\begin{equation}
 \label{eq:goodcoordinates}
\begin{array}{ccc}
\Gamma(X',\End(E))  & \longrightarrow &  \Gamma(X, \End(E)) \oplus \Gamma_0(Z',\End(E)) \\
   s & \longmapsto & (s-\iota_*\circ p_0\circ \iota^* s \;,\;  p_0\circ \iota^* s),
\end{array}
\end{equation}
with inverse map $(s_X, s_Z) \mapsto (\pi^*s_X + \iota_* s_Z)$. This splits the Lie algebra of gauge transformations, and will be used to identify contributions coming from $X$ and from $Z'$ in the $\varepsilon$-expansion of the linearisation, which we describe in the next section. From now on, by abuse of notations, we will consider the spaces $\Gamma(X, \End(E))$ and $\Gamma_0(Z',\End(E))$ as subspaces of $\Gamma(X',\End(\pi^*E))$, and denote $s=s_X+s_Z$ the decomposition of an element $s\in \Gamma(X',\End(E))$.

\subsubsection{Decomposition of the Laplace operator}
\label{sec:decomposeLaplace}
 We extend $\lambda$ to a closed $(1,1)$-form over $X'$ as in \cite[Section 3.3]{voisin1} and consider the family of K\"ahler metrics on $X^\prime$:
 $$\omega_\varepsilon=\pi^*\omega+\varepsilon\lambda\in c_1(L_\varepsilon),\: 0 <\varepsilon\ll 1.$$
 Let $A$ be a Hermitian connection on $E$,
  which we pull back to $X^\prime$ and extend to the bundle $\End(\pi^*E)$. We will now study the Laplace operator
\begin{eqnarray*}
\Delta_\varepsilon s =i\Lambda_\varepsilon (\bar\partial_{A}\partial_{A}-\partial_{A}\bar\partial_{A})s
\end{eqnarray*}
acting on the various components of $s=s_X+s_Z \in\Gamma(X^\prime,\End(E))$, where $\Lambda_\varepsilon$ is the Lefschetz operator for the metric $\omega_\varepsilon$. For this, we need to introduce an elliptic operator on $Z'$. The \textit{vertical Laplace operator}, denoted
$$
\Delta_{\mathcal{V}} : \Gamma_0 \left( Z', \End(E) \right) \to \Gamma_0 \left( Z',\End(E)\right),
$$
is the operator defined by the following procedure. Let $\sigma\in\Gamma_0(Z',\End(E))$. Over a point $x \in Z$, take the restriction $\sigma_x$ of $\sigma$ to $Z'_x=\pi^{-1}(x)$, and consider $\sigma_x$ as a map to $\mathbb{C}^p$ with components $\sigma_x^i$ in a trivialisation $\pi^*\End(E)_x \cong \mathbb{C}^p$ of the restriction of $\pi^* \End(E)$ to the fibre $Z'_x$ of $Z' \to Z$. Define
$$\left( \Delta_{\mathcal{V}} \left( \sigma\right) \right)^i_x = \Delta_{ (\lambda)_{|Z'_x} } \left( \sigma_x^i \right),$$
for $\Delta_\lambda$ the Laplacian of the K\"ahler form $\lambda$ on $Z'_x$. Then glue together to form a section of $\pi^*\End(E)$. As in \cite[Section 4.1]{SekTip}, one easily obtains that this construction is independent on the trivialisation chosen, and sends smooth sections to smooth sections. In the following Lemma, the supscript $l$ (or $l+2$) stands for the Sobolev completion with respect to some $L^{2,l}$ Sobolev norm, where those norms can be produced out of the metrics $\omega,\lambda$ and any metric $h$ on $E$, together with the covariant derivatives given by $A$.
\begin{lemma}\cite[Section 4.1]{SekTip}
 \label{lem:verticallaplacianinvertible}
 The vertical Laplacian
 $$
\Delta_{\mathcal{V}} : \Gamma_0 \left( Z', \End(E) \right)^{l+2} \to \Gamma_0 \left( Z',\End(E)\right)^l
$$
is invertible.
\end{lemma}
 In the following statements, if $\mathcal{A}$ denotes a second order operator acting on sections, then in an expression of the form
$$\mathcal{A}(\sigma)= \sigma_0+\varepsilon \sigma_1 + \ldots +\varepsilon^{d-1}\sigma_{d-1}+\mathcal{O}(\varepsilon^d)$$
the term $\mathcal{O}(\varepsilon^d)$
will stand for $\sigma_d\cdot\varepsilon^d$, where $\sigma_d$ is a section whose $L^{2,l}$ Sobolev norm is bounded by the $L^{2,l+2}$ Sobolev norm of $\sigma$.
\begin{lemma}
If $s_Z=\iota_*\sigma_Z$ for $\sigma_Z\in\Gamma(Z^\prime,\End(E))$, then
\begin{eqnarray*}
(p_0\circ \iota^*)\Delta_\varepsilon(\iota_*\sigma_Z)=\varepsilon^{-1}\Delta_\mathcal{V}\sigma_Z +\mathcal{O}(1).
\end{eqnarray*}
\end{lemma}
\begin{proof}
We introduce the operator $D$ given by
\begin{eqnarray*}
Ds_Z=i(\bar\partial_{A}\partial_{A}-\partial_{A}\bar\partial_{A})s_Z.
\end{eqnarray*}
The Laplacian $\Delta_\varepsilon$ satisfies on $X^\prime$ :
$$\Delta_\varepsilon s_Z\,\omega_\varepsilon^n=n Ds_Z\wedge \omega_\varepsilon^{n-1},$$
or equivalently
\begin{eqnarray*}
\Delta_\varepsilon s_Z = \frac{n\ Ds_Z\wedge (\omega+\varepsilon\lambda)^{n-1}}{(\omega+\varepsilon\lambda)^n}.
\end{eqnarray*}
We note that $\omega$ is a K\"ahler form on $X$, but on $X^\prime$ is degenerate along the fibre directions of the submanifold $Z'$. Then $(i^*\omega)^{m+1}=0\in\Omega^{2(m+1)}(Z^\prime)$, and at $x\in Z^\prime \subseteq X^\prime$, $\omega^{m+2}=0$.
Then, expanding $(\omega+\varepsilon\lambda)^{n-1}$ and $(\omega+\varepsilon\lambda)^n$ gives
\begin{eqnarray*}
\iota^*\Delta_\varepsilon s_Z
= (n-m-1)\varepsilon^{-1} \frac{Ds_Z\wedge \omega^{m+1}\wedge\lambda^{n-m-2}}{\omega^{m+1}\wedge\lambda^{n-m-1}} +\mathcal{O}(1).
\end{eqnarray*}
Restricting to $Z^\prime$, the connection $1$-forms of $A$ vanish, so $\iota^*Ds_Z=i\partial\bar\partial \sigma_Z$, acting on the coefficient functions of $\sigma_Z$. On the other hand, by considering a convenient orthonormal frame at $x\in Z^\prime$, we see that $\iota^*\Delta_\varepsilon\iota_*\sigma_Z=
\varepsilon^{-1}\Delta_\mathcal{V}\sigma_Z+\mathcal{O}(1)$.
\end{proof}
In the next lemma, we denote $\Delta_\varepsilon s_Z=(\Delta_\varepsilon s_Z)_X+(\Delta_\varepsilon s_Z)_Z$ the decomposition according to (\ref{eq:goodcoordinates}).
\begin{lemma}
For $s_Z=\iota_*\sigma_Z$ with $\sigma_Z\in\Gamma(Z^\prime,\End(E))$, we have
\begin{eqnarray*}
(\Delta_\varepsilon s_Z)_X=\mathcal{O}(1).
\end{eqnarray*}
\end{lemma}
\begin{proof}
 By definition, $(\Delta_\varepsilon s_Z)_X=\pi^*\phi$ for some $\phi\in\Gamma(X,\End(E))$. As we also have
\begin{eqnarray*}
 (\Delta_\varepsilon s_Z)_X &=& (\mathrm{Id}-\iota_*(p_0\circ\iota^*))\Lambda_\varepsilon D s_Z,
\end{eqnarray*}
we deduce that the section $\phi$ is the continuous extension of $\pi_*(\mathrm{Id}-\iota_*(p_0\circ\iota^*))\Lambda_\varepsilon D s_Z$ across $Z\subseteq X$. On $X^\prime\setminus Z^\prime$ we have
 \begin{eqnarray*}
\Lambda_\varepsilon Ds_Z &=& n\frac{Ds_Z\wedge (\omega^{n-1}+\mathcal{O}(\varepsilon))}{\omega^n+\mathcal{O}(\varepsilon)}=\mathcal{O}(1).
\end{eqnarray*}
  As $\pi_*(\mathrm{Id}-\iota_*(p_0\circ\iota^*))$ is $\mathcal{O}(1)$, the result follows.
\end{proof}

From the previous two lemmas, in the decomposition $$s=s_X+s_Z,$$
$\Delta_\varepsilon s_Z$ also lies in the subspace $\Gamma_0(Z^\prime,\End(E))\subseteq\Gamma(X^\prime,\End(E))$ to higher order in $\varepsilon$. For $s_X\in\Gamma(X,\End(E))$,
$$\Delta_\varepsilon s_X=(\Delta_\varepsilon s_X)_X+(\Delta_\varepsilon s_X)_Z$$
where $(\Delta_\varepsilon s_X)_Z=\iota_*(p_0\circ\iota^*)\Delta_\varepsilon s_X$. We first consider $\iota^*\Delta_\varepsilon s_X$.

\begin{lemma}
For $s_X=\pi^*\sigma_X\in\Gamma(X,\End(E))\subseteq\Gamma(X^\prime,\End(E))$,
\begin{eqnarray*}
\iota^*\Delta_\varepsilon s_X=(m+1)\frac{Ds_X\wedge\omega^m\wedge\lambda^{n-m-1}}{\omega^{m+1}\wedge\lambda^{n-m-1}}+\mathcal{O}(\varepsilon).
\end{eqnarray*}
\end{lemma}
\begin{proof}
Firstly, $s_X=\pi^*\sigma_X$, and the connection $A$ is pulled back from $X$, so $Ds_X$ is basic for the projection to $X$ and $Ds\wedge \omega^{m+1}=0$ at points in $Z^\prime$. Secondly, we note that $\omega^{m+1}\wedge\lambda^{n-m-1}$ is a volume form on $X^\prime$, in a neighbourhood of $Z^\prime$. Then, the result follows similarly to the previous lemma.
\end{proof}
For the final term $(\Delta_\varepsilon s_X)_X$, we introduce $\Delta_X$ the Laplace operator of $A$ on $\End (E) \to (X,\omega)$:
$$
\begin{array}{cccc}
\Delta_{X} : &  \Gamma \left( X, \End(E) \right) &\to& \Gamma \left(X,\End(E)\right)\\
            & \sigma & \mapsto & i\Lambda_\omega (\bar\partial_{A}\partial_{A}-\partial_{A}\bar\partial_{A})\sigma.
\end{array}
$$
\begin{lemma}
For $s_X=\pi^*\sigma_X\in\Gamma(X,\End(E))\subseteq\Gamma(X^\prime,\End(E))$,
\begin{eqnarray*}
(\Delta_\varepsilon s_X)_X =\pi^*(\Delta_{X}\sigma_X)+\mathcal{O}(\varepsilon).
\end{eqnarray*}
\end{lemma}

\begin{proof}
There is $\phi\in\Gamma(X,\End(E))$ such that
$(\Delta_\varepsilon s_X)_X=\pi^*\phi$.
The element $\phi$ can be identified as the lowest order term in the asymptotic expansion in $\varepsilon$ of $(\Delta_\varepsilon\pi^*\sigma_X)_X$. However, we have at $x\in X'\setminus Z'$ :
\begin{eqnarray*}
\Delta_\varepsilon\pi^*\sigma_X=n\frac{D\pi^*\sigma_X\wedge (\omega+\varepsilon\lambda)^{n-1}}{(\omega+\varepsilon\lambda)^n} = n\pi^*\frac{D\sigma_X\wedge\omega^{n-1}}{\omega^n} +\mathcal{O}(\varepsilon)
\end{eqnarray*}
so we see that the lowest order term in the expansion of $(\Delta_\varepsilon\pi^*\sigma_X)_X$ is $\Delta_{X}\sigma_X$.
\end{proof}

Summarizing the above calculations, with respect to the decomposition $s=s_X+s_Z$ produced by (\ref{eq:goodcoordinates}), the operator $\Delta_\varepsilon$ takes the form
\begin{equation}
 \label{eq:linearopmatrix}
\left(
\begin{array}{cc}
\Delta_X & 0\\
\mathcal{L} & \varepsilon^{-1}\Delta_\mathcal{V}
\end{array}
\right)
\end{equation}
plus higher order terms, for some second order operator $\mathcal{L}$. In the next section, we will apply the previous lemmas and the resulting form for $\Delta_\varepsilon$ to the pullback of a HYM connection $A_0$ on the graded object $\mathrm{Gr}(E)$ of $E$.

\section{The perturbation argument}
\label{sec:perturbation}
The goal of this section is to reduce the problem of finding a zero for the operator $s\mapsto i\Lambda_{\omega_\varepsilon}(F_{A^{\exp(s)}})-c_\varepsilon\mathrm{Id}$ in a gauge group orbit to a finite dimensional problem. The ideas here go back to \cite{donaldson-largesym,gabor-deformations}, and our framework will be that of \cite{BuSchu}.

\subsection{Kuranishi slice}
\label{sec:kuranishi}
We start from a simple semi-stable and sufficiently smooth holomorphic vector bundle $E$ on $(X,L)$, with $L=[\omega]$. Denote by $\mathrm{Gr}(E)=\bigoplus_{i=1}^\ell\mathcal{G}_i$ the associated polystable graded object, with stable components $\mathcal{G}_i$. We let $\overline{\partial}_0$ be the Dolbeault operator of $\mathrm{Gr}(E)$. The automorphism group $G:=\Aut(\mathrm{Gr}(E))$ is a reductive Lie group with Lie algebra $\mathfrak{g}:=\mathfrak{aut}(\mathrm{Gr}(E))$ and compact form $K\subset G$, with $\mathfrak{k}:=\mathrm{Lie}(K)$. The Dolbeault operator $\overline{\partial}_E$ on $E$ is given by
$$
\overline{\partial}_E=\overline{\partial}_0+\gamma
$$
where $\gamma \in\Omega^{0,1}(X,\mathrm{Gr}(E)^*\otimes \mathrm{Gr}(E))$ can be written
$$
\gamma=\sum_{i<j} \gamma_{ij}
$$
for (possibly vanishing) $\gamma_{ij} \in\Omega^{0,1}(X,\mathcal{G}_j^* \otimes \mathcal{G}_i)$. Elements
$$g:=g_1\Id_{\mathcal{G}_1}+\ldots,+g_\ell \Id_{\mathcal{G}_\ell}\in G,$$
for $(g_i)\in(\mathbb{C}^*)^\ell$, act on $\overline{\partial}_E$ and produce isomorphic holomorphic vector bundles in the following way :
\begin{equation}
\label{eq:explicitaction}
g\cdot \overline{\partial}_E= \overline{\partial}_0 + \sum_{i<j} g_ig_j^{-1}\gamma_{ij}.
\end{equation}
In particular, for $g=(t^\ell,t^{\ell-1},\ldots,t)$, letting $t\mapsto 0$, we can see $E$ as a small complex deformation of $\mathrm{Gr}(E)$. Our starting point to produce HYM connections on $E'=\pi^*E$ over $X'$ will then be the HYM connection $A_0$ on $\mathrm{Gr}(E)\to X$ given by the Chern connection of $(\overline{\partial}_0,h_0)$, where $h_0$ is an Hermite-Einstein metric on the polystable bundle $\mathrm{Gr}(E)$. Rather than working with the single bundle $E$, we will consider the family of bundles given by the $G$-action on Dolbeault operators. This will require the following proposition, whose proof follows as in \cite{kuranishi} (see also \cite{BuSchu,doan} for a detailed treatment). We introduce the notation
$$
V:=H^{0,1}(X,\End(\mathrm{Gr}(E)))
$$
for the space of harmonic $(0,1)$-forms with values in $\mathrm{Gr}(E)$, where the metrics used to compute adjoints are $\omega$ on $X$ and $h_0$ on $\mathrm{Gr}(E)$. Note that the $G$-action on $E$ induces a linear representation $G \to \mathrm{GL}(V)$.
\begin{proposition}
 \label{prop:firstslice}
 There exists a holomorphic $K$-equivariant map
 $$
 \Phi : B \to \Omega^{0,1}(X,\End(\mathrm{Gr}(E)))
 $$
 from a ball around the origin $B\subset V$ such that :
 \begin{enumerate}
  \item $\Phi(0)=0$;
  \item $Z:=\lbrace b\in B\,\vert\: (\overline{\partial}_0+\Phi(b))^2=0\rbrace$ is a complex subspace of $B$;
  \item if $(b,b')\in Z^2$ lie in the same $G$-orbit, then $\overline{\partial}_0+\Phi(b)$ and $\overline{\partial}_0+\Phi(b')$ induce isomorphic holomorphic bundle structures;
  \item The $\mathscr{G}^{\mathbb{C}}(\mathrm{Gr}(E))$-orbit of any small complex deformation of $\mathrm{Gr}(E)$ intersects $\Phi(Z)$.
 \end{enumerate}
\end{proposition}
Here, $\mathscr{G}^{\mathbb{C}}(\mathrm{Gr}(E))= \Gamma \left( \GL \left(\mathrm{Gr}(E), \mathbb{C} \right) \right)$ stands for the full gauge group of $\mathrm{Gr}(E)$. The space $Z$ corresponds to the space of integrable Dolbeault operators in the image of $\Phi$, and $\Phi(B)$ is a {\it slice} for the gauge group action on the set of Dolbeault operators nearby $\overline{\partial}_0$. We will then lift the slice to the space $\Omega^{0,1}(X',\End(\pi^*\mathrm{Gr}(E)))$ on the  blown-up manifold $X'$, and denote $\widehat{\Phi}$ the map
$$
\pi^*\circ \Phi : B \to \Omega^{0,1}(X',\End(\mathrm{Gr}(E))),
$$
where to ease notations we omitted $\pi^*$ for the pulled back bundle. The map $\widehat{\Phi}$ might no longer provide a slice for the gauge-group action on $X'$, but what matters for us is that its image will contain all elements in the $G$-orbit of $\pi^*\overline{\partial}_E$ close to $\pi^*\overline{\partial}_0$.

\subsection{Perturbing the slice}
\label{sec:perturbslice}
 The next step will be to perturb $\widehat{\Phi}$ to reduce our problem to a finite dimensional one. The strategy to do this in family with respect to the parameter $\varepsilon$ was inspired by \cite{dervanstabilityconditions,dervan-sektnan-blowups,SekTip22}.

 Given the metrics $\omega$ on $X\setminus Z$, $\lambda$ on $Z'$, and $h=\pi^*h_0$ on $E$, together with the covariant derivatives given by $\nabla_{A_0}$, we can introduce $L^{2,l}$ Sobolev norms on spaces of sections. We will denote by $\mathcal{E}^l$ the $L^{2,l}$ Sobolev completion of any space of sections $\mathcal{E}$. In what follows, $l\in\mathbb{N}^*$ will be assumed large enough for elements in $\mathcal{E}^l$ to admit as much regularity as required.

\begin{proposition}
 \label{prop:perturbingslice}
 Up to shrinking $B$, there is $\varepsilon_0 >0$ and a continuously diffenrentiable map
 $$
 \check\Phi : [0,\varepsilon_0)\times B \to \Omega^{0,1}(X',\End(\mathrm{Gr}(E)))^l
 $$
 such that for all $(\varepsilon,b)\in[0,\varepsilon_0) \times B$, if $\check A_{\varepsilon,b}$ is the Chern connection of $(\pi^*\overline{\partial}_0+ \check \Phi(\varepsilon,b),h)$ :
 \begin{enumerate}
  \item $\pi^*\overline{\partial}_0+\check \Phi(\varepsilon,b)$ and $\pi^*\overline{\partial}_0+\widehat{\Phi}(b)$ induce isomorphic holomorphic structures.
  \item $\Lambda_\varepsilon i F_{\check A_{\varepsilon,b}} \in \mathfrak{k}$.
 \end{enumerate}
\end{proposition}
\begin{remark}
 By elliptic regularity, elements in the image of $\check \Phi$ will actually be smooth. However, regularity of the map $\check \Phi$ is with respect to the $L^{2,l}$ Sobolev norm.
\end{remark}
We will use the implicit function theorem to prove Proposition \ref{prop:perturbingslice}, and will need the following lemma, where we still denote $A_0$ its pullback to $\pi^*\mathrm{Gr}(E)$, and use the notation $A_0^{s_X+\varepsilon s_Z}$ for $A_0^{\exp(s_X+\varepsilon s_Z)}$.
\begin{lemma}
\label{lem:differentiable}
 The map :
 \begin{eqnarray*}
\Psi : [0,\varepsilon_0)\times \Gamma(X,\End_H(E))^{l+2}\times \Gamma_0(Z^\prime,\End_H(E))^{l+2} &\longrightarrow & \Omega^{0}(X^\prime,\End_H(E))^l,\\
(\varepsilon\, ,\, s_X\, ,\, s_Z) &\mapsto & \Lambda_\varepsilon F_{A^{s_X+\varepsilon s_Z}_0}-c_\varepsilon \mathrm{Id}
\end{eqnarray*}
is continuously differentiable.
\end{lemma}
Above, the topological constants $c_\varepsilon$ are given by
\begin{eqnarray*}
c_\varepsilon = \frac{2\pi n}{\mathrm{vol}_{\omega_\varepsilon}(X^\prime)}\frac{\left( c_1(E)\cup [\omega_\varepsilon]^{n-1}\right)[X^\prime]}{\mathrm{rank}(E)}.
\end{eqnarray*}
\begin{proof}
Note first that for $\varepsilon=0$, $\Psi(0,s_X,s_Z)=\pi^*(\Lambda_\omega F_{A_0^{s_X}} - c_0 \Id_E)$ and is well defined.
Then, recall that if $f=\exp(s)$ for $s\in\Gamma(X^\prime,\mathrm{End}_H(E))$, the curvature of $f\cdot A_0$ is given by
\begin{eqnarray*}
F_{A_0^s}=F_{f\cdot A_0}= F_{A_0} +(\bar\partial\partial -\partial\bar\partial)s+(\partial s-\bar \partial s)\wedge (\partial s-\bar \partial s),
\end{eqnarray*}
where $\partial$ and $\bar \partial$ stand for the $(1,0)$ and $(0,1)$ components of $d_{A_0}$ (see e.g. \cite{BuSchu}[Section 1]).
In particular, taking $s=s_X+\varepsilon s_Z$,
\begin{eqnarray*}
F_{A_0^s} &=& F_{A_0} + (\bar \partial\partial-\partial\bar \partial)s_X+\varepsilon (\bar \partial\partial-\partial\bar \partial)s_Z
+(\partial s_X-\bar \partial s_X)\wedge (\partial s_X-\bar \partial s_X) \\
&&+\varepsilon(\partial s_X-\bar \partial s_X)\wedge (\partial s_Z-\bar \partial s_Z) +\varepsilon(\partial s_Z-\bar \partial s_Z)\wedge (\partial s_X-\bar \partial s_X) \\
&&+\varepsilon^2(\partial s_Z-\bar \partial s_Z)\wedge (\partial s_Z-\bar \partial s_Z).
\end{eqnarray*}
That is, ignoring the first term $F_{A_0}$, there are six remaining terms that we denote $F_{A^s}^i$, for $i=1,\ldots 6$. For each term we consider the factors coming from $Z^\prime$ and from $X$ (using (\ref{eq:goodcoordinates})) in $\Lambda_\varepsilon F_{A^s}^i$ and can conclude that $\Psi$ is smooth. For example, for the term $F_{A^s}^2=\varepsilon(\bar\partial \partial -\partial\bar\partial)s_Z$,
\begin{eqnarray*}
\Lambda_\varepsilon F^2_{A^s} &=& n\frac{\varepsilon Ds_Z\wedge (\omega+\varepsilon\lambda)^{n-1}}{(\omega+\varepsilon\lambda)^n},\\
\iota^*\Lambda_\varepsilon F^2_{A^s} &=& n\frac{\varepsilon Ds_Z\wedge \left( {n-1\choose m+1} \omega^{m+1}\wedge (\varepsilon\lambda)^{n-m-2}+\mathcal{O}(\varepsilon^{n-m-1})\right)}{{n\choose m+1} \omega^{m+1}\wedge (\varepsilon\lambda)^{n-m-1}+\mathcal{O}(\varepsilon^{n-m})},\\
&=& (n-m-1)\frac{Ds_Z\wedge \left(\omega^{m+1}\wedge\lambda^{n-m-1}+\mathcal{O}(\varepsilon)\right)}{\omega^{m+1}\wedge\lambda^{n-m-1}+\mathcal{O}(\varepsilon)},
\end{eqnarray*}
noting that here $\mathcal{O}(\varepsilon)$ denotes a polynomial in $\varepsilon$ with coefficients $2n$-forms on a neighbourhood of $Z^\prime$, such that $\mathcal{O}(0)=0$. We also note that $\omega^{m+1}\wedge\lambda^{n-m-1}$ is a volume form on a neighbourhood of $Z^\prime$. We conclude that
$$(\Lambda_\varepsilon F^2_{A^s})_Z=\iota_*(p_0\circ \iota^*)\Lambda_\varepsilon F^2_{A^s}$$
is a smooth function of $(\varepsilon,s_Z)$ with values in $\Gamma_0(Z^\prime,\End(E))$.

The $X$-component of $\Lambda_\varepsilon F^2_{A^s}$,
\begin{eqnarray*}
(\Lambda_\varepsilon F^2_{A^s})_X &=& (\mathrm{Id}-\iota_*(p_0\circ\iota^*))\Lambda_\varepsilon F^2_{A^s},
\end{eqnarray*}
 is of the form $\pi^*\phi$ for some $\phi\in\Gamma(X,\End(E))$. The section $\phi$ is given as the continuous extension of $\pi_*(\mathrm{Id}-\iota_*(p_0\circ\iota^*))\Lambda_\varepsilon F^2_{A^s}$ across $Z\subseteq X$. On $X^\prime\setminus Z^\prime$ we have
 \begin{eqnarray*}
\Lambda_\varepsilon F^2_{A^s} &=& n\frac{Ds_Z\wedge (\omega^{n-1}+\mathcal{O}(\varepsilon))}{\omega^n+\mathcal{O}(\varepsilon)},
\end{eqnarray*}
 which depends smoothly on $s_Z$ and $\varepsilon$. As $\pi_*(\mathrm{Id}-\iota_*(p_0\circ\iota^*))$ is linear, $\phi$ depends smoothly on these variables too.

 Using that $s_X$ is a pulled back section, at points in $Z'$ we have $D s_X\wedge \omega^{m+1}=0$, from which we deduce $\iota^*\Lambda_\varepsilon F_{A_s}^1=\mathcal{O}(1)$ and $\iota^*\Lambda_\varepsilon F_{A_s}^3=\mathcal{O}(1)$. This shows, as for $(\Lambda_\varepsilon F_{A_s}^2)_Z$, that $(\Lambda_\varepsilon F_{A_s}^1)_Z$ and $(\Lambda_\varepsilon F_{A_s}^3)_Z$ are $\mathcal{C}^1$. The other terms $F_{A^s}^i$ can be dealt with in a similar manner.
\end{proof}

\begin{proof}[Proof of Proposition \ref{prop:perturbingslice}]
 For $b\in B$, we will denote by $A_b$ the Chern connection associated to $(\pi^*\overline{\partial}_0+ \widehat{ \Phi}(b),h)$, where $h=\pi^*h_0$. Note that in particular $A_0$ is the pullback of a HYM connection on $\mathrm{Gr}(E)$. The aim is to apply the implicit function theorem to perturb $A_b$ along gauge orbits in order to satisfy point $(2)$ of the statement. The key will be to consider small perturbations along the exceptional divisor. Recall the splitting from Section \ref{sec:decompositions} induced by the operator $\iota_*$:
 $$
 i\Gamma(X',\End_H(\mathrm{Gr}(E),h))=i\Gamma(X,\End_H(\mathrm{Gr}(E),h))\oplus i\Gamma_0(Z',\End_H(\mathrm{Gr}(E),h)),
 $$
 that we will simply denote
 $$
 \Gamma(X')=\Gamma(X)\oplus \Gamma_0(Z').
 $$
For $(s_X,s_Z)\in \Gamma(X)\oplus \Gamma_0(Z')$, and $ \varepsilon $ small enough, we define
$$
A_b(\varepsilon,s_X,s_Z)=A_b^{s_X+\varepsilon s_Z},
$$
 where $s_X+\varepsilon s_Z$ stands for $\pi^*s_X+\varepsilon\,\iota_*s_Z\in\Gamma(X')$. By the regularity of $\widehat{\Phi}$, the assignment $(b,\varepsilon,s_X,s_Z)\mapsto A_b(\varepsilon,s_X,s_Z)-A$  (resp. $(b,\varepsilon,s_X,s_Z)\mapsto F_{A_b(\varepsilon,s_X,s_Z)}$) is smooth from $B\times [0,\varepsilon_0)\times \Gamma(X')^l$ to $\Omega^1(X',\End(E))^{l-1}$ (resp. $\Omega^2(X',\End(E))^{l-2}$), for any $\varepsilon_0$ small enough. Arguing as in Lemma \ref{lem:differentiable}, using the fact that the perturbations along $Z'$ are $\mathcal{O}(\varepsilon)$, we deduce that the operator
 $$
 \begin{array}{cccc}
 \widehat{\Psi} : & B\times [0,\varepsilon_0) \times \Gamma(X')^l  & \to & \Gamma(X')^{l-2} \\
       &       (b,\varepsilon,s_X,s_Z) & \mapsto & \Lambda_\varepsilon i F_{A_b(\varepsilon,s_X,s_Z)}-c_\varepsilon \Id_E
 \end{array}
 $$
 is a $\mathcal{C}^1$ map. As $A_0$ is HYM on $\mathrm{Gr}(E)\to X$, we have $\widehat{\Psi}(0)=0$. By the various lemmas of Section \ref{sec:decomposeLaplace}, its differential in the $(s_X, s_Z)$ direction at zero is given by the map
 $$
 \begin{array}{ccc}
  \Gamma(X)^l\times \Gamma_0(Z')^l & \to & \Gamma(X)^{l-2}\times\Gamma_0(Z')^{l-2}\\
         (s_X, s_Z) & \mapsto & \left[ \begin{array}{cc}
                                        \Delta_X s_X & 0 \\
                                        * & \Delta_{\mathcal{V}} s_Z
                                        \end{array}
                                        \right]
 \end{array}
 $$
which, from Lemma \ref{lem:linop} and Lemma \ref{lem:verticallaplacianinvertible}, has cokernel $i\mathfrak{k}\times \lbrace 0 \rbrace$. Then, by a standard projection argument onto some orthogonal complement of $i\mathfrak{k}$,  we can apply the implicit function theorem and obtain a $\mathcal{C}^1$ map $(\varepsilon,b)\mapsto s(\varepsilon,b)$ such that $\widehat{\Psi}(b,\varepsilon,s(\varepsilon,b))$ lies in $\mathfrak{k}$, and conclude the proof by setting
$$\check\Phi(\varepsilon,b)=(A_b(\varepsilon,s(\varepsilon,b)))^{0,1}-A^{0,1}.
$$
\end{proof}
We will now explain that for each $\varepsilon\in[0,\varepsilon_0)$, the map
\begin{equation}
 \label{eq:finitedimmoment}
\begin{array}{cccc}
 \mu_\varepsilon : & B & \to & \mathfrak{k} \\
           & b &  \mapsto & \Lambda_\varepsilon i F_{\check A_{\varepsilon,b}} - c_\varepsilon \Id_E
\end{array}
\end{equation}
is a moment map for the $K$-action on $B$, for suitable symplectic forms $\Omega_\varepsilon$ on $B$. Recall from \cite{AtBot,donaldson85} that for $\varepsilon\in(0,\varepsilon_0)$, the gauge action of $\mathscr{G}^{\mathbb{C}}(\pi^*\mathrm{Gr}(E),h)$ on the affine space $\overline{\partial}_0+\Omega^{0,1}(X',\End(\mathrm{Gr}(E)))$ is hamiltonian for the symplectic form given, for $(a,b)\in \Omega^{0,1}(X',\End(\mathrm{Gr}(E)))^2$, by
\begin{equation}
 \label{eq:momentmapdonaldson}
\Omega^{D}_\varepsilon(a,b)=\int_{X'} \mathrm{trace}(a\wedge b^*)\wedge \frac{\omega_\varepsilon^{n-1}}{(n-1)!},
\end{equation}
with equivariant moment map $\overline{\partial} \mapsto \Lambda_\varepsilon F_{A_{\overline{\partial}}}$ where $A_{\overline{\partial}}$ stands for the Chern connection of $(\overline{\partial},h)$. Here, we identified the Lie algebra of $\mathscr{G}^{\mathbb{C}}(\mathrm{Gr}(E),h)$ with its dual by mean of the invariant pairing
\begin{equation}
 \label{eq:pairing}
\langle s_1,s_2\rangle_\varepsilon := \int_{X'} \mathrm{trace}(s_1\cdot s_2^*)\; \frac{\omega_\varepsilon^n}{n!}.
\end{equation}
Note that the above expressions admit continuous extensions for $\varepsilon=0$ when we restrict to the $\mathscr{G}^{\mathbb{C}}(\mathrm{Gr}(E),h_0)$ action on $\overline{\partial}_0+\Omega^{0,1}(X,\End(\mathrm{Gr}(E)))$ and integrate over $(X,\omega)$.
\begin{remark}
 We used above the Chern correspondence, for $h$ fixed, between Dolbeault operators and hermitian connections to express the infinite dimensional moment map picture on the space of Dolbeault operators.
\end{remark}
\begin{proposition}
 \label{prop:symplecticembedding}
 Up to shrinking $\varepsilon_0$ and $B$, for all $\varepsilon\in[0,\varepsilon_0)$, the map $\overline{\partial}_0+\check\Phi(\varepsilon,\cdot)$ is a $K$-equivariant map from $B$ to $\overline{\partial}_0+\Omega^{0,1}(X',\End(\mathrm{Gr}(E)))$ whose image is a symplectic submanifold for $\Omega^D_\varepsilon$.
\end{proposition}

\begin{proof}
The equivariance follows easily from Proposition \ref{prop:firstslice} and from the construction of $\check \Phi$ in the proof of Proposition \ref{prop:perturbingslice}. For $\varepsilon=0$, the map $\check\Phi (0,\cdot)$ is obtained by perturbing $\widehat{\Phi}=\pi^*\circ\Phi$. But $\Phi$ is complex analytic with, by construction, injective differential at the origin (see e.g. the orginal proof \cite{kuranishi} or \cite{doan}). So is $\widehat{\Phi}$, and thus $\widehat{\Phi}(B)$ is a complex subspace of $\Omega^{0,1}(X',\End(\pi^*\mathrm{Gr}(E)))$. We deduce that, up to shrinking $B$, $\widehat{\Phi}$ induces an embedding of $B$ such that the restriction of $\Omega^D_0$ to $\widehat{\Phi}(B)$ is non-degenerate (recall that $\Omega^D_0$ is a K\"ahler form on the space of Dolbeault operators on $X$). As $\widehat{\Phi}(\varepsilon,\cdot)$ is obtained by a small and continuous perturbation of $\widehat{\Phi}$, and as being a symplectic embedding is and open condition, the result follows.
\end{proof}
From this result, we deduce that the map $\mu_\varepsilon$ defined in (\ref{eq:finitedimmoment}) is a moment map for the $K$-action on $B$ with respect to the pulled back symplectic form
$$\Omega_\varepsilon:=\check\Phi(\varepsilon,\cdot)^*\Omega^D_\varepsilon,
$$
and where we use the pairing $\langle \cdot,\cdot \rangle_\varepsilon$ defined in (\ref{eq:pairing}) to identify $\mathfrak{k}$ with its dual. From the discussion of Section \ref{sec:kuranishi}, $E$ is obtained as a small complex deformation of $\mathrm{Gr}(E)$, and thus by Proposition \ref{prop:firstslice}, $\overline{\partial}_E$ is gauge equivalent to an element $\overline{\partial}_b:=\overline{\partial}_0+ \Phi(b)$. Then, from properties of the maps $\Phi$ and $\check\Phi$, for all $\varepsilon\in[0,\varepsilon_0)$ and for all $g\in G$, $\pi^*\overline{\partial}_E$ will be gauge equivalent to $\pi^*\overline{\partial}_0+\check\Phi(\varepsilon,g\cdot b)$, provided $g\cdot b \in B$. As a zero of $\mu_\varepsilon$ corresponds to a HYM connection on $(X',\omega_\varepsilon)$, we are left with the problem of finding a zero for $\mu_\varepsilon$ in the $G$-orbit of $b$.

\section{Proof of the main theorem}
\label{sec:proofmain}
We carry on with notations from the last section, and our goal now is to prove Theorem \ref{theo:intro}. This is where we will need to assume that in $\mathrm{Gr}(E)=\bigoplus_{i=1}^\ell \mathcal{G}_i$, all stable components $\mathcal{G}_i$ are non isomorphic. This implies that
$$
\mathfrak{g}=\mathfrak{aut}(\mathrm{Gr}(E))=\bigoplus_{i=1}^\ell \mathbb{C}\cdot\Id_{\mathcal{G}_i}
$$
and thus its compact form $\mathfrak{k}$ is abelian, with $K$ a compact torus.

\subsection{The local convex cone associated to the $K$-action}
\label{sec:reducing}
In order to prove the existence of a zero of $\mu_\varepsilon$ in $\mathcal{Z}:=G\cdot b\cap B$, we start by describing, at least locally, the images of $\mathcal{Z}$ by the maps $(\mu_\varepsilon)_{\varepsilon\in [0,\varepsilon_0)}$. In this section, relying on \cite{SekTip22}, we will see that those images all contain translations of (a neighbourhood of the apex of) the same convex cone.

By simplicity of $E$, the stabiliser of $b$ under the $K$-action is reduced to the $S^1$-action induced by gauge transformations of the form $e^{i\theta}\Id_E$. As those elements fix all the points in $B$, elements in $S^1\cdot \Id_E$ will play no role in the arguments that follow. Hence, we will work instead with the quotient torus $K_0:=K/S^1\cdot \Id_E$. Note that the constants $c_\varepsilon$ that appear in the maps $\mu_{\varepsilon}$ in (\ref{eq:finitedimmoment}) are chosen so that $\langle \mu_\varepsilon, \Id_E \rangle_\varepsilon=0$. As the $\mu_\varepsilon$ take vakues in $\mathfrak{k}$, this is equivalent to say $\mathrm{trace}(\mu_\varepsilon)=0$. Hence, setting $\mathfrak{k}_0\subset \mathfrak{k}$ to be the set of trace free elements in $\bigoplus_{i=1}^\ell i\mathbb{R} \cdot \Id_{\mathcal{G}_i}$, we will consider the family of moment maps $\mu_\varepsilon:B\to\mathfrak{k}_0$ for the $K_0$-action, and we may, and will, assume that the stabiliser of $b$ is trivial.  Then, by using the inner product $\langle \cdot, \cdot \rangle_\varepsilon$ to identify $\mathfrak{k}_0\simeq\mathfrak{k}_0^*$, we can see the maps $\mu_\varepsilon$ as taking values in $\mathfrak{k}_0^*$ :
 $$
 \mu_{\varepsilon}^* : B \to \mathfrak{k}_0^*.
 $$

 There is a weight decomposition of $V$ under the abelian $K$-action
 \begin{equation}
  \label{eq:weightdecomposition}
 V:= \bigoplus_{m\in M} V_m
 \end{equation}
 for $M\subset \mathfrak{k}_0^*$ the lattice of characters of $K_0$. In the matrix blocks decomposition of $V=H^{0,1}(X,\End(\mathrm{Gr}(E)))$ induced by $\mathrm{Gr}(E)=\bigoplus_{i=1}^\ell\mathcal{G}_i$, using the product hermitian metric $h_0$, we have
 $$
 V=\bigoplus_{1\leq i,j\leq \ell} H^{0,1}(X,\mathcal{G}_i^*\otimes\mathcal{G}_j).
 $$
 The action of $g\in K_0$ on $V_{ij}:=H^{0,1}(X,\mathcal{G}_i^*\otimes\mathcal{G}_j)$ is, by Equation (\ref{eq:explicitaction}):
 \begin{equation}
 \label{eq:action2}
 g\cdot \gamma_{ij}=g_ig_j^{-1} \gamma_{ij}.
 \end{equation}
Thus, in the weight space decomposition (\ref{eq:weightdecomposition}), $V_{ij}$ is the eigenspace with weight
 \begin{equation}
  \label{eq:weightsmij}
 m_{ij}:= (0,\ldots,0,1,0,\ldots, 0, -1,0,\ldots,0)
 \end{equation}
 where $+1$ appears in $i$-th position and $-1$ in the $j$-th position. If we decompose $b$ accordingly as
 \begin{equation}
  \label{eq:indicicesdecompositionb}
 b=\sum_{ij} b_{ij},
 \end{equation}
 where $b_{ij}\in V_{ij}$ is non zero, as $\overline{\partial}_E=\overline{\partial}_0 +\gamma$ with $\gamma$ upper triangular, or equivalently as $E$ is obtained as successive extentions of the stable components $\mathcal{G}_i$'s, only indices $(i,j)$ with $i < j$ will appear in (\ref{eq:indicicesdecompositionb}).
From now on, we will restrict our setting to
 $$
 B\cap \bigoplus_{b_{ij}\neq 0} V_{ij},
 $$
which we still denote by $B$. That is, we only consider weight spaces that appear in the decomposition of $b$. Similarily, we use the notation $V$ for $\bigoplus_{b_{ij}\neq 0} V_{ij}$.

To sum up, we are in the following setting :
\begin{enumerate}
 \item[($R_1$)] The compact torus $K_0$ acts effectively and holomorphically on the complex vector space $V$;
 \item[($R_2$)] There is a continous family of symplectic forms $(\Omega_{\varepsilon})_{0\leq \varepsilon < \varepsilon_0}$ on $B\subset V$ around the origin, with respect to which the $K_0$-action is hamiltonian;
 \item[($R_3$)] The point $b\in B$ has trivial stabiliser, $0$ in its $K_0^\mathbb{C}$-orbit closure, and for all weight $m_{ij}\in M$ appearing in the weight space decomposition of $V$, $b_{ij}\neq 0$.
 \item[($R_4$)] The restriction of the symplectic form $\Omega_0$ to the $K_0^\mathbb{C}$-orbit of $b$ is non-degenerate.
\end{enumerate}
This last point follows as in the proof of Proposition \ref{prop:symplecticembedding}.
We set $$\overline{\mathcal{Z}}:=B\cap(\overline{K_0^\mathbb{C}\cdot b}).$$
We also introduce
$$
\sigma:= \sum_{b_{ij}\neq 0} \mathbb{R}_+ \cdot m_{ij}\subset \mathfrak{k}_0^*
$$
with $\lbrace m_{ij},\: b_{ij}\neq 0 \rbrace$ the set of weights that appear in the decomposition of $b\in V$,
and for $\eta >0$
$$
\sigma_\eta := \sum_{b_{ij}\neq 0} [0,\eta ) \cdot m_{i j}\subset \mathfrak{k}_0^*.
$$
Note that by the local version of Atiyah and Guillemin--Sternberg's convexity theorem, there exists $\eta >0$ such that $\mu_\varepsilon^*(0)+\sigma_\eta\subset\mu_\varepsilon^*(B)$ for all $\varepsilon$ small enough (see the equivariant Darboux Theorem \cite[Theorem 3.2]{Dwivedi} combined with the local description of linear hamiltonian torus actions \cite[Section 7.1]{Dwivedi}). By \cite[Proposition 4.6]{SekTip22}, the properties $(R_1)-(R_4)$ listed above actually imply :
 \begin{proposition}
 \label{prop:sigma-eta-image-orbit}
  Up to shrinking $B$ and $\varepsilon_0$, there exists $\eta>0$ such that for all $\varepsilon\in [0,\varepsilon_0)$,
  $$
   \mu_{\varepsilon}^*(0)+ \mathrm{Int}(\sigma_\eta)  \subset \mu^*_{\varepsilon}(\mathcal{Z})
  $$
  and
  $$
  \mu_{\varepsilon}^*(0)+ \sigma_\eta  \subset \mu^*_{\varepsilon}(\overline{\mathcal{Z}}).
  $$
 \end{proposition}
 \begin{remark}
 The fact that the interior of $ \mu_{\varepsilon}^*(0)+ \sigma_\eta$ is included in the image of the $K_0^\mathbb{C}$-orbit of $b$ by $\mu_\varepsilon^*$ is not stated explicitely in \cite{SekTip22}, but follows from the discussion at the beginning of the proof of \cite[Proposition 4.6]{SekTip22}.
 \end{remark}

\subsection{Solving the problem}
\label{sec:solving}
From Proposition \ref{prop:sigma-eta-image-orbit}, to prove the existence of a zero of $\mu_\varepsilon$ in $\mathcal{Z}$, it is enough to show that $-\mu_\varepsilon^*(0)\in \mathrm{Int}(\sigma_\eta)$, which reduces to $-\mu_\varepsilon^*(0)\in \mathrm{Int}(\sigma)$ for small enough $\varepsilon$. Arguing as in \cite[Lemma 4.8]{SekTip22}, $\sigma$ and its dual $$\sigma^\vee:=\lbrace v\in \mathfrak{k}_0\:\vert\: \langle m, v \rangle\geq 0\; \forall m\in\sigma \rbrace$$ are strongly convex rationnal polyhedral cones of dimension $\ell-1$. Note that here the pairing $\langle \cdot,\cdot \rangle$ is the natural duality pairing. By duality, $\sigma=(\sigma^\vee)^\vee$, and we are left with proving
$$
-\mu_\varepsilon^*(0)\in \mathrm{Int}((\sigma^\vee)^\vee).
$$
The cone $\sigma^\vee$ can be written
$$
\sigma^\vee=\sum_{\underline{a}\in\mathcal{A}} \mathbb{R}_+\cdot v_{\underline{a}}
$$
for a finite set of generators $\lbrace v_{\underline{a}} \rbrace_{\underline{a}\in\mathcal{A}}\subset \mathfrak{k}_0$. Hence, our goal now is to show that for all $\underline{a}\in\mathcal{A}$, $\langle \mu_\varepsilon^*(0), v_{\underline{a}} \rangle < 0$, which by construction is equivalent to
\begin{equation}
 \label{eq:toprove}
\langle \mu_\varepsilon(0), v_{\underline{a}} \rangle_\varepsilon< 0,
\end{equation}
under the asumption that for any $\mathcal{F}\in\mathfrak{E}_{[\omega]}$,
\begin{equation}
\label{eq:wehave}
\mu_{L_\varepsilon}(\mathcal{F})\underset{\varepsilon\to 0}{<}\mu_{L_\varepsilon}(E).
\end{equation}
We will then study in more details Equations (\ref{eq:toprove}) and (\ref{eq:wehave}). In order to simplify the notations, in what follows, we will assume that all the stable components of $\mathrm{Gr}(E)$ have rank one, so that $\mathrm{trace}(\Id_{\mathcal{G}_i})=1$ for $1\leq i \leq \ell$. The general case can easily be adapted, and is left to the reader.

\subsubsection{Condition (\ref{eq:toprove}) : generators of the dual cone}
We will give here a more precise form for the generators $\lbrace v_{\underline{a}}\rbrace_{\underline{a}\in\mathcal{A}}$ of $\sigma^\vee$. Recall from \cite[Section 1.2]{fulton} the method to find such generators : as $\sigma$ is $\ell-1$-dimensional, each of its facets is generated by $\ell-2$ elements amongst its generators $(m_{ij})$. Then, a generator $v_{\underline{a}}$ for $\sigma^\vee$ will be an ``inward pointing normal'' to such a facet. Hence, if $$v_{\underline{a}}=\sum_{i=1}^\ell a_i \Id_{\mathcal{G}_i}$$ is a generator of $\sigma^\vee$, there exists a set $\mathcal{S}:=\lbrace m_{ij} \rbrace$ of $\ell-2$ generators of $\sigma$  such that
$$
\forall\; m_{ij}\in\mathcal{S},\: \langle m_{ij}, v_{\underline{a}} \rangle=0.
$$
Moreover, $v_{\underline{a}}\in \mathfrak{k}_0$ should be trace free, and as we assume here $\mathrm{rank}(\mathcal{G}_i)=1$ for all stable components, it gives
$$
\sum_{i=1}^\ell a_i =0.
$$
\begin{lemma}
 \label{lem:dualgenerators}
 Up to scaling $v_{\underline{a}}$, there exists a partition $\lbrace 1,\ldots,\ell \rbrace =I^-\cup I^+$ such that for all $i\in I^-$, $a_i=-\frac{1}{\sharp I^-}$ and for all $i\in I^+$, $a_i=\frac{1}{\sharp I^+}$, where $\sharp$ stands for the cardinal of a set.
\end{lemma}
\begin{proof}
 The key is to observe that if $m_{ij},m_{jk}\in \mathcal{S}^2$, then $m_{ik}\notin \mathcal{S}$. Indeed, by (\ref{eq:weightsmij}), $m_{ij}+m_{jk}=m_{ik}$, and those are generators of the cone. Equivalently, if $m_{ij},m_{ik}\in\mathcal{S}^2$, then $m_{jk}\notin \mathcal{S}$. We then assign an oriented graph $G_{\underline{a}}$ to $v_{\underline{a}}$. The vertices are labelled $a_1$ to $a_\ell$, and we draw an oriented edge from $a_i$ to $a_j$ if $a_i=a_j$ and $i<j$. For each $m_{ij}\in\mathcal{S}$, $\langle m_{ij}, v_{\underline{a}} \rangle =0$ gives $a_i=a_j$. Hence, $G_{\underline{a}}$ has at least $\ell-2$ edges. To prove the result, it is enough to show that $G_{\underline{a}}$ has $2$ connected components. Indeed, we can then set $I^-=\lbrace i\,\vert a_i <0 \rbrace$ and $I^+=\lbrace i\,\vert a_i >0 \rbrace$. All elements $a_i$ for $i\in I^-$ will correspond to the same connected component and be equal, and similarily with $i\in I^+$. As $\sum_{i=1}^\ell a_i =0$, we obtain the result by rescaling.

 Proving that $G_{\underline{a}}$ has two connected components is then routine. It has $\ell$ vertices and $\ell-2$ oriented edges, with the rule that if there is an edge from $a_i$ to $a_j$ and an edge from $a_i$ to $a_k$, then there is no edge from $a_j$ to $a_k$. We consider the number of edges that start from $a_1$. If there are $\ell-2$ of those, then the connected component of $a_1$ has at least $\ell-1$ vertices, and we are left with at most $1$ singleton for the other component. The fact that $v_{\underline{a}}$ is trace free imposes that there are at least $2$ connected components, and we are done in that case. Then, if there are $\ell-2-k$ edges from $a_1$, its connected component has at least $\ell-1-k$ elements, and we are left with at most $k+1$ vertices and $k$ edges for the other components. But its easy to show, by induction on $k$, that the rule stated above implies that there will be at most $1$ connected component for such a graph with $k+1$ vertices and $k$ edges, and we are done.
\end{proof}
We can now translate condition (\ref{eq:toprove}), by Lemma \ref{lem:dualgenerators}, it is equivalent to
\begin{equation}
 \label{eq:toprove2}
 \frac{\sum_{i\in I^+}\langle \mu_\varepsilon(0),\Id_{\mathcal{G}_i}\rangle_\varepsilon}{\sharp I^+}  < \frac{\sum_{i\in I^-}\langle \mu_\varepsilon(0),\Id_{\mathcal{G}_i}\rangle_\varepsilon}{\sharp I^-}.
\end{equation}

\subsubsection{Condition (\ref{eq:wehave}) : one parameter degenerations}
We will associate to each generator $v_{\underline{a}}$ of $\sigma^\vee$ a subsheaf $\mathcal{F}\in\mathfrak{E}_{[\omega]}$. Geometrically, the idea is that $v_{\underline{a}}\in\mathfrak{k}_0$ generates a one-parameter subgroup of $K_0$ and a degeneration of $E$ to $\mathcal{F}\oplus E/\mathcal{F}$, to which is assigned the Hilbert--Mumford weight $\mu_{L_\varepsilon}(\mathcal{F})-\mu_{L_\varepsilon}(E)<0$. We let $v_{\underline{a}}=\sum_{i=1}^{\ell} a_i \Id_{\mathcal{G}_i}\in\sigma^\vee$ a generator as above, and define
$$
\mathcal{F}_{\underline{a}}=\bigoplus_{i\in I^+} \mathcal{G}_i,
$$
as a {\it smooth} complex vector bundle, and will show that $\overline{\partial}_E (\mathcal{F}_{\underline{a}})\subset \Omega^{0,1}(X',\mathcal{F}_{\underline{a}})$. This implies that $\mathcal{F}_{\underline{a}}\in \mathfrak{E}_{[\omega]}$ as a {\it holomorphic} vector bundle, with Dolbeault operator the restriction of $\overline{\partial}_E$. Recall that $\overline{\partial}_E=\overline{\partial}_0 +\gamma=\overline{\partial}_0+ \sum_{b_{ij}\neq 0} \gamma_{ij}$, that is, by choice of $b$, the weights that appear in the weight decomposition of $\gamma$ are the same as those that appear in the decomposition of $b$. In the matrix blocks decomposition given by $\bigoplus_{i=1}^\ell\mathcal{G}_i$, the operator $\overline{\partial}_0$ is diagonal, and thus sends $\mathcal{F}_{\underline{a}}$ to $\Omega^{0,1}(X',\mathcal{F}_{\underline{a}})$. We need to show that for each $j\in I^+$, $\gamma (\mathcal{G}_j)\subset \Omega^{0,1}(X',\mathcal{F}_{\underline{a}})$. As $v_{\underline{a}}\in\sigma^\vee$, it satisfies, for all generator $m_{ij}$ of $\sigma$ :
$$
\langle m_{ij}, v_{\underline{a}} \rangle \geq 0,
$$
that is, for all $(i,j)$ with $i<j$ and $b_{ij}\neq 0$,
$$
a_i - a_j \geq 0.
$$
As $j\in I^+$, this implies $a_i\geq a_j > 0$. Hence, if $i < j$ is such that $b_{ij}\neq 0$, then $i\in I^+$. Equivalently, for $i<j$, $i\in I^-$ implies $\gamma_{ij}=0$, and thus we see that $\gamma(\mathcal{G}_j)\subset \Omega^{0,1}(X',\mathcal{F}_{\underline{a}})$, and hence $\overline{\partial}_E (\mathcal{F}_{\underline{a}})\subset \Omega^{0,1}(X',\mathcal{F}_{\underline{a}})$.

Then we have $\mathcal{F}_{\underline{a}}\in \mathfrak{E}_{[\omega]}$ and Condition (\ref{eq:wehave}) gives
$$
\mu_{L_\varepsilon}(\mathcal{F}_{\underline{a}}) \underset{\varepsilon\to 0}{<} \mu_{L_\varepsilon}(E),
$$
which, by the see-saw property of slopes (see e.g. \cite[Corollary 3.5]{SekTip} ), gives
$$
\mu_{L_\varepsilon}(\mathcal{F}_{\underline{a}}) \underset{\varepsilon\to 0}{<} \mu_{L_\varepsilon}(E/\mathcal{F}_{\underline{a}})
$$
and thus (recall we assume $\mathrm{rank}(\mathcal{G}_i)=1$):
\begin{equation}
\label{eq:wehave2}
\frac{\sum_{i\in I^+}\mu_{L_\varepsilon}(\mathcal{G}_i)}{\sharp I^+}  \underset{\varepsilon\to 0}{<} \frac{\sum_{i\in I^-}\mu_{L_\varepsilon}(\mathcal{G}_i)}{\sharp I^-} .
\end{equation}

\subsubsection{Conclusion}
Recall that Equation (\ref{eq:wehave2}) means that in the $\varepsilon$-expansion of $\frac{\sum_{i\in I^+}\mu_{L_\varepsilon}(\mathcal{G}_i)}{\sharp I^+}  - \frac{\sum_{i\in I^-}\mu_{L_\varepsilon}(\mathcal{G}_i)}{\sharp I^-} $, the first non-zero term is strictly negative. By Chern--Weyl theory, using the fact that $A_0$ and $\check A_{\varepsilon,0}$ are gauge-equivalent by point $(2)$ of Proposition \ref{prop:perturbingslice}, we have
$$
\begin{array}{ccc}
\mu_{L_\varepsilon}(\mathcal{G}_i) & = & c_1(\mathcal{G}_i)\cdot [\omega_\varepsilon]^{n-1} \\
               & = & \frac{1}{2\pi}\langle \mu_\varepsilon(0), \Id_{\mathcal{G}_i} \rangle_\varepsilon + \frac{c_\varepsilon}{2\pi}\langle \Id_E, \Id_{\mathcal{G}_i} \rangle_\varepsilon.
\end{array}
$$
Hence Inequality (\ref{eq:wehave2}) implies Inequality (\ref{eq:toprove2}), for $\varepsilon$ small enough, which concludes the existence of $b_\varepsilon\in \mathcal{Z}$ such that $\mu_\varepsilon(b_\varepsilon)=0$. Then, by construction, the associated connections $\check A_{\varepsilon,b_\varepsilon}$ provide HYM connections with respect to $\omega_\varepsilon$ on bundles gauge equivalent to $E$, where the gauge equivalences are given by elements in the finite dimensional Lie group $\Aut(\mathrm{Gr}(E))$. To conclude the proof of Theorem \ref{theo:intro}, it then remains to show that the connections $\check A_{\varepsilon,b_\varepsilon}$ converge to $\pi^*A_0=\check A_{0,0}$ in any $L^{2,l}$ Sobolev norm. By construction of $\check A_{\varepsilon,b}$ in Proposition \ref{prop:perturbingslice}, it is enough to prove that $b_\varepsilon$ converges to $0$ when $\varepsilon \to 0$. Recall from \cite[Theorem 3.2 and Section 7.1]{Dwivedi} that $B$ can be chosen so that $\mu_{\varepsilon}^*$ is given by
\begin{equation}
 \label{eq:expressionmuep}
\mu_{\varepsilon}^*(b')=\mu_{\varepsilon}^*(0)+\sum_{ij} \vert\vert b_{ij}'\vert\vert^2_{\varepsilon} \cdot m_{ij},
\end{equation}
for some norm $\vert\vert \cdot \vert\vert_{\varepsilon}$ that depends continously on $\varepsilon$. As $\mu_\varepsilon(0)\underset{\varepsilon\to 0}{\to}\mu_0(0)=0$, the equation $\mu_\varepsilon^*(b_\varepsilon)=0$ implies that for all $(i,j)$, $ \vert\vert (b_\varepsilon)_{ij}\vert\vert_{\varepsilon} \underset{\varepsilon\to 0}{\to} 0$. As the norms $\vert\vert\cdot\vert\vert_\varepsilon$ vary continuously, they are mutually bounded, and thus $b_\varepsilon\underset{\varepsilon\to 0}{\to}0$, which concludes proof of Theorem \ref{theo:intro}.

\subsubsection{Proof of the corollaries}
We comment now on the various corollaries stated in the introduction. First, Corollary \ref{cor:stablecase} is a direct application of Theorem \ref{theo:intro}, where $E=\mathrm{Gr}(E)$ as a single stable component. Corollary \ref{cor:restrictionnumericalcriterion} also follows directly, using Formula (\ref{eq:formulaAchim}). What remains is to show Corollary \ref{cor:semistablecase}. The only remaing case to study is when for all $\mathcal{F}\in\mathfrak{E}_{[\omega]}$,  $\mu_{L_\varepsilon}(\mathcal{F})\underset{\varepsilon\to 0}{\leq}\mu_{L_\varepsilon}(E)$, with at least one equality. In that situation, the discussion in the last two sections shows that $-\mu_\varepsilon(0)\in \sigma$ will lie in the boundary of $\sigma$. Hence, by Proposition \ref{prop:sigma-eta-image-orbit}, there is a boundary point $b'\in \overline{\mathcal{Z}}$ in the orbit closure of $b$ with $\mu_\varepsilon(b')=0$. This point corresponds to a HYM connection on a vector bundle that is then polystable for the holomorphic structure given by $\check A_{\varepsilon,b'}^{0,1}$, with respect to $L_\varepsilon$. As this bundle correspond to a boundary point in the complex orbit of $b$, it admits a small complex deformation to $\pi^*E$. As semi-stability is an open condition, we deduce that $\pi^*E$ is itself semi-stable for $L_\varepsilon$.

\bibliographystyle{plain}

redpajama_set_name: RedPajamaArXiv


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