text:
\section{Introduction}

Let $X$ be a compact K\"ahler manifold of dimension $n$. Let $\omega$ be a K\"ahler form on $X$. For every closed positive $(p,p)$-current $S$ on $X$, we recall that the mass $\|S\|$ of $S$ is given by $\int_X S \wedge \omega^{n-p}$.  A cohomology $(p,p)$-class $\alpha$ is said to be \emph{pseudoeffective} if it is the class of a closed positive $(p,p)$-current. For  pseudoeffective $(p,p)$-classes $\alpha_1$ and $\alpha_2$ on $X$, we write $\alpha_1 \le  \alpha_2$ if $\alpha_2- \alpha_1$ is pseudoeffective. For a pseudoeffective $(p,p)$-class $\alpha$, we  use the notation $\|\alpha\|$ to denote the norm of a closed positive current $S$ representing $\alpha$. This quantity is independent of the choice of $S$.

Let $1\le m \le n$ be an integer. Let $\alpha_1, \ldots, \alpha_m$ be  big nef cohomology classes on $X$. Let $T_j$ be a closed positive current of bi-degree $(1,1)$ in $\alpha_j$ for $1 \le j \le m$. The non-pluripolar product $\langle T_1 \wedge \cdots \wedge T_m \rangle$ plays an important role in complex geometry; see \cite{BT_fine_87,BEGZ,Lu-Darvas-DiNezza-logconcave,GZ-weighted,Viet-generalized-nonpluri} and references therein. It generalizes the classical product of $(1,1)$-currents of bounded potentials (see \cite{Bedford_Taylor_82}). A key phenomenon about that notion is that the non-pluripolar products don't preserve the mass, \emph{i.e,} in general, we have
\begin{align}\label{ine-sosanhmassT1denTmomega1m}
\big \| \langle T_1 \wedge \cdots \wedge T_m \rangle \big \| \le \big \| \alpha_1 \wedge \cdots \wedge \alpha_m \big \|.
\end{align}
We indeed have a much stronger property that the cohomology class of the current  $\langle T_1 \wedge \cdots \wedge T_m \rangle$ is less than or equal to $\alpha_1 \wedge \cdots \wedge \alpha_m$, see \cite[Theorem 1.1]{Viet-generalized-nonpluri} and also  \cite{BEGZ,Lu-Darvas-DiNezza-mono,WittNystrom-mono} for the case where $m=n$.

When the equality in (\ref{ine-sosanhmassT1denTmomega1m}) occurs, the currents $T_1,\ldots,T_m$ are said to be \emph{of full mass intersection}. The last notion is at the heart of the theory of non-pluripolar products. Characterizing such currents is hence important. So far we have known  that the positivity of Lelong numbers of $T_j$'s is an obstruction to being of full mass intersection; see  \cite{Lu-Darvas-DiNezza-singularitytype,GZ-weighted,Vu_lelong-bigclass} and references therein. We are interested in understanding this property from a quantitative point of view. Although we think that this question is worth studying, there has not been much research in this direction.  To our best knowledge, the first available result is probably \cite[Theorem 1.2]{Vu_lelong-bigclass} which treated the case when $m=n$; see also \cite[Corollary 7.6]{Demailly_regula_11current} for a related result.  Unfortunately, the method in \cite{Vu_lelong-bigclass} is not good enough to obtain appropriate quantitative estimates when $m<n$, see comments after Theorem \ref{the-self-intersec} below. The following is our main result giving such an quantitative estimate for every $m$ in the case of self-intersection.

\begin{theorem} \label{the-self-intersec} Let $\cali{N}_0$ be a compact subset in the intersection of the big and nef cones of $X$.  Let $\alpha$ be a cohomology $(1,1)$-class in $\cali{N}_0$. Let $T$ be a closed positive current in $\alpha$. Let $1\le m \le n$ be an integer.  Let $\cali{V}$ be  the set of  maximal irreducible analytic subsets $V$ of dimension at least $n-m$ of $X$ such that the generic Lelong number $\nu(T,V)$ of $T$ along $V$ is strictly positive. Then, there exists a constant $C>0$ independent of $\alpha$ such that
\begin{align}\label{ine-obstructionselfinter}
\big\| \alpha^m - \{\langle T^m \rangle \}  \big\| \ge C \bigg(\sum_{V \in \cali{V}}\nu(T,V)^{n- \dim V} \vol(V) \bigg)^{2^m}.
\end{align}
\end{theorem}

Here we put $\vol(V):= \int_V \omega^{\dim V}$.  The \emph{maximality} of $V$ in the above result means that in the set of  irreducible analytic subsets $V'$ of $X$ such that $\nu(T,V')>0$, we consider the partial order given by the inclusion of sets, and $V$ is maximal if it is so with respect to that order.  Theorem \ref{the-self-intersec} no longer holds if one enlarges $\cali{V}$ to include non-maximal analytic sets. One can see it as follows: take for example $T$ to be a current with analytic singularities along analytic set $V_0$ of dimension $>n-m$, for a suitable choice of $V_0$ (e.g, $V_0$ is biholomorphic to a complex projective space), we see that the volume of an irreducible set $V$ of dimension $n-m$ in $V_0$ can be as big as we want.

To get motivated about Theorem \ref{the-self-intersec}, one can consider an ideal classical example where $T$ has analytic singularities along  an irreducible analytic set $V$ of dimension $n-m$, in this case, the right-hand side of (\ref{ine-obstructionselfinterkahler}) can be replaced by $\nu(T,V)^m \vol(V)$. When  $\cali{N}_0$ is compact in the K\"ahler cone of $X$, we have a more precise estimate; see Theorem \ref{the-kahlerself-intersec} for details.

We underline that  the arguments used in \cite[Theorem 1.2]{Vu_lelong-bigclass} and  \cite[Theorem 1.2]{Viet-generalized-nonpluri} are not sufficient to get (\ref{ine-obstructionselfinter})  because the proofs there use desingularizations of $V$ and the blowup along $V$; hence the lower bounds obtained there depend intrinsically on $V$.   We don't know what should be an optimal lower bound for the left-hand side of (\ref{ine-obstructionselfinter}).

As a direct consequence of Theorem \ref{the-self-intersec},  we infer that if $\{\langle T^m \rangle \}= \alpha^m$, then for every (not necessarily maximal) irreducible analytic subset $V$ of dimension at least $n-m$, we have $\nu(T,V)=0$. This recovers a previous known result in \cite{Vu_lelong-bigclass} (see Theorem 1.3 there). As a byproduct of our method, we also generalize some results from \cite{Lu-Darvas-DiNezza-singularitytype}, see Corollaries \ref{cor-Tcongvoitheta} and \ref{cor-Tcongvoitheta2} below. \emph{Finally we note that Theorem \ref{the-self-intersec} is new even when  $m=n$ and $\alpha$ is K\"ahler.}

In order to prove Theorem \ref{the-self-intersec}, we will prove a reserved Alexandrov-Fenchel inequality for non-pluripolar products, see Proposition \ref{pro-theoremmain} below. This will reduce the question to the K\"ahler case. Next, we establish a generalization of Demailly's lower bound for Lelong numbers of intersection of currents to the setting of the theory of density currents introduced by Dinh-Sibony in \cite{Dinh_Sibony_density}, see Corollary \ref{cor-sosanhlelong}. This is a key in our treatment of the K\"ahler case. Moreover, we emphasize that the notion of relative non-pluripolar products in \cite{Viet-generalized-nonpluri} will play a crucial role in our proof. It serves as a bridge from density currents to non-pluripolar products.

In the next section, we prove the above-mentioned lower bound for Lelong numbers of density currents. In Section \ref{sec-AF}, we establish a reversed Alexandrov-Fenchel type inequality for non-pluripolar products. Theorems \ref{the-self-intersec} is proved in Sections \ref{sec-the1}.
\\

\noindent
\textbf{Acknowledgments.} The author would like to thank Tien-Cuong Dinh and Nessim Sibony  for fruitful discussions. This research  is supported by a postdoctoral fellowship of the Alexander von Humboldt Foundation.

\section{Lelong numbers of intersection of currents} \label{sec-lelong-density}

We first recall some basic properties of density currents. The last notion was introduced in \cite{Dinh_Sibony_density}.

Let $X$ be a complex K\"ahler manifold of dimension $n$ and $V$ a smooth complex submanifold of $X$ of dimension $l.$  Let $T$ be a closed positive $(p,p)$-current on $X,$ where $0 \le p \le n.$  Denote by $\pi: E\to V$ the normal bundle of $V$ in $X$ and $\overline E:= \P(E \oplus \C)$ the projective compactification of $E.$ By abuse of notation, we also use $\pi$ to denote the natural projection from $\overline E$ to $V$.

Let $U$ be an open subset of $X$ with $U \cap V \not = \varnothing.$  Let $\tau$ be  a smooth diffeomorphism  from $U$ to an open neighborhood of $V\cap U$ in $E$ which is identity on $V\cap U$ such that  the restriction of its differential $d\tau$ to $E|_{V \cap U}$ is identity.  Such a map is called \emph{an admissible map}.  Note that in \cite{Dinh_Sibony_density}, to define an admissible map,  it is required furthermore that $d\tau$ is $\C$-linear at every point of $V$. This difference doesn't affect what follows.  When $U$ is a small enough tubular neighborhood of $V,$ there always exists an admissible map $\tau$ by \cite[Lemma 4.2]{Dinh_Sibony_density}. In general, $\tau$ is not holomorphic.  When $U$ is a small enough local chart, we can choose a holomorphic admissible map by using suitable holomorphic coordinates on $U$.   For $\lambda \in \C^*,$ let $A_\lambda: E \to E$ be the multiplication by $\lambda$ on fibers of $E.$  We recall the following crucial result.

\begin{theorem} \label{th-dieukienHVconictangenetcurrent} (\cite[Theorem 4.6]{Dinh_Sibony_density})
Let $\tau$ be an admissible map defined on a tubular neighborhood of $V$. Then,  the family $(A_\lambda)_* \tau_* T$ is of mass uniformly bounded in $\lambda$, and  if $S$ is a limit current of the last family as $\lambda \to \infty$, then  $S$ is a current on $E$ which can be extended trivially through $\overline E \backslash E$ to be a closed positive current on $\overline E$  such that the cohomology class $\{S\}$ of $S$ in $\overline E$ is independent of the choice of $S$, and $\{S\}|_V= \{T\}|_V$,  and $\|S\| \le C \|T\|$ for some constant $C$ independent of $S$ and $T$.
\end{theorem}

We call $S$  \emph{a tangent current to $T$ along $V$}, and its cohomology class is called \emph{the total tangent class of $T$ along $V$} and is denoted by $\kappa^V(T)$. By \cite[Theorem 4.6]{Dinh_Sibony_density} again,  if
$$S=\lim_{k\to \infty} (A_{\lambda_k})_* \tau_* T$$ for some sequence $(\lambda_k)_k$ converging to $\infty$, then  for every open subset $U$ of $X$ and  every admissible map $\tau': U' \to E$ , we also have
$$S=\lim_{k\to \infty} (A_{\lambda_k})_* \tau'_* T.$$
This is equivalent to saying that tangent currents are independent of the choice of the admissible map $\tau$.

\begin{definition} (\cite{Dinh_Sibony_density}) Let $F$ be a complex manifold and $\pi_F: F \to V$ a holomorphic submersion. Let $S$  be  a positive current of bi-degree $(p,p)$  on $F$. \emph{The h-dimension} of $S$ with respect to $\pi_F$ is the biggest integer $q$ such that $S \wedge \pi_F^* \theta^q \not =0$ for some Hermitian metric $\theta$ on $V$.
\end{definition}

By a bi-degree reason, the h-dimension of $S$ is in $[\max\{l- p,0\}, \min\{\dim F -p,l\}]$.
We have the following description of currents with minimal h-dimension.

\begin{lemma} \label{le-minimalhdimension} (\cite[Lemma 3.4]{Dinh_Sibony_density})  Let $\pi_F: F \to V$ be a submersion. Let $S$ be a closed positive  current of bi-degree $(p,p)$ on $F$ of h-dimension $(l -p)$ with respect to $\pi_F$. Then $S= \pi^* S'$ for some closed positive current $S'$ on $V$.
\end{lemma}

By \cite{Dinh_Sibony_density}, the h-dimensions of tangent currents to $T$ along $V$ are the same and this number is called \emph{the tangential h-dimension of $T$ along $V$}.

Let $m\ge 2$ be an integer. Let $T_j$ be a closed positive current  of bi-degree $(p_j, p_j)$ for $1 \le j \le m$ on $X$ and let  $T_1 \otimes \cdots \otimes T_m$ be the tensor current of $T_1, \ldots, T_m$ which is a current on $X^m.$  A \emph{density current} associated to $T_1, \ldots,  T_m$ is a tangent current to $\otimes_{j=1}^m T_j$ along the diagonal $\Delta_m$ of $X^m.$ Let $\pi_m: E_m \to \Delta$ be the normal bundle of $\Delta_m$ in $X^m$. Denote by $[V]$ the current of integration along $V$.  When $m=2$ and $T_2 =[V]$, the density currents of $T_1$ and  $T_2$ are naturally identified with the  tangent currents to $T_1$ along $V$ (see \cite[Lemma 2.3]{Vu_density-nonkahler}).

The unique cohomology class of density currents associated to $T_1,\ldots,T_m$ is called \emph{the total density class of $T_1, \ldots, T_m$}. We denote the last class by $\kappa(T_1,\ldots, T_m)$. The tangential h-dimension of  $T_1 \otimes \cdots \otimes T_m$ along $\Delta_m$ is called \emph{the density h-dimension} of  $T_1, \ldots T_m$.

\begin{lemma} \label{le-classDSproduct} (\cite[Section 5]{Dinh_Sibony_density})  Let $T_j$ be a closed positive current of bi-degree $(p_j,p_j)$ on $X$ for $1 \le j \le m$ such that $\sum_{j=1}^m p_j \le n$.  Assume that the density h-dimension of $T_1, \ldots, T_m$ is minimal, \emph{i.e}, equal to $n- \sum_{j=1}^m p_j$. Then the total density class of  $T_1, \ldots, T_m$ is equal to $\pi_{m}^*(\wedge_{j=1}^m\{T_j\})$.
\end{lemma}

We recall the following result.

\begin{theorem} \label{th-sosanhVV'densityDS} (\cite[Proposition 4.13]{Dinh_Sibony_density}) Let  $V'$ be a submanifold of  $V$ and  let $T$ be a closed positive current on $X$. Let $T_\infty$ be a tangent current to $T$ along $V$. Denote by $s$ the density h-dimension of $T_\infty$ and $[V']$. Then, the density h-dimension of $T$ and $[V']$  is at most $s$, and we have
$$\kappa_s(T,[V']) \le \kappa_s(T_\infty, [V']).$$
The inequality still holds if we replace $s$ by the tangential h-dimension of $T$ along $V$.
\end{theorem}

As a consequence, we obtain the following result generalizing the well-known lower bound of Lelong numbers of intersection of $(1,1)$-currents due to Demailly \cite[Page 169]{Demailly_analyticmethod} in the compact setting.

\begin{corollary}\label{cor-sosanhlelong} Let $T_j$ be a closed positive current on $X$ for $1 \le j \le m$.  Then, for every $x \in X$ and for every density current $S$ associated to $T_1, \ldots, T_{m}$, we have
\begin{align}\label{inesosanhSTjlelong}
\nu(S, x) \ge \nu(T_1, x) \cdots \nu(T_m,x).
\end{align}
\end{corollary}

\proof  Let $x \in X$. Let $\Delta_m$ be the diagonal of $X^m$.  Let   $\pi: E \to \Delta_m$ be the natural projection from the normal bundle of the diagonal $\Delta_m$ of $X^m$ in $X^m$. We identify $x$ with a point in $\Delta_m$ via the natural identification $\Delta_m$ with $X$.  Put $T:= \otimes_{j=1}^m T_j$ and $V':=\{x\}$. By \cite[Lemma 2.4]{Meo-auto-inter}, we have $\nu(T,x) \ge \nu(T_1,x) \cdots \nu(T_m,x).$   By \cite[Proposition 5.6]{Dinh_Sibony_density}, we have
$$\kappa_0(S, [V'])= \nu(S,x), \quad \kappa_0(T,[V'])= \nu(T,x)$$
(notice here $\dim V' =0$).  This combined by Theorem \ref{th-sosanhVV'densityDS} applied to $X^m$, $T:= \otimes_{j=1}^m T_j$, $\Delta_m$ the diagonal of $X^m$ and $V':=\{x\}$  implies
$$\nu(S,x)  \ge \nu(T,x) \ge \nu(T_1,x) \cdots \nu(T_m,x).$$
Hence, the desired inequality follows.   The proof is finished.
\endproof

In the next part, we will use density currents to study the loss of mass of non-pluripolar products.  We need to recall basic properties of relative non-pluripolar products.  Let $T_1, \ldots, T_m$ be closed positive $(1,1)$-currents on $X$. By \cite{Viet-generalized-nonpluri}, the $T$-relative non-pluripolar product $\langle \wedge_{j=1}^m T_j \dot{\wedge} T\rangle$ is defined  in a way similar to that of  the usual non-pluripolar product. For readers' convenience, we explain briefly how to do it.

Write $T_j= \ddc u_j+ \theta_j$, where $\theta_j$ is a smooth form and $u_j$ is a $\theta_j$-psh function. Put
$$R_k:=\bold{1}_{\cap_{j=1}^m \{u_j >-k\}} \wedge_{j=1}^m (\ddc \max\{u_j,-k\} + \theta_j)\wedge T$$
for $k \in \N$.  By the strong quasi-continuity of bounded psh functions (\cite[Theorems 2.4 and 2.9]{Viet-generalized-nonpluri}), we have
$$R_k= \bold{1}_{\cap_{j=1}^m \{u_j >-k\}} \wedge_{j=1}^m (\ddc \max\{u_j,-l\} + \theta_j)\wedge T$$
for every $l \ge k \ge 1$. Although it is not an immediate fact, one can check that  $R_k$ is positive (see \cite[Lemma 3.2]{Viet-generalized-nonpluri}).

As in \cite{BEGZ}, we have that $R_k$ is of mass bounded uniformly in $k$ and $(R_k)_k$ converges to a closed positive current as $k \to \infty$. This limit is denoted by $\langle \wedge_{j=1}^m T_j \dot{\wedge} T\rangle$.  The  last product is, hence,  a well-defined closed positive current of bi-degree $(m+p,m+p)$; and it is  symmetric with respect to $T_1, \ldots, T_m$ and homogeneous. We refer to  \cite[Proposition 3.5]{Viet-generalized-nonpluri} for more properties of relative non-pluripolar products.

For every closed positive $(1,1)$-current $P$, we denote by $I_P$ the set of $x \in U$ so that local potentials of $P$ are equal to $-\infty$ at $x$. Note that $I_P$ is  a locally complete pluripolar set.  It is clear from the definition that $\langle \wedge_{j=1}^m T_j \dot{\wedge} T\rangle$ has no mass on $\cup_{j=1}^m I_{T_j}$. Furthermore, for every locally complete pluripolar set $A$ (\emph{i.e,} $A$ is locally equal to $\{\psi= -\infty\}$ for some psh function $\psi$), if $T$ has no mass on $A$, then so does $\langle \wedge_{j=1}^m T_j \dot{\wedge} T\rangle$.  The following is deduced from \cite[Proposition 3.5]{Viet-generalized-nonpluri}.

\begin{proposition}\label{pro-sublinearnonpluripolar}
$(i)$ For $R:= \langle \wedge_{j=l+1}^m T_j \dot{\wedge} T \rangle$,  we have $\langle \wedge_{j=1}^m T_j \dot{\wedge} T \rangle = \langle \wedge_{j=1}^l T_j \dot{\wedge} R \rangle$.

$(ii)$ For every complete pluripolar set $A$, we have
$$\bold{1}_{X \backslash A}\langle  T_1 \wedge T_2 \wedge \cdots \wedge T_m \dot{\wedge} T\rangle= \big\langle  T_1 \wedge T_2 \wedge \cdots \wedge T_m \dot{\wedge}(\bold{1}_{X \backslash A} T)\big\rangle.$$
In particular,  the equality
$$\langle \wedge_{j=1}^m T_j \dot{\wedge} T \rangle = \langle \wedge_{j=1}^m T_j \dot{\wedge} T' \rangle$$
holds, where $T':= \bold{1}_{X \backslash \cup_{j=1}^m I_{T_j}} T$.
\end{proposition}

Here is a crucial property of relative non-pluripolar products.

\begin{theorem} \label{th-monoticity} (\cite[Theorem 1.1]{Viet-generalized-nonpluri})  Let $T'_j$ be closed positive $(1,1)$-current in the cohomology class of $T_j$ on $X$ such that $T'_j$ is less singular than $T_j$ for $1 \le j \le m$. Then we have
$$\{\langle T_1 \wedge \cdots \wedge T_m \dot{\wedge} T \rangle \} \le \{\langle T'_1 \wedge \cdots \wedge T'_m \dot{\wedge} T \rangle\}.$$
\end{theorem}

Weaker versions of the above result were proved in  \cite{BEGZ,Lu-Darvas-DiNezza-mono,WittNystrom-mono}.  Regarding the relation between relative non-pluripolar products and density currents, the following fact was proved in \cite[Theorem 3.5]{Viet-density-nonpluripolar}, see also \cite{VietTuanLucas,Viet_Lucas}.

\begin{theorem} \label{the-phanbucuarestricteddenstyvanonpluri}  Let $R_\infty$ be a density current associated to $T_1, \ldots, T_m,T$. Then we have
\begin{align} \label{ine_TjS2}
\pi^* \langle \wedge_{j=1}^m T_j  \dot{\wedge} T \rangle \le R_\infty,
\end{align}
where $\pi$ is the natural projection from the normal bundle of the diagonal $\Delta$ of $X^{m+1}$ to $\Delta$, and as usual we identified $\Delta$ with $X$.
\end{theorem}

We will need the following to estimate the density h-dimension of currents.

\begin{proposition} \label{pro-uocluongmassonsmallsetdensity} (\cite[Proposition 3.6]{Viet-density-nonpluripolar}) Let $A$ be a Borel subset of $X$. Assume that for every density current $R_J$ associated to $(T_j)_{j \in J},T$ with $m_J:=|J|<m$, we have that $R_J$ has  no mass on the set $$\pi_{m_J+1}^*(\cap_{j \not \in J} \{x \in A: \nu(T_j, x) >0\}).$$
 Then for every density current $S$ associated to $T_1, \ldots, T_m,T$, the h-dimension of the current $\bold{1}_A S$ is equal to $n-p-m$.
\end{proposition}

Let $P$ and $T$ be closed positive current of bi-degree $(1,1)$ and  $(p,p)$ respectively on $X$, where $1 \le p \le n-1$.

\begin{lemma} \label{le-truonghom=1lelong} Assume that $T$ has no mass on $I_{P}$.  Then, the cohomology class
$$\gamma:= \{P\} \wedge \{T\}- \{\langle P \dot{\wedge} T\rangle \}$$
 is pseudoeffective and  we have
\begin{align}\label{ine-uocluonggammaVPT}
\| \gamma\| \ge \sum_{V} \nu(P,V)\nu(T,V) \vol(V),
\end{align}
where the sum is taken over every irreducible subset $V$ of dimension at least $n-p-1$ in $X$.
\end{lemma}

Here $\nu(P,V)$ and $\nu(T,V)$ denote the generic Lelong numbers of $P$ and $T$ along $V$, respectively.

\proof  Let $\cali{V}$ be the set of irreducible analytic subsets $V$ of dimension at least $n-p-1$ in $X$ such that $\nu(T,V)>0$ and $\nu(P,V)>0$. We note that in (\ref{ine-uocluonggammaVPT}), it is enough to consider  $V \in \cali{V}$. We will see below that $\cali{V}$ has at most countable elements.

Observe that if $\nu(P,x)>0$, then $x \in I_P$. Hence, by hypothesis, the trace measure of $T$ has no mass on the set $\{x \in X: \nu(P,x)>0\}$. This allows us to apply Proposition \ref{pro-uocluongmassonsmallsetdensity} to $P$ and $T$ to obtain that the density h-dimension of $P$ and $T$ is minimal. Using this and Lemma \ref{le-classDSproduct} gives
\begin{align}\label{eq-densityclassPT}
\kappa(P,T)= \pi^* (\{P\} \wedge \{T\}),
\end{align}
where $\pi$ is the natural projection from the normal bundle of the diagonal $\Delta$ of $X^2$ to $\Delta$.

Let $S$ be a density current associated to $P$ and $T$. Since the h-dimension of $S$ is minimal, using Lemma \ref{le-minimalhdimension}, we get that there exists a current $S'$ on $X$ such that $S= \pi^* S'$ (recall $\Delta$ is identified with $X$). Since the relative non-pluripolar product is dominated by density currents (Theorem \ref{the-phanbucuarestricteddenstyvanonpluri}), the current $S' - \langle P \dot{\wedge} T \rangle$ is closed and positive. Moreover, by (\ref{eq-densityclassPT}), the cohomology class of the last current is equal to $\gamma$. It follows that $\gamma$ is pseudoeffective.

It remains to prove (\ref{ine-uocluonggammaVPT}).  Let $V \in \cali{V}$. By definition, the generic Lelong number of $T$ along $V$ is positive. Since $T$ is of bi-degree $(p,p)$, the dimension of $V$ must be at most $n-p$. Hence, we have two possibilities: either $\dim V= n-p-1$ or $\dim V= n-p$. The latter case cannot happen because $T$ would have mass on $V$ which is contained in $I_P$ (for $\nu(P,V)>0$), this contradicts the hypothesis that $T$ has no mass on $I_P$.  Hence, for every $V \in \cali{V}$, we have $\dim V= n-p-1$. We also deduce that for $V,V' \in \cali{V}$, then $V \not \subset V'$.

Applying Corollary \ref{cor-sosanhlelong} to $P,T$ and generic $x \in V$ gives
$$\nu(S,V) \ge \nu(P,V) \nu(T,V).$$
This combined with the fact that $\dim V= n-p-1$ implies  $S \ge \nu(P,V) \nu(T,V) \, [V]$.
We deduce that
$$S \ge \sum_{V \in\cali{V}}\nu(P,V) \nu(T,V) \, [V].$$
The desired assertion follows. The proof is finished.
\endproof

\section{Reversed Alexandrov-Fenchel inequalities} \label{sec-AF}

We first recall an integration by parts formula for relative non-pluripolar products from \cite{Viet-convexity-weightedclass} generalizing those given in \cite{BEGZ,Lu-cap-compare,Xia}.

Let $X$ be a compact K\"ahler manifold. Recall that a \emph{dsh} function on $X$ is the difference of two quasi-plurisubharmonic (quasi-psh for short) functions on $X$ (see \cite{DS_tm}). These functions are well-defined outside pluripolar sets. Let $v$ be a dsh function on $X$. The last function is said to be \emph{bounded} in $X$ if there exists a constant $C$ such that $|v| \le C$ on $X$ (outside certain pluripolar set).

Let $T$ be a closed positive current on $X$. We say that $v$ is \emph{$T$-admissible} if  there exist  quasi-psh functions $\varphi_1, \varphi_2$ on $X$ such that $v= \varphi_1- \varphi_2$  and $T$ has no mass on $\{\varphi_j=-\infty\}$
for $j=1,2$. In particular, if $T$ has no mass on pluripolar sets, then every dsh function is $T$-admissible.

Assume now that $v$ is  \emph{bounded} $T$-admissible.    Let $\varphi_{1}, \varphi_{2}$ be quasi-psh functions such that $v= \varphi_{1}- \varphi_{2}$ and $T$ has no mass on $\{\varphi_{j}=-\infty\}$ for $j=1,2$. Let
$$\varphi_{j,k}:= \max\{\varphi_{j}, -k \}$$
for every $j=1,2$ and $k \in \N$. Put $v_k:= \varphi_{1,k}- \varphi_{2,k}$ and
$$Q_k:= d v_k \wedge \dc v_k \wedge T=\ddc v_k^2  \wedge T - v_k\ddc v_k \wedge T.$$
By the plurifine locality with respect to $T$ (\cite[Theorem 2.9]{Viet-generalized-nonpluri}) applied to the right-hand side of the last equality, we have
\begin{align}\label{eq-localplurifineddc}
\bold{1}_{\cap_{j=1}^2 \{\varphi_{j}> -k\}} Q_k =\bold{1}_{\cap_{j=1}^2\{\varphi_{j}> -k\}} Q_{k'}
\end{align}
for every $k'\ge k$. By \cite[Lemma 2.5]{Viet-convexity-weightedclass},  the mass of $Q_k$ on $X$ is bounded uniformly in $k$.  This combined with  (\ref{eq-localplurifineddc}) implies that there exists a positive current $Q$ on $X$ such that for every bounded Borel form $\Phi$ with compact support on $X$ such that
$$\langle Q,\Phi \rangle  = \lim_{k\to \infty} \langle Q_k, \Phi\rangle.$$
We define $\langle d v \wedge \dc v \dot{\wedge} T \rangle$ to be the current $Q$.  This agrees with the classical definition if $v$ is the difference of two  bounded quasi-psh functions. One can check  that this definition is independent of the choice of $\varphi_1, \varphi_2$.

Let $w$ be another bounded $T$-admissible dsh function.  If $T$ is of bi-degree $(n-1,n-1)$, we can also define the current $\langle  dv \wedge \dc w \dot{\wedge} T\rangle$ by a similar procedure as above.  We put
$$\langle \ddc v \dot{\wedge} T \rangle:= \langle \ddc \varphi_1 \dot{\wedge} T\rangle- \langle \ddc \varphi_2 \dot{\wedge} T\rangle$$
which is independent of the choice of $\varphi_1,\varphi_2$. By $T$-admissibility, we have
$$\langle  \ddc (v+w)  \dot{\wedge} T\rangle=\langle  \ddc v\dot{\wedge} T\rangle+\langle  \ddc w   \dot{\wedge} T\rangle.$$
Here is an integration by parts formula for relative non-pluripolar products.

\begin{theorem} \label{th-integra} (\cite[Theorem 2.6]{Viet-convexity-weightedclass})  Let  $T$ a closed positive current of bi-degree $(n-1,n-1)$ on $X$. Let $v$ and $w$ be bounded $T$-admissible dsh functions on $X$. Then,  we have
\begin{align}\label{eq-intebyparts}
\int_X w \langle  \ddc v \dot{\wedge} T\rangle=\int_X v \langle \ddc w\dot{\wedge} T\rangle=- \int_X \langle dw \wedge \dc v \dot{\wedge} T \rangle.
\end{align}
\end{theorem}

Let $\theta$ be  a closed smooth $(1,1)$-form on $X$. Given a $\theta$-psh function $u$, we use the usual notation that
$$\theta_u:= \ddc u+ \theta.$$
Let $T$ be a closed positive current of bi-degree $(n-2,n-2)$ on $X$. Let $u, v, \varphi$ be $\theta$-psh functions such that $u\le \varphi$ and $v \le \varphi$.

The following can be regarded as an inequality of reversed Alexandrov-Fenchel type. Several related estimates for mixed Monge-Amp\`ere operators were obtained in the local setting; see \cite{Demailly-Hiep,Kim-Rashkovskii}.

\begin{proposition}\label{pro-theoremmain} Assume that
\begin{align}\label{eq-hypo-fullmasthetau}
\int_X \langle \theta_\varphi^2\dot{\wedge} T\rangle = \int_X \langle \theta_u \wedge \theta_\varphi \dot{\wedge} T\rangle,
\end{align}
and $T$ has no mass on $\{u= -\infty\}\cup\{v= - \infty\}$. Then, we have
\begin{align}\label{ine-reversedAF}
\int_X \big(\langle \theta_\varphi \wedge \theta_v \dot{\wedge} T\rangle - \langle \theta_u \wedge \theta_v \dot{\wedge} T\rangle\big)  \le \bigg( \int_X \big(\langle \theta_\varphi^2\dot{\wedge} T\rangle -  \langle \theta_u^2 \dot{\wedge} T\rangle\big) \bigg)^{1/2} \bigg(  \int_X \big(\langle \theta_\varphi^2 \dot{\wedge} T\rangle -  \langle \theta_v^2\dot{\wedge} T\rangle\big)  \bigg)^{1/2}
\end{align}
\end{proposition}

Notice that by the monotonicity of relative non-pluripolar products, the left-hand side of (\ref{ine-reversedAF}) is non-negative.

\proof    By hypothesis, one has
\begin{align}\label{eq-massofTonuinftry}
\bold{1}_{\{u= -\infty\} \cup \{v= -\infty\}}T=0.
\end{align}
Note here that
$$\{\varphi = -\infty\} \subset \{u= -\infty\}\cap \{v= -\infty\}.$$
 Let
$$u_{k}:= \max\{u, \varphi-k\}- (\varphi -k)$$
and
 $$\psi_k:= k^{-1}\max\{u+ v-  2\varphi, - k \}+1=k^{-1}\max\{u+ v,  2\varphi - k \}- k^{-1} 2 \varphi +1$$
Define $v_k$ similarly. We have $\psi_k= 0$ on $ \{u \le \varphi -k\} \cup \{v \le \varphi -k\}$.  Note also that $0 \le u_{k}, v_k \le k$ for $j=1,2$ (hence $u_k,v_k,\psi_k$ are bounded $T$-admissible dsh functions by (\ref{eq-massofTonuinftry})) and
$$\ddc \psi_k + k^{-1}\eta\ge 0,$$
where $\eta:=  2 \theta_{\varphi}$.   We can check that
\begin{align*}
\langle \theta_{u} \wedge \theta_{v}  \dot{\wedge} T \rangle& =\lim_{k \to \infty} \psi_k \big\langle (\ddc u_{k}+\theta_{\varphi})\wedge (\ddc v_{k}+\theta_{\varphi}) \dot{\wedge} T \big\rangle.
\end{align*}
Put
$$B_k:= \int_X \psi_k \big\langle \theta_{\varphi}\wedge (\ddc v_{k}+\theta_{\varphi}) \dot{\wedge} T \big\rangle - \int_X \psi_k \big\langle (\ddc u_{k}+\theta_{\varphi})\wedge (\ddc v_{k}+\theta_{\varphi}) \dot{\wedge} T \big\rangle$$
and
$$A:= \int_X \big(\langle \theta_\varphi \wedge \theta_v \dot{\wedge} T\rangle - \langle \theta_u \wedge \theta_v \dot{\wedge} T\rangle\big)$$
By (\ref{eq-massofTonuinftry}) and Proposition \ref{pro-sublinearnonpluripolar} $(ii)$, we have
$$A= \lim_{k \to \infty} B_k.$$
Observe
\begin{align*}
B_k &=  - \int_X \psi_k \langle \ddc u_k  \wedge (\ddc v_{k}+\theta_{\varphi}) \dot{\wedge} T \big\rangle\\
 &=   - \int_X \psi_k \langle \ddc u_k  \wedge \ddc v_{k} \dot{\wedge} T \big\rangle   - \int_X \psi_k \langle \ddc u_k  \wedge \theta_{\varphi} \dot{\wedge} T \big\rangle.
\end{align*}
Denote by $I_1, I_2$ the first and  second term in the right-hand side of the last equality. Using (\ref{eq-massofTonuinftry}) gives
\begin{align*}
\lim_{k \to \infty} I_2 &= \lim_{k \to \infty} \bigg(- \int_X \psi_k \langle (\ddc u_k+ \theta_\varphi)  \wedge \theta_{\varphi} \dot{\wedge} T \big\rangle+ \int_X \psi_k \langle \theta_{\varphi}^2 \dot{\wedge} T \big\rangle\bigg)\\
&=  -  \int_X\langle   \theta_{u}\wedge \theta_{\varphi} \dot{\wedge} T \big\rangle+\int_X \langle \theta_{\varphi}^2 \dot{\wedge} T \big\rangle=0
 \end{align*}
 by (\ref{eq-hypo-fullmasthetau}).
 Thus we get
 \begin{align}\label{ine-buocdautienABD}
B_k  =I_1+ o_{k \to \infty}(1).
\end{align}
 Theorem \ref{th-integra} applied to the formula defining $I_1$ gives
\begin{align}\label{eq-biendoiI1}
- I_1  &= \int_X  u_{k} \big\langle   \ddc \psi_k \wedge \ddc v_{k} \dot{\wedge} T \big\rangle\\
\nonumber
&=  -\int_X \big \langle d u_{k} \wedge \dc v_{k} \wedge   \ddc \psi_k  \dot{\wedge} T \big\rangle\\
\nonumber
&= -\int_X  \big \langle d u_k \wedge \dc v_k \wedge   (\ddc \psi_k+k^{-1}\eta)  \dot{\wedge} T \big\rangle + k^{-1}\int_X \big \langle d u_k \wedge \dc v_k \wedge   \eta  \dot{\wedge} T \big\rangle.
\end{align}
Denote by $J_1,J_2$ the first and second term in the right-hand side of the last equality. We treat $J_2$. By integration by parts (Theorem \ref{th-integra}), we obtain
\begin{align} \label{eq-biendoiJ2}
J_2 &= - k^{-1} \int_X   u_k \langle \ddc v_k \wedge   \eta  \dot{\wedge} T \rangle
&=  - k^{-1} \int_X   u_k \langle (\ddc v_k+\theta_{\varphi}) \wedge   \eta  \dot{\wedge} T \rangle+ o_{k \to \infty}(1)
\end{align}
because of (\ref{eq-massofTonuinftry}) and the fact that $u_k/k$ converges to $0$ on $\{u>- \infty\}$ and to  $-1$ otherwise.  Using this and noticing that $u_k\le 0$,  we infer
\begin{align} \label{ine-uocluongJ2}
\liminf_{k\to \infty} J_2 \ge  0.
\end{align}
On the other hand, using the Cauchy-Schwarz inequality, we obtain
\begin{align}\label{ine-uocliongJ1}
J_1^2  \le  \int_X \big \langle d u_k \wedge \dc u_k \wedge   (\ddc \psi_k+k^{-1}\eta)  \dot{\wedge} T \big\rangle   \int_X\big \langle d v_k \wedge \dc v_k \wedge   (\ddc \psi_k+k^{-1}\eta)  \dot{\wedge} T \big\rangle.
\end{align}
Denote by $J_{11}, J_{12}$ the first and second term in the right-hand side of the last inequality. Put
$$J'_{11}:=k^{-1} \int_X\big \langle d u_k \wedge \dc u_k \wedge  \eta  \dot{\wedge} T \big\rangle.$$
Using integration by parts and arguing as in (\ref{eq-biendoiJ2}), we obtain
\begin{align} \label{ine-uocluongJphay11sua}
J'_{11} &=-k^{-1} \int_X  u_k \big \langle \ddc u_k \wedge  \eta  \dot{\wedge} T \big\rangle\\
\nonumber
&= - k^{-1} \int_X  u_k \big \langle (\ddc u_k+ \theta_\varphi) \wedge  \eta  \dot{\wedge} T \big\rangle+ o_{k \to\infty}(1)\\
\nonumber
&= \int_X \big \langle (\ddc u_k+ \theta_\varphi) \wedge  \eta  \dot{\wedge} T \big\rangle - \int_X (u_k/k +1)  \big \langle (\ddc u_k+ \theta_\varphi) \wedge  \eta  \dot{\wedge} T \big\rangle+ o_{k \to\infty}(1) \\
\nonumber
&= \int_X \big \langle \theta_\varphi \wedge  \eta  \dot{\wedge} T \big\rangle - \int_X (u_k/k +1)  \big \langle \theta_u \wedge  \eta  \dot{\wedge} T \big\rangle+ o_{k \to\infty}(1)
\end{align}
because $u_k/k+1=0$ on $\{u \le \varphi -k\}$. Letting  $k \to \infty$ in (\ref{ine-uocluongJphay11sua}) gives
\begin{align}\label{ine-uocluongJphay11}
\lim_{k\to \infty} J'_{11}= \int_X   \big \langle \theta_{\varphi} \wedge   \eta  \dot{\wedge} T \big\rangle-\int_X   \big \langle \theta_{u} \wedge   \eta  \dot{\wedge} T \big\rangle=2 \int_X   \big \langle \theta_{\varphi}^2  \dot{\wedge} T \big\rangle- 2\int_X   \big \langle \theta_{u} \wedge   \theta_\varphi  \dot{\wedge} T \big\rangle.
\end{align}
Using Theorem \ref{th-integra} again and arguing as in (\ref{ine-uocluongJphay11sua}), we have
\begin{align*}
J_{11} &=\int_X \big \langle d u_k \wedge \dc u_k \wedge  \ddc \psi_k \dot{\wedge} T \big\rangle+ J'_{11}\\
&= -\int_X \psi_k \big \langle \ddc u_k \wedge \ddc u_k  \dot{\wedge} T \big\rangle+ J'_{11}\\
&= - \int_X \psi_k \big \langle  (\ddc u_k+\theta_\varphi)^2  \dot{\wedge} T \big\rangle + 2 \int_X \psi_k \big \langle \ddc u_k \wedge \theta_{\varphi} \dot{\wedge} T \big\rangle + \int_X \psi_k \big \langle \theta_{\varphi}^2 \dot{\wedge} T \big\rangle   + J'_{11}\\
&= \int_X \big\langle \theta_\varphi^2  \dot{\wedge} T \big\rangle -\int_X \big\langle \theta_u^2  \dot{\wedge} T \big\rangle+ 2 \int_X \big\langle \theta_u \wedge \theta_\varphi  \dot{\wedge} T \big\rangle- 2\int_X \big\langle \theta_\varphi^2  \dot{\wedge} T \big\rangle + o_{k \to\infty}(1)+ J'_{11}
\end{align*}
This combined with (\ref{ine-uocluongJphay11}) yields that
$$\limsup_{k \to \infty} J_{11} \le  \int_X \big \langle \theta^2_{\varphi} \dot{\wedge} T \big\rangle- \int_X \big \langle \theta_{u}^2 \dot{\wedge} T \big\rangle.$$
By similar computations, we also get
$$\limsup_{k \to \infty} J_{12} \le  \int_X \big \langle \theta^2_{\varphi} \dot{\wedge} T \big\rangle- \int_X \big \langle \theta_{v}^2 \dot{\wedge} T \big\rangle.$$
 This together with (\ref{eq-biendoiI1}) and (\ref{ine-buocdautienABD}) gives
$$A=\limsup_{k \to \infty} B_k=\limsup_{k \to \infty} I_1= -  \liminf_{k \to \infty} -I_1 \le C,$$
 where $C$ is the right-hand side of (\ref{ine-reversedAF}). This finishes the proof.
\endproof

\begin{corollary}\label{cor-theoremmain} Let $T$ be closed positive current of bi-dimension $(m,m)$ on $X$ and let  $\theta$ be  a closed smooth $(1,1)$-form on $X$. Let $u, v, \varphi$ be $\theta$-psh functions such that $u,v \le \varphi$, and  $T$ has no mass on $\{u= -\infty\}$ and $\{v= - \infty\}$. Let $M$ be a positive constant greater than $\int_X \theta_\varphi^m \dot{\wedge} T$.  Then, there exists a constant $C_M$ depending only on $M,m,n$ such that
\begin{align}\label{ine-thetauvtachldinhluong}
\bigg| \int_X \langle \theta_\varphi^{m-l} \wedge \theta_v^l  \dot{\wedge} T\rangle - \int_X \langle \theta_u^{m-l} \wedge \theta_v^l \dot{\wedge} T\rangle \bigg | \le C_M \bigg(\int_X \langle \theta_\varphi^m\dot{\wedge} T\rangle - \int_X \langle \theta_u^m \dot{\wedge} T\rangle\bigg)^{2^{-l}}.
\end{align}
In particular, if
$$\int_X \langle \theta_\varphi^m\dot{\wedge} T\rangle = \int_X \langle \theta_u^m \dot{\wedge} T\rangle,$$
then, for every integer $0 \le l \le m$, we have
$$\int_X \langle \theta_\varphi^{m-l} \wedge \theta_v^l  \dot{\wedge} T\rangle=  \int_X \langle \theta_u^{m-l} \wedge \theta_v^l \dot{\wedge} T\rangle.$$
\end{corollary}

\proof In what follows, we denote by $C_M$ a positive constant depending only on $M$ and $n$, and the value of $C_M$ might vary from line to line. Put
$$I:= \int_X \langle \theta_\varphi^m\dot{\wedge} T\rangle - \int_X \langle \theta_u^m \dot{\wedge} T\rangle.$$
 By the monotonicity of relative non-pluripolar products, we get
\begin{align}\label{eq-hypo-fullmasthetaucor}
0 \le \int_X \langle \theta_\varphi^m\dot{\wedge} T\rangle - \int_X \langle \theta_\varphi^{m-l} \wedge \theta_u^{l} \dot{\wedge} T\rangle \le I
\end{align}
for every $0 \le l \le m$. Observe that  the desired assertion in the case where $m=2$ is a direct consequence of Proposition \ref{pro-theoremmain}.  We prove  by induction on $l'$ that for every $0\le l_1, l_2\le m$ with $l_1+l_2 \le l'$ we have
\begin{align}\label{eq-induclwedgemthetavarphi}
\int_X \langle \theta_\varphi^{m-l_2} \wedge \theta_v^{l_2} \dot{\wedge} T\rangle- \int_X \langle \theta_\varphi^{m-l_1-l_2} \wedge \theta_u^{l_1} \wedge \theta_v^{l_2}  \dot{\wedge} T\rangle \le C_M \, I^{2^{-l_2}},
\end{align}
for some constant $C_M$ depending only on $M,m$ and $n$.

The desired inequality is a special case of (\ref{eq-induclwedgemthetavarphi}) when $l_1= m-l_2$.  When $l'=0$, the inequality (\ref{eq-induclwedgemthetavarphi}) is clear. Suppose that  (\ref{eq-induclwedgemthetavarphi}) holds for every $l_1,l_2$ with  $l_1+ l_2 \le l'-1$. We need to prove it for $l'$ in place of $l'-1$.  To this end, we now use another induction on $0 \le l_2 \le l'$ to prove the statement $(*)$ that (\ref{eq-induclwedgemthetavarphi}) holds for every $l_1$ with $l_1+ l_2 \le l'$. When $l_2=0$, the statement $(*)$ is a direct consequence of (\ref{eq-hypo-fullmasthetaucor}). Assume now that $(*)$ holds for $l_2-1$. We now prove it for $l_2$.    Let
$$T':=\langle \theta_\varphi^{m-l_1-l_2} \wedge \theta_u^{l_1-1} \wedge \theta_v^{l_2-1} \dot{\wedge} T\rangle.$$
 We have
 $$  \langle \theta_\varphi^{m-l_1-l_2} \wedge \theta_u^{l_1} \wedge \theta_v^{l_2} \dot{\wedge} T\rangle= \langle \theta_u \wedge \theta_v \dot{\wedge} T'\rangle,$$
 and
\begin{align*}
\int_X \langle \theta_u^2 \dot{\wedge} T'\rangle =\int_X \langle \theta_\varphi^{m-l_1-l_2} \wedge \theta_u^{l_1+1} \wedge \theta_v^{l_2-1} \dot{\wedge} T\rangle
\end{align*}
and
\begin{align*}
\int_X \langle \theta_\varphi^2 \dot{\wedge} T'\rangle= \int_X \langle \theta_\varphi^{m-l_1-l_2+2}  \wedge \theta_u^{l_1-1} \wedge \theta_v^{l_2-1} \dot{\wedge} T\rangle.
\end{align*}
By this and the induction hypothesis on $l_2$ that $(*)$ holds for $l_2-1$ and the monotonicity of relative non-pluripolar products, we obtain that
\begin{align*}
\int_X \langle \theta_\varphi^2 \dot{\wedge} T'\rangle- \int_X \langle \theta_u^2 \dot{\wedge} T'\rangle &\le \int_X \langle \theta_\varphi^{m-l_2+1} \wedge \theta_v^{l_2-1} \dot{\wedge} T\rangle-   \int_X \langle \theta_\varphi^{m-l_1-l_2} \wedge \theta_u^{l_1+1} \wedge \theta_v^{l_2-1} \dot{\wedge} T\rangle\\
&\le C_M I^{2^{-l_2+1}}.
\end{align*}
Note that $T'$ has no mass on $\{u= -\infty\}$ and $\{v= - \infty\}$ because $T$ does so; see \cite[Lemma 2.1]{Viet-generalized-nonpluri}.  Applying Proposition \ref{cor-theoremmain} to $u,v,\varphi, T'$ gives
$$  \int_X\langle \theta_\varphi \wedge \theta_v \dot{\wedge} T'\rangle-  \int_X\langle \theta_u \wedge \theta_v \dot{\wedge} T'\rangle \le C_M \bigg( \int_X \langle \theta_\varphi^2 \dot{\wedge} T'\rangle- \int_X \langle \theta_u^2 \dot{\wedge} T'\rangle\bigg)^{1/2} \le C_M \, I^{2^{-l_2}}.$$
We deduce that
$$ \int_X \langle \theta_\varphi^{m-l_1-l_2+1} \wedge \theta_u^{l_1-1} \wedge \theta_v^{l_2} \dot{\wedge} T\rangle-   \int_X\langle \theta_\varphi^{m-l_1-l_2} \wedge \theta_u^{l_1} \wedge \theta_v^{l_2} \dot{\wedge} T\rangle \le C_M \, I^{2^{-l_2}}.$$
On the other hand, using the induction hypothesis on $l'-1$ gives
$$  \int_X\langle \theta_\varphi^{m-l_2} \wedge \theta_v^{l_2} \dot{\wedge} T\rangle - \int_X \langle \theta_\varphi^{m-l_1-l_2+1} \wedge \theta_u^{l_1-1} \wedge \theta_v^{l_2} \dot{\wedge} T\rangle \le C_M \, I^{2^{-l_2}}.$$
Summing up the last two inequalities gives the desired assertion $(*)$ for $l_2$. In other words, (\ref{eq-induclwedgemthetavarphi}) holds for every $l_1,l_2$ with $l_1+ l_2 \le l'$. This is what we want to prove. The proof is finished.
\endproof

We say that a  closed positive $(1,1)$-current $P$ is \emph{a current with analytic singularities associated to a coherent analytic ideal sheaf $\cali{S}$ on $X$} if locally on $X$, every local potential $u$ of $P$ satisfies
$$u=c \log \sum_{j=1}^M |f_j|+O(1),$$
where $c> 0$ is a constant, and  $\{f_1,\ldots, f_M\}$ are local generators of $\cali{S}$; see \cite[Definition 9.1]{Boucksom_l2}. By Demailly's analytic approximation of psh functions, every big cohomology class contains such a current $P$ which is K\"ahler.

We fix a smooth K\"ahler form $\omega$ on $X$ with $\int_X \omega^n =1$. For every positive $(p,p)$-current $S$, the mass of $S$ is defined by
$$\|S\|:= \int_X S \wedge \omega^{n-p}.$$
Similarly, we can define the mass norm of a pseudoeffective cohomology classes.

\begin{theorem} \label{the-congthemomega}  Let $1 \le m \le n$ be an integer. Let $\alpha$ be a big cohomology $(1,1)$-class in $X$. Let $T$ and $T'$ be closed positive currents in $\alpha$ such that $T'$ is more singular than $T$. Assume that there exists a closed positive current $P$ with analytic singularities associated to a coherent ideal sheaf such that $P$ is more singular than $T$.  Let $\theta$ be a closed continuous real $(1,1)$-form. Let $M>0$ be constant greater than $\|\alpha\|$, $\|\langle \alpha^m \rangle\|$ and $\|\theta\|_{\cali{C}^0}$ such that $P \ge M^{-1} \omega$.   Then, there exists a constant $C_M>0$ depending only on $M,m,\omega,X$ such that
\begin{align}\label{eq-TTcongvoiomega}
 \big \| \{\langle (T'+\theta)^m \rangle\} -\{\langle (T+\theta)^m \rangle\} \big\| \le  C_M\big\|  \{\langle T'^m \rangle\}- \{\langle T^m \rangle\}\big\|^{2^{-m}}.
\end{align}
In particular, if $T+\theta$ and $T'+ \theta$ are positive, then there holds
\begin{align}\label{state-tuonduongTcongtheta}
\{\langle T'^m \rangle\}= \{\langle T^m \rangle\}  \quad \text{if and only if}\quad  \{\langle (T'+\theta)^m \rangle\}= \{\langle (T+\theta)^m \rangle\}.
\end{align}
\end{theorem}

Examples of $T$ in the above result are currents with minimal singularities in $\alpha$. The assertion (\ref{state-tuonduongTcongtheta})  was known when $\alpha$ is K\"ahler (see \cite{DiNezza-stability,Viet-generalized-nonpluri}) and seems to be new for big classes even when $m=n$.

\begin{proof} As above we denote by $C_M$ a positive constant depending only on $M,m,\omega,X$ whose value can vary from line to line.  We can choose such a  $C_M$ big enough such that $\omega':=\theta+ C_M \omega \ge 0$.  By the multi-linearity of non-pluripolar products, it is sufficient to prove (\ref{eq-TTcongvoiomega}) for $\omega'$ in place of $\theta$. In order to do so, we only need to check that
\begin{align}\label{ine-lmhieuTTphay}
 \big \| \{\langle T'^l \rangle\}- \{\langle T^l \rangle \}\big \| \le C_M \big \| \{\langle T'^m \rangle\}- \{\langle T^m \rangle \}\big \|^{2^{-m}}
\end{align}
 for every $1 \le l \le m$.

Principalizing the ideal sheaf associated to $P$ (see \cite{Hironaka-desingular,Wlodarczyk}), we get a smooth modification $\sigma: \widehat X \to X$ such that $\widehat P:= \sigma^*P= P_1+ P_2$, where $P_1$ is  a linear combination with nonnegative coefficients of currents of integration along smooth hypersurfaces, and $P_2$ is a closed positive smooth form.
Since $P \ge M^{-1} \omega$, we get
\begin{align}\label{ine-P2omefa}
P_2 \ge M^{-1} \sigma^* \omega
\end{align}
Let $\widehat T:= \sigma^* T$ and $\widehat T':= \widehat T'$. Since non-pluripolar products have no mass on analytic sets, using the hypothesis, we have
$$\int_{\widehat X}\langle \widehat T^m \rangle \wedge \sigma^* \omega^{n-m}=\int_{X}\langle T^m \rangle \wedge \omega^{n-m}, \quad \int_{X}\langle T'^m \rangle \wedge \omega^{n-m}=\int_{\widehat X}\langle \widehat T'^m \rangle \wedge \sigma^* \omega^{n-m}.$$
Using this and applying Corollary \ref{cor-theoremmain} to potentials of $\widehat T, \widehat T', \widehat P$ give
$$\int_{\widehat X} \big(\langle \widehat T^l \wedge \widehat P^{m-l} \rangle- \langle \widehat T'^l \wedge \widehat P^{m-l} \rangle\big) \wedge \sigma^* \omega^{n-m} \le C_M \| \langle T^m \rangle - \langle T'^m \rangle\|^{2^{-m+l}}$$
for every $1 \le l \le m$.

Let $R$ be a closed positive current representing the cohomology class $\{\langle \widehat T^l \rangle\}-\{\langle \widehat T'^l \rangle\}$. By the above inequality  and the smoothness of $P_2$, we obtain
$$\int_{\widehat X} R \wedge P_2^{m-l} \wedge \sigma^* \omega^{n-m} \le C_M \| \langle T^m \rangle - \langle T'^m \rangle\|^{2^{-m+l}}.$$
This combined with (\ref{ine-P2omefa}) gives
$$\int_{\widehat X} R \wedge \sigma^* \omega^{n-l} \le C_M \| \langle T^m \rangle - \langle T'^m \rangle\|^{2^{-m+l}}.$$
It follows that
\begin{align*}
\int_X \big(\langle T^l \rangle - \langle T'^l \rangle \big) \wedge \omega^{n-l} &=\int_{\widehat X} \big(\langle \widehat T^l \rangle- \langle \widehat T'^l \rangle) \wedge \sigma^* \omega^{n-l}\\
&=\int_{\widehat X} R \wedge \sigma^* \omega^{n-l} \le C_M \| \langle T^m \rangle - \langle T'^m \rangle\|^{2^{-m+l}}.
\end{align*}
The desired assertion follows because $\|\langle T^m \rangle\| \le \|\langle \alpha^m \rangle\|$.

We now prove (\ref{state-tuonduongTcongtheta}). The implication $\Rightarrow$ is clear. The other one is obtained from the first one by applying to $T+\theta, T'+\theta$ in place of $T, T'$ respectively, and $-\theta$ in place of $\theta$.  The proof is complete.
\end{proof}

\begin{corollary} \label{cor-Tcongvoitheta} Let $1 \le m \le n$ be an integer. Let $\alpha$ be a big cohomology $(1,1)$-class in $X$. Let $T$ be  a closed positive current in $\alpha$ such that $\{\langle T^m \rangle\}= \langle \alpha^m \rangle$. Then we have
$$\{\langle T^l \rangle\}= \langle \alpha^l \rangle$$
for every $1\le l \le m$. In particular, if $\alpha$ is big nef and $\theta$ is a closed continuous real $(1,1)$-form representing a cohomology class $\gamma$, then
\begin{align}\label{eq-congthetabignef}
\{\langle (T+ \theta)^m\rangle \} = (\alpha +\gamma)^m.
\end{align}
\end{corollary}

The equality (\ref{eq-congthetabignef}) was proved in  \cite[Theorem 1.1]{Lu-Darvas-DiNezza-singularitytype} in the case where $m=n$. Note that in the last case, if additionally $T+\theta \ge 0$, then using \cite[Corollary 6.4]{Demailly_regula_11current} and the fact that the Lelong numbers of $T$ vanish, we get that $\alpha+\gamma$ is nef.

\begin{proof}
Let $T_{\min}$ be a current with minimal singularity in $\alpha$. By Theorem \ref{the-congthemomega} for $T$ and $T_{\min}$ and the hypothesis, we get
$$\{\langle T+\omega)^m\rangle\}= \{\langle (T_{\min}+ \omega)^m\rangle\}.$$
Expanding the sums in both sides and using the fact that $\{\langle T^l\rangle\}\le \langle T_{\min}^l \rangle$ for every $1\le l \le m$, we get the first  desired equality.
The second desired equality follows immediately from the first one. This finishes the proof.
\end{proof}

\begin{corollary} \label{cor-Tcongvoitheta2} Let $\alpha_1$ and $\alpha_2$ be big nef cohomology classes  on $X$. Let $T_j$ be a closed positive current in $\alpha_j$ for $j=1,2$. Then,
\begin{align}\label{eq-congu1u2theta}
\big\{\big\langle (T_1+T_2)^m\big\rangle \big\} = \langle (\alpha_1+\alpha_2)^m \rangle
\end{align}
 if and only if
\begin{align}\label{eq-congu1u2thetatach}
\{\langle T_j^m\rangle \} = \langle \alpha_j^m \rangle
\end{align}
 for $j=1,2$.
\end{corollary}

The las result was obtained in  \cite[Theorem 1.3 and Corollary 4.2]{Lu-Darvas-DiNezza-mono} when $m=n$.

\begin{proof}[Proof of Corollary  \ref{cor-Tcongvoitheta2}]
At this point, the proof is similar to that of \cite[Corollary 4.2]{Lu-Darvas-DiNezza-mono}. The implication from (\ref{eq-congu1u2theta}) to (\ref{eq-congu1u2thetatach}) is clear thanks to the monotonicity. Assume now (\ref{eq-congu1u2thetatach}).

Let $\omega'$ be a big enough K\"ahler form such that $\{\omega'\}+ \alpha_j$ is K\"ahler for $j=1,2$. Using this, the fact that $\alpha_j$ is big nef, and Theorem  \ref{the-congthemomega}, we deduce that
$$\{\langle (T_j+\omega')^m\rangle \} = (\alpha_j + \{\omega'\})^m $$
for $j=1,2$. Now the convexity of the class of currents of full mass intersection in K\"ahler cohomology classes (see, for example, \cite[Theorem 1.3]{Viet-generalized-nonpluri} or \cite{DiNezza-stability}) gives that
$$\big\{ \big\langle (T_1+T_2+ 2 \omega')^m \big\rangle \big\} = (\alpha_1 +\alpha_2+ 2\{\omega'\})^m.$$
By this and Theorem  \ref{the-congthemomega} again, we get the desired equality. The proof is finished.
\end{proof}

\section{Proof of Theorem \ref{the-self-intersec}} \label{sec-the1}

In this section, we will give a proof of Theorem \ref{the-self-intersec}.
The following result implies Theorem \ref{the-self-intersec} in the case where $\alpha$ is K\"ahler.

\begin{theorem}\label{the-kahlerself-intersec} Let $\cali{K}_0$ be a compact subset in the K\"ahler cone of $(1,1)$-classes on $X$. Let $\alpha\in \cali{K}_0$ and  let $T$ be a closed positive current in $\alpha$. Let $1\le m \le n$ be an integer.  Let $\cali{V}$ be  the set of  maximal irreducible analytic subsets $V$ of dimension at least $n-m$ of $X$ such that the generic Lelong number $\nu(T,V)$ of $T$ along $V$ is strictly positive.
 Then, we have
\begin{align}\label{ine-obstructionselfinterkahler}
\big\| \alpha^m - \{\langle T^m \rangle \}  \big\| \ge C\sum_{V \in \cali{V}}\big(\nu(T,V)\big)^{n- \dim V} \, \vol(V),
\end{align}
for some constant $C>0$ independent of $\alpha$ (but depending on $\cali{K}_0$).
\end{theorem}

By the proof below, one can see that if the sum in the right-hand of  (\ref{ine-obstructionselfinterkahler}) is taken only on $V \in \cali{V}$ such that the dimension of $V$ is equal to $n-m$, then the factor $C$ can be replaced by  $1$.

\proof By Demailly's analytic approximation of psh functions (see \cite{Demailly_analyticmethod}), it is enough to check (\ref{ine-obstructionselfinterkahler}) when $T$ has analytic singularities (note that the maximality of $V$ is preserved here). Hence, the polar locus $I_T$ of $T$ is an analytic subset on $X$.

Let $0 \le l \le n-1$ be an integer. Let $\cali{V}_l$ be the subset of $\cali{V}$ consisting of $V$ of dimension $l$.
Let $V \in \cali{V}_l$. By the definition of $\cali{V}$, we have $l \ge n-m$. Put $S:= \langle T^{n-l-1}\rangle$. Note that $\langle T^{n-l} \rangle = \langle T \dot{\wedge} S \rangle$ by Proposition \ref{pro-sublinearnonpluripolar}.  Using this and  monotonicity of non-pluripolar products, we get
\begin{align}\label{ine-chuyenveVdimllossmass}
\big\| \alpha^m - \{\langle T^m \rangle \}  \big\| \ge \big\| \alpha^m - \{\langle T^{n-l} \rangle \wedge \alpha^{l+m-n}\}\big\| \gtrsim  \big\| \alpha^{n-l} - \{\langle T \dot{\wedge}S \rangle\}\big\|.
\end{align}

Let $1 \le s \le n-l-1$ be an integer. Since $V$ is of dimension $l$ and $V$ is a maximal irreducible analytic subset of $X$ such that $\nu(T,V)>0$, we see that the self-intersection $T^{s}$ is classically well-defined in an open neighborhood $U$ of $V \backslash \Sing(I_T)$ in $X$, where $\Sing (I_T)$ is the singular part of the analytic set $I_T$. Moreover, $T^s$ has no mass on $V$ because $s < n-l$. By \cite[Page 169]{Demailly_analyticmethod} or Corollary \ref{cor-sosanhlelong}, we have
\begin{align}\label{ine-TswelldefinedU}
\nu(T^s, V\cap U) \ge  \nu(T,V)^s.
\end{align}
Now using the fact that $T$ has analytic singularities and the maximality of $V$, we get
$$\langle T^s \rangle =\bold{1}_{U \backslash V} T^s = T^s$$
on $U$ (we choose $U$ such that $U$ doesn't intersect the singularity of $I_T$). Using these observations for $s=n-l-1$ gives
\begin{align}\label{ine-TswelldefinedU2}
\nu(S, V) \ge  \nu(T,V)^{n-l-1}.
\end{align}
Now recall that non-pluripolar products have no mass on pluripolar sets. In particular, $S$ has no mass on $I_T$.
This allows us to apply Lemma \ref{le-truonghom=1lelong} to $T$ and $S$. As a result, we obtain
$$\big \| \alpha \wedge \{S\} - \{\langle T \dot{\wedge}S \rangle \}\big\| \ge \sum_{V \in \cali{V}_l} \nu(S,V) \nu(T,V) \vol(V) \ge \sum_{V \in \cali{V}_l} \nu(T,V)^{n-l} \vol(V)$$
by (\ref{ine-TswelldefinedU2}). On the other hand, by monotonicity of non-pluripolar products again, we get $\{S\} \le \alpha^{n-l-1}$. This combined with the fact that $\alpha$ is K\"ahler implies $\alpha \wedge \{S\} \le \alpha^{n-l}$. Hence, we deduce that
$$\big \| \alpha^{n-l} - \{\langle T \dot{\wedge}S \rangle \}\big\| \ge \sum_{V \in \cali{V}_l} \nu(T,V)^{n-l} \vol(V).$$
This coupled with (\ref{ine-chuyenveVdimllossmass}) gives (\ref{ine-obstructionselfinterkahler}). The proof is finished.
\endproof

We now prove Theorem \ref{the-self-intersec}. Let the notations be as in the statement of that result.

\begin{proof}[End of the proof of Theorem \ref{the-self-intersec}]   Let $T_{\min}$ be a current with minimal singularities in $\alpha$. Let $\theta$ be a smooth  closed $(1,1)$-form such that $\beta:= \alpha+ \{\theta\}$ is K\"ahler. We can choose $\theta$ to depend only on $\|\alpha\|$.   Applying Theorem \ref{the-congthemomega} to $T, T_{\min}, \theta$ gives
$$ \big \| \{\langle (T_{\min}+\theta)^m \rangle\} -\{\langle (T+\theta)^m \rangle\} \big\| \le  C\big\|  \alpha^m- \{\langle T^m \rangle\}\big\|^{2^{-m}},$$
for some constant $C$ depending only on $\cali{N}_0$. Using this and  the fact that $\alpha$ is big nef, we infer
$$ \big \|  \beta^m  -\{\langle (T+\theta)^m \rangle\} \big\| \le  C\big\|  \alpha^m- \{\langle T^m \rangle\}\big\|^{2^{-m}}.$$
Now applying Theorem \ref{the-kahlerself-intersec} to $\beta, T+\theta$ shows that the left-hand side of the last inequality is greater than or equal to
$$\frac{1}{m}\sum_{V \in \cali{V}}\big(\nu(T,V)\big)^{n- \dim V} \, \vol(V).$$
Hence (\ref{ine-obstructionselfinter}) follows. The proof is complete.
\end{proof}

redpajama_set_name: RedPajamaArXiv


text:
Home/bulgaria/Post Bank bought Piraeus Bank Bulgaria :: Investor.bg
Post Bank bought Piraeus Bank Bulgaria :: Investor.bg
bulgaria November 7, 2018 bulgaria, България
Eurobank Ergasias S.A. (Eurobank) announced the conclusion of an agreement on the purchase of Piraeus Bank Bulgaria by its subsidiary Eurobank Bulgaria ("Postbank").
Since September 2018, Piraeus Bank Bulgaria has assets of 1.7 billion euros, a loan portfolio of 820 million euros and 1.3 billion euros of deposits. The Bank operates in retail banking and mainly in corporate banking through a branch network of 70 affiliates and more than 900 employees.
The contract will consolidate Postbank's position on the Bulgarian banking market, so the bank will have an expected market share of over 10% and the third largest loan portfolio. Upon completion of the contract, Postbank will have assets in the total amount of more than 5 billion, with a capital of more than 600 million euros, and after realization of the expected synergies, profit before reservation and net profit of over 150 euros and more than 70 million euros. In addition, the bank will maintain its strong capital adequacy and liquidity coefficients.
The value of the shares is EUR 75 million, after the conclusion of the transaction, the impact on the core capital of Eurobank will be minimal with 14 basis points, and after realization of the expected synergies, the transaction is expected to positively affect the capital.
Stavros Ioannou, deputy executive director of Eurobanka, said: "We are pleased with the agreement reached with Piraeus Bank for the purchase of Piraeus Bank Bulgaria by our bank in Bulgaria Postbank. The acquisition follows the Eurobank's strategy of focusing on the expansion of its international markets, which it regards Eurobank Bulgaria will be the third largest bank in the country after the acquisition. Our goal is to better serve our clients, support the Bulgarian economy and the Greek business in our neighboring country. go to our colleagues from Piraeus Bank for the good cooperation that has led to the conclusion of this agreement and welcome our new colleagues from Piraeus Bank Bulgaria. "
The completion of the transaction is subject to the approval of the competent regulatory and supervisory authorities and is expected to be completed in the first quarter of 2019.
Barclais Bank Plc is a financial adviser to the Eurobank Group in terms of the agreement, and legal services are from legal firms Milbank, Tveed, Hadlei & McCloi LLP and Karatzas & Partners.
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The car turned into a ditch next to KCM, inside there is no one!
The lustful mayor of Strelcha wanted to have sex with a famous chalga diva.
Analysts: Tesla's performance in Europe is worrying
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Plovdiv residents drove authorities crazy with outrageous outbursts amid the pandemic, police stormed restaurant after restaurant VIDEO

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Firstly we can see how to get back blocked WeChat account from WeChat support, see below steps.
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1.Open browser on your PC and navigated to WeChat official site.
2.Also you can go to Account Recovery page and follow its instructions to have a try.
Remark: When you have lost or deleted some of your WeChat data on iPhone occasionally, or your WeChat account is blocked and you need to view some WeChat data, iRefone can help you with it.


text:
丝绸之路项目
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What are the Spice Routes?
The Spice Routes, also known as Maritime Silk Roads, is the name given to the network of sea routes that link the East with the West. They stretch from the west coast of Japan, through the islands of Indonesia, around India to the lands of the Middle East - and from there, across the Mediterranean to Europe. It is a distance of over 15,000 kilometres and, even today, is not an easy journey. From our very earliest history, people have travelled the Spice Routes. At first, they probably ventured only short distances from their home ports but over the centuries their ships sailed further and further across seas and oceans. They braved treacherous seas and eventual hostile reception on arrival in an unknown land. These journeys were not undertaken purely in the spirit of adventure - the driving force behind them was trade. The Spice Routes were, and still are, first and foremost trade routes. Since ancient times, trade has had an important role in human life. When we buy something we are trading, exchanging one item (usually money) for another. However, our purchase is the final link in a long chain of buyers and sellers: from the supplier of raw materials, to the manufacturer, to the wholesaler, to the shop - and if the goods we buy come from abroad there may be several other stages in between.
The journey of the goods between all these links in the chain is what is called a trade route. In the case of the Spice Routes the links were formed by traders buying and selling goods from port to port. The principal and most profitable goods they traded in were spices - giving the routes their name. As early as 2000 BC, spices such as cinnamon from Sri Lanka and cassia from China found their way along the Spice Routes to the Middle East. Other goods were exchanged too - cargoes of ivory, silk, porcelain, metals and dazzling gemstones brought great profits to the traders who were prepared to risk the dangerous sea journeys. But precious goods were not the only points of exchange between the traders. Perhaps more important was the exchange of knowledge: knowledge of new peoples and their religions, languages, expertise, artistic and scientific skills. The ports along the Maritime Silk Roads (Spice Routes) acted as melting pots for ideas and information. With every ship that swept out with a cargo of valuables on board, fresh knowledge was carried over the seas to the ship's next port of call.
Today, it may seem strange that the demand for spices was the main reason for such large-scale trade across such long distances. One probably thinks of them simply as flavouring for food. Yet, the word "spice" comes from the Latin species, which means an item of special value, as compared to ordinary articles of trade. Travelling these long distances becomes understandable if one considers the fact that many of the important spices had ritual and medical values and could only grow in the tropical East, from South of China to Indonesia as well as southern India and Sri Lanka. In particular, they grew in the Moluccas or, as they are better known, the Spice Islands. These are a chain of mountainous islands strung out in the Pacific Ocean between Sulawesi (Celebes) and New Guinea. From here came the fragrant spices of cloves and nutmeg which grew nowhere else in the world. To reach the spice markets found across Asia and Europe, the spices had to be transported thousands of kilometres over the seas.
One may never discover how people came to know and value these spices which grew so far away. As trading links from Indonesia fanned out through south and central Asia, they met with links that spread from the Middle East and the north. Goods were exchanged and traders would return to their homeland carrying the beautifully scented, exotic spices. Perhaps it was their strangeness and rarity that led great medicinal and spiritual values to be attributed to them. From ancient times, spices were burned as incense in religious ceremonies, purifying the air and carrying the prayers of the people heavenward to their gods. They were also added to healing ointments and to potions drunk as antidotes to poisons. To hide the many household smells, people burned spices daily in their homes. They were used as cooking ingredients very early on - not only to add flavour but also to make the food, which was often far from fresh, palatable, particularly in hot climates.
Myths and legends were woven around these exotic substances. They were linked to strange beasts like the phoenix, giant eagles, serpents and dragons. In the Fifth Century BC, the Greek historian Herodotus wrote how the spice cassia grew in a lake "infested by winged creatures like bats, which screeched alarmingly and were very pugnacious". Some of these stories were probably created by the traders who, wishing to protect their profits, tried to hide the sources of the spices.
The profits to be made from spices were considerable. They were small and dried, and consequently could be transported easily. The wealth of the spice trade brought great power and influence and, over the centuries, bloody battles were fought to win control of it and the routes along which it took place.
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text:
Bolivar Technical College
1135 North Oakland Avenue
Bolivar, Missouri 65613
Recent News About Bolivar Technical College View More
Which U.S. schools receive the least federal student aid?
By Local Labs News Service | Dec 22, 2019
Stevens-The Institute of Business & Arts, Lake Career and Technical Center and New Dimensions School of Hair Design reported receiving the least federal financial aid in the state during the 2018-19 academic year among U.S. schools.
Which Missouri schools receive the least federal student aid?
Stevens-The Institute of Business & Arts, Lake Career and Technical Center and New Dimensions School of Hair Design reported receiving the least federal financial aid during the 2018-19 academic year among Missouri schools.
33 percent of applicants to Bolivar Technical College were admitted for fall 2018
By Local Labs News Service | Sep 29, 2019
At Bolivar Technical College, 56 percent of undergraduate students are nontraditional students - age 25 or older - and 95 percent are female, according to the latest disclosure from the U.S. Department of Education.
Bolivar Technical College Upcoming Events View More


text:
Academic Fraud May Have Been Alleged In Redacted Manning Court Documents
Much has been written in recent days about an old court case between a former University of Tennessee athletic trainer and Peyton Manning in which Jaimie Naughwright accused Manning of sexually assaulting her in 1994 while she was examining a potential stress fracture on his foot. Manning, although initially denying the incident took place, later admitted he had mooned a fellow teammate in Naughwright's presence. She insisted he dropped his pants and pressed his butt against her head and rested his scrotum and penis on her head.
A 74-page court document filed by Wainwright's attorney in a case later settled out-of-court stated facts in support of his client's case, but a number of pages were redacted from that document by the court at Manning's attorney's insistence. The New York Daily News' Shaun King, who has written extensively about the events from Manning's college years, has promised to publish information about the redacted pages but has so far failed to do so. NBC Sports, by connecting a few dots, suggests the redacted information may have alleged academic fraud against Manning:
Via Hobson, an October 2002 deposition of former Tennessee athletic director Doug Dickey contains an exchange in which Naughright's lawyer asked whether Naughright had spoken to associate A.D. Carmen Tegano "about the possibility of Peyton Manning having committed academic fraud" in a 1994 course taught by Tegano, which featured Naughright as a guest lecturer.
That question appears at page 63 of Dickey's deposition transcript. However, at no point during or after the question or before or during the answer does Manning's lawyer request that the matter be treated as confidential. At page 77, after Naughright's lawyer presents to Dickey two documents for inspection and questioning, Manning's lawyer requests that the questions and answers regarding the documents be placed in the "confidential record" of the case.
While it's possible that, only 14 pages before asserting confidentiality regarding documents relating to the secret 1994 incident, Manning's lawyers failed to make the same request regarding questions on the same topic, most lawyers don't allow such matters to slip through the legal equivalent of the five hole. The two documents that triggered the confidentiality request at page 77 of the Dickey deposition had been introduced only one week earlier during the deposition of Jamie Naughright. Thus, Manning's lawyer was well aware of the sensitivity of the 1994 allegation — and the lawyers undoubtedly were ready to ensure that any sensitive information would be kept out of the public record of the case.
That said, it's one thing for the lawyers to have asked general questions about possible academic fraud; it's quite another for the lawyers to have produced a pair of documents that arguably (or actually) substantiated it.
Until someone discloses the details of the 1994 incident that was removed from the 74-page document published last week by the Daily News, it will remain exactly what it has been for the last six days: A mystery.
Posted by Gary R. Welsh at 11:48 AM
Labels: Peyton Manning
hey gary, Last night Marco Rubio dismissed the "conspiracy" nuts at Breitbart during an appearance on Fox News, telling host Neil Cavuto that his campaign doesn't even credential Breitbart reporters for his events. Boom!
The local media's silence on Manning's foibles is deafening after their haranguing of Tom Brady over the amount of air in his footballs.
Anon. 12:48, That really has nothing to do with this post. I did, however, see Rubio's comments. I was taken aback that he was dismissive of what the head of the ICE union had to say about his conversations with him. He seemed to think he shouldn't be taken care of because he was a mere ICE agent who represented their official union as opposed to an ICE official, or an appointee of the Obama administration. The ICE union has been very critical of the Obama administration appointees, whom they have accused of ordering them not to enforce immigration laws. I don't see the conspiracy angle in listening to the concerns of the front-line workers as opposed to the high-level political appointees.
I think anon 12:48 is trying to suggest that there is some sort of conspiracy against Indy's hero, Peyton Manning. That's nonsense. The recent filing isn't even about Manning. He was merely mentioned in it, but because he has never come clean and because he holds himself out as holier than Thou and because he appears to have attempted to smear someone of character, it's not going away. It's really quite remarkable if you think about it. NOBODY who knows what happened has come out in support of Indy's hero. EVERYBODY has come out in support of Dr. Naughright. Everybody. It never plays out that way. Superstars always have their apologists, but nobody who knows what happened in that locker room has come out and said leave my boy Peyton alone. Nobody.
This is so reminiscent of Lance Armstrong's attempt to destroy Frankie Andreu's wife and it absolutely disgusts me. Manning disgusts me. The local fools who continue to defend him disgust me more. Gannett and our local fawning jock sniffers in the media and government disgust me most of all. The truth will eventually come out. It always does. You can only cover this nonsense up for so long. Ask Lance.
Anon 901, while I suspect Manning has been a boob in his younger years, I find it hard to compare him to Lance Armstrong.
Lance went out to systematically destroy good people and deserved to be taken down - and hard. I know the cancer community doesn't want to hear anything bad about Mr Live Strong, but as a survivor? I want nothing to do with anything he touched. I saw LIVE STRONG diaries at the cancer center radiotherapy waiting room and put it down.
I am happy the guy beat cancer, but karma's a bitch.


NAIROBI, July 24 (Xinhua) -- Kenya's Lawrence Cherono says he will do everything in his power to retain his Amsterdam Marathon crown and deny Ethiopia's Kenenisa Bekele the title.
The two are expected to compete at this year's race, which will be held on Oct 21 and Cherono believes he has played the underdog card before and surprised his critics when he won in the Dutch capital last year.
"Last year, I was very happy with the result. I immediately knew that I will come under focus this year and though I had not known who to face, I will be happy to battle it out with Bekele and win again," Cherono said on Tuesday in Eldoret. Of the nine marathons Cherono has contested, he has won four and finished on the podium in eight.
Alongside winning in Amsterdam last year he also finished second in Rotterdam with a time of 2:06:21. The Amsterdam Marathon has always attracted a strong group of elite runners and the 2018 race will be no different.
Bekele, a multiple world and Olympic champion announced last week that he will skip the big city marathons to compete in Amsterdam, which is an IAAF Gold Label road race. The Ethiopian distance runner owns the second-fastest marathon performance in history on a record-eligible course, having clocked a national record of 2:03:03 to win the 2016 Berlin Marathon.
His time is just six seconds shy of Dennis Kimetto's world record (2:02:57). Bekele, who will be contesting his first marathon on Dutch soil, will be up against Cherono and a horde of other top Kenyan and Ethiopian road racers.
"Kenenisa Bekele is one of the world's best long-distance runners," said race director Cees Pronk. "We are incredibly proud that Bekele will be lining up at the start on Oct 21. Bekele decided to run in Amsterdam because he has experienced the expert organization of the event and knows first-hand that the athletes always come first."
However, while Bekele might be focused on the world record, Olympic marathon champion Eliud Kipchoge might just make his work harder if he is successful in his bid to beat that time in Berlin in September.


text:
Liverpool data platform for fashion retailers secures "six-figure" investment
by Rachael Hesno
Margin Guardian, a Liverpool-headquartered data insights platform, has secured a "six-figure" investment.
The platform has secured the investment from Northern Powerhouse Investment Fund (NPIF) – Maven Equity Finance, managed by Maven Capital Partners.
Margin Guardian, co-founded in 2018 by Richard Blanchard, Stephen Goodheart and Stewart Cox, provides a range of fashion retailers with data insight and targeting services. It plans to use the funding to further invest in its platform, commercialise and develop an android version of its app and bolster its marketing activities.
Richard Blanchard, CEO of Margin Guardian, said: "We believe retailers increasingly need advanced data insight to maintain competitiveness and profit. Our unique purchase intent data will allow them to do just that. We are excited to be partnering with Maven on our journey as they truly understand the data revolution underway."
Dean Cox, Senior Investment Manager at Maven Equity Finance, added: "MG was founded by a strong team of retail experts and the technology they've created addresses a genuine need for both retailers and consumers. MG have received encouraging traction from several high-profile retailers, and we're pleased to support Richard, Stephen and Stewart at such a pivotal point in the company's growth journey. We look forward to working with the team to help drive MG forward."
The NPIF project is supported by the European Union using funding from the European Regional Development Fund (ERDF).
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Health Minister should meet with Halton's Regional Chair
(Queen's Park) – Ontario's Health and Long-Term Minister Eric Hoskins needs to meet with Halton Regional Chair Gary Carr to discuss Regional Council's serious concerns about public health funding and the proposed restructuring of public health boards.
That was the message Wellington-Halton Hills MPP Ted Arnott took to the floor of the Ontario Legislature during Question Period on November 15th.
"Halton Regional Council is expressing serious concerns about the report of the Minister's Expert Panel on Public Health, which is entitled Public Health within an Integrated Health System." Mr. Arnott said in the House. "What's more, they are speaking up about the fact that this Government continues to shortchange the Region of Halton for public health: to the tune of almost 10 million dollars a year."
In response, the Minister said that he would be willing to look at the request from the Regional Chair, but he did not commit to a meeting any time soon.
"I know there is a request in from the Regional Chair of Halton, Mr. Speaker. I'm happy to have a look at that," Eric Hoskins replied.
Mr. Arnott is hopeful the Minister will set aside time in the near future to meet with Halton Region's leadership to hear Regional Council's concerns and seek to find ways to address them.
"Our elected Municipal colleagues are amongst our most important partners. We need to work together. It's in the public interest that we cooperate with our municipal partners," Mr. Arnott added.
(Attached are Mr. Arnott's remarks in the Ontario Legislature and the responses from the Minister of Health and Long-Term Care, November 15th, 2017)
Official Report of Debates (Ontario Hansard) – Nov, 15, 2017
Mr. Ted Arnott: My question is for the Minister of Health and Long-Term Care. Halton regional council is expressing serious concerns about the report of the Minister's Expert Panel on Public Health, which is entitled Public Health Within an Integrated Health System. What's more, they're speaking up about the fact that this government continues to shortchange the region of Halton for public health to the tune of almost $10 million a year. The regional council, by way of resolution, has authorized regional chair Gary Carr to seek a meeting with the Minister of Health and Long-Term Care. His original request was sent July 11, almost four months ago. When will the Minister of Health and Long-Term Care meet with the Halton regional chair?
Hon. Eric Hoskins: I appreciate the question. Our 36 public health units across the province play such an essential role in ensuring that Ontarians stay safe and live and thrive in a safe environment. Their work is absolutely essential, which is part of the reason why, since coming into office, we've doubled our funding to public health. In fact, a couple of years ago we changed the funding formula, based on multi-year consultation with the sector—with the front-line public health workers and others—so that the funding formula was not a simple per capita formula as before but it actually reflected the socio-demographic economic conditions and the need expressed on public health for each of the various localities.
I know that there is a request in from the regional chair of Halton, Mr. Speaker. I'm happy to have a look at that, and I'm happy to address any further issues in the supplementary.
Mr. Ted Arnott: The minister said he'd be happy to look at the request, but he didn't say that he would meet with the regional chair. I can't believe that the Minister of Health would refuse to meet with the regional chair of Halton, which has been one of the fastest-growing areas in the province.
Our elected municipal colleagues are amongst our most important partners. We need to work together. It's in the public interest that we co-operate with our municipal partners, and I hope the minister will do that. The recommendations of the expert panel on public health include re-drawing geographic boundaries of public health agencies to conform with LHIN boundaries. However, in some cases, this means that regional municipalities will be separated into more than one public health agency.
At the same time, municipalities will lose their voice on public health boards but will still be expected to substantially fund the public health programs through local property tax dollars. The region of Halton believes that accountability will go out the window. How will the minister ensure that the accountability of public health boards to municipalities is not compromised?
Hon. Eric Hoskins: Several years ago, as we spent and invested that time and consultation on the funding formula coming out of the Patients First Act, we committed to having a similar process and consultation to look at the structure and governance of public health in this province, something which hasn't been looked at for many, many years—I think probably since the download that the PCs did for public health. Obviously, we have uploaded a considerable amount of the download to municipalities that the PCs were responsible for.
The expert panel was asked to look at structure and governance. They consulted to do that. They made the report available over the summer. The day after I received the report, I think, I immediately met with AMO, the MOU table with the municipalities. We are now in a consultation process across the board, including with Halton and public health units and municipalities, to get their input on the report.
Fall Economic Statement Falls Flat College strike finally ended by Ontario Legislature


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Date: February 9, 2018 Author: Jeff Thorburn Categories: features
I've only encountered Dennis Busenitz once in real life, and it happens to have been almost exactly eight years ago to the day of me writing this. Normally I have an awful memory when it comes to dates, but in this case I can remember the specific situation in which I met him, and the time. It was February of 2010, and I was in San Francisco with Kevin Lowry and some other friends. We stayed at the Green Tortoise Hostel, a place that was basically made into a de facto Canadian skateboarder's embassy by the late Darrell Smith many years ago. The reason I can remember the month and year so well is because of how extraordinary it was. Firstly, one afternoon we all watched Canada win the gold medal in Olympic hockey while sitting at a strip club near our hostel. It was the nearest place we could find that was showing the game. I remember walking out of that place absolutely euphoric from what we watched. The hockey game, that is. Secondly, this was just two months before my son was born, which turned out to be a life-changer, as you may have guessed. Lastly, my birthday occurred while I was there.
I didn't plan to write about Dennis' part in Roll Forever because of the timing though, but as I sat down to write this intro it all just clicked for me: the Olympics are starting in South Korea today, my birthday is next week, and my son turns eight years old in two months. Suffice to say, it was a memorable trip.
Now, to the subject at hand, Dennis Busenitz. One day we met up with Keegan Sauder and Lee Yankou, fellow Canadians that were living in San Francisco at the time, to start our day at 3rd and Army, a must hit spot in the city. As our crew pulled up in a couple of vehicles, I could see one person skating the spot alone, and I immediately knew it was Dennis Busenitz. Not only because of his recognizable style, but because of his van parked right there, and his dog Gary chasing behind him. I was as much in awe of seeing him in person as I was of the absolute normalcy of what he was doing: skating alone, just like all of us have many times. But you don't imagine a top pro doing that, skating alone in the morning with his dog trailing behind him. The guy was tied to some of the best brands in skateboarding, brands that are based in the Bay Area. He had an innovative and well-received pro model shoe that had just come out the year before with adidas. I was actually wearing those shoes. So if you didn't know any better, you'd expect Dennis Busenitz to be skating with a group and a couple of people with cameras documenting it all. Seeing him out there trying to piece together new tricks and lines all by himself would be like seeing the 2010 Golden Goal scorer himself, Sidney Crosby, show up unexpectedly at an outdoor rink in Quebec to run through some training drills with a junior player.
Let us muse on Dennis Busenitz in Real's 2005 video, Roll Forever, one of the first free DVDs I remember being released. —Jeff Thorburn
-It's sometime before 2005, and we've got a couple of Real's young bucks laying around on some bricks and stretching. Flash forward over a decade, and one of these guys will swell up and become very active on Instagram, labeling himself "SKATEBOARDER | LIFTER | MODEL | COACH", while the other will reluctantly join Instagram to appease sponsors, post random things infrequently, and continue to look, dress, and skate almost exactly the same.
-Look at that trail of sweat. Will Dennis every become as synonymous with sweating at Gershon has? Also, maybe we should have seen what was coming with Nick's coaching on that stretch.
-A Fallen-era Frank Gerwer drives a set of Spitfire's right into Dennis' ribs in a failed attempt at a No-Comply over him. Dennis shows his true German-colours with a completely dry reaction: "Ah, that kinda hurts."
-Here we have the juxtaposition of the San Francisco-by-way-of-Kansas-by-way-of-Germany guy wearing a Metallica t-shirt, asking if they "Think it's hard to get rights for Biggie?" while a folk rock song kicks in for his part. The song is by Old Canes, who are from Lawrence, Kansas, which is about 3 hours from where Dennis spent his teenage years in Newton, after moving there from Germany. Dennis would end up skating to another one of their songs in a section from Real's Life and Times video in 2006. "Yes, America, it's what I drink."
-The intro to the video and then the individual intros to the parts in this video are a really cool way to make the whole thing cohesive and offer up a sample of what that person is like. I still remember some of what the guys said before their parts. Anyway, it's worth watching the whole video if you are nostalgic, interested in some history, or just a sap like me.
-And then we just get thrown into the San Francisco hills by way of a Nollie down that double-set.
-You know you are blasting on transition when you are going higher over the lip than the quarterpipe is tall.
-As mentioned in previous musings, I'm a big fan of hitting the same spot back-to-back but in a different way. This counts.
-While technically the Alley-Oop Frontside 360 over the hip is the main event, I'm much more interested in the Kickflip opener and the casually-carved ride away.
-I love the awkward tricks Dennis has always been willing to mess with in the hills of SF. Downhill manuals, man. And then the Switch Flip off the curb. The way he 180s out of the Manual makes me think of Ocean Howell in the Ipath promo. Maybe it's a combination of style and location. Man, that's a part to be mused on one of these days.
-Skating in a tank top like this makes me think of Van Wastell. Any excuse to pull up some Van footage is a welcome excuse. I'm noticing just how far out Dennis flicks his front foot on Kickflips.
-This looks like a scene from a comic strip.
-This is Dennis in the middle of flirting with a mohawk. He seems to be over that phase, but still skates in a lot of jackets, something you don't see very often.
-One of the things that I believe a lot of people like about Dennis is just how normal and hard-working he is. He's never acted like a tortured artist or genius. He's down to try hard and represent his sponsors. He handles skating like a job that he enjoys. He's always been progressive, not just in his tricks, but in his approach to how skating can look. I can't tell you with certainty the name of anyone that has won Tampa Pro before besides Dennis.
-I dig the eagerness he's rolling with in the natural bank/QP line. He's pushing the instant he completes the little Half-Cab slide around after the first trick, and keeps going to get the Switch 360 Flip at the end. You know he probably kept going after that too.
-The Backside Nosegrind on the stair is such an SF type spot, and most would just roll with the single, but leave it to Dennis to keep going and screech a Bluntslide on the next stair. He's always expanding the zone like this in lines, which somehow reminds me of this scene from The Fresh Prince of Bel-Air.
-The next two clips had to have been put back to back intentionally to show the contrasting outfits. On the Frontside Noseslide on the bank he looks like the farm-boy skater that wears big headphones and smokes weed at the skatepark everyday, with those basketball shorts and a Metallica t-shirt, whereas in the Switch Flip on the bank he's got a nice sweater over top of a collared shirt.
-It might be interesting for some people now to be seeing the mix of shoes he's wearing in this part, including the Vans Half Cabs seen below. Dennis joined the revamped adidas Skateboarding program in 2006 and had his first pro model shoe come out in 2009. It's weird how excited I get just watching this commercial again and thinking about the time when this shoe came out. It just looked so different from all the Half Cab rip-offs that brands were doing at that time.
-There's something Dennis said in a 2012 interview with Jenkem that has always stood out to me:
Is there any good advice you've gotten about skating?
Well my friend Matt, I guess around the time of my Seeing Double part was like, "C'mon Dennis, good part but you know, smith grinds, tailslides… you can do more!" I'll never forget that. A well needed constructive criticism. It was funny I think. He just told me to do more than the bare minimum to get by I guess.
His response really explains a lot of what drives Dennis. But after rewatching his parts in Real to Reel and Seeing Double, I'm kind of blown away that this advice made such a mark on him. To me, he's pushing it pretty hard in those parts, and he hasn't really stopped since. But I guess it's this advice that is keeps stoking the fire in him to consistently put out great parts.
-Watch the ride away from that Bluntslide on the bank and tell me he didn't keep going to do a Switch Tre after.
-Actually, it's possible that Dennis is driven not just by his friend Matt's comment, but also by a desire to keep up with the fabled-but-fictitious Gary Headlock, as he explained to TWS in 2010.
Your dog's name is Gary. Could you tell me about how that name came about?
Well, I was pretty green behind the ears on the first skate trip I ever went on with Deluxe, so everyone would kind of f--k with me. They'd make up shit and I'd believe them, because I was so naïve. So, Frank Gerwer was telling me about this incredible skater named Gary Headlock that was doing the craziest shit that nobody had ever heard of and he had a sidekick named Ducky Donzworth, I think. And, they'd tell me Gary was doing stuff like switch 360 flip frontside crooked grinds down an eight-stair handrail and it was the opener in the newest 411VM. I just hadn't seen it yet, so the dumb kid that I was, I was like, "What the f--k? No way! That's so gnarly." I don't believe anything Frank tells me now, but Gary Headlock has remained this infamous, fictitious ripper joke now.
-The China Banks bench carve. In case you somehow can't tell, that bench is extremely long. That's a spot he's returned to many times over the years to approach in different ways and innovate on, even to this day.
-Fitting shoes for the spot. Wearing the Huf 3s at a spot that I definitely associated with Keith Hufnagel.
-Again, never content with just a single trick, unlike some previously mused on skaters.
-While I'm sure it's in the Bay Area, the Polejam Backside 180 always felt like a Miami clip to me. It's probably because of how bright it is and the lack of torso and leg covers.
-Everything about the downhill line, starting with the Kickflip Manual, seems nonchalant, just a lead up to the Nollie Heelflip, which was itself quite chill.
-If you've seen his Seeing Double part, you'll know that Dennis has some serious chops in a pool or tranny park. You'll also know that he and Jim Beam don't mix well.
-I'd like to know what the plan was on this scorching fast downhill Manual.
-And while it's likely got an anti-climatic ending, the cut on this curving powerslide has always left me wondering, kind of like Gonz in the intersection.
-I've always liked that his Nollie Flips are sort of mobbed. And this line is a nice follow-up to the earlier Nollie and then Ollie.
-A rather tame Fakie Manual at the famed 3-up-3-down spot in SF that Dennis would continue to hammer on over the years.
-He seemed to be really feeling Nollies down things at this time.
-Quick hops, very Keith Hufnagel-esque.
-A ketchup and mustard clothed Tony Trujillo blazes through with a Backside Tailslide and Dennis backs him up with a Lipslide on a spot we saw in Bobby Puleo's mused-on Static II part.
-Expanding the zone again with an Ollie over the wall.
-There's a quick snippet of an awful looking sliding slam down a hill that looks like it comes from his final clip.
-A heavily mohawk'd Backside Noseblunt Slide at another spot we saw Puleo skate.
-And then an endless ride-away into the sunset perfectly timed to the music.
-That's it, the building of a perfect storm that would lead to a German-Kansan developing one of the most popular pro-model skate shoes of all time and becoming one of the quintessional examples of a modern day professional skateboarder.
Read previous musings: Gino Iannucci in "Yeah Right!"; Anthony Papplalardo in "Mosaic"; Jason Dill in "Mosaic"; Alien Workshop in "Mosaic"; Bobby Puleo in "Static II"; Nick Jensen in "Lost and Found"