text:
\section{Notation and Conventions and Basic Definitions}
\begin{Convention} By $\mathbb N$\, we denote the natural numbers, by $\mathbb N_0$\, the set of nonnegative integers.
\end{Convention}
We use multi index notation: if $x_1,...,x_n$\, is a set of variables, we denote  $\underline{x}^{\underline{m}}: x_1^{m_1}\cdot x_2^{m_2}\ldots x_n^{m_n}$\, where  $\underline{m}:=(m_1,m_2,\ldots ,m_n)$\, is a multi-index of lenght $n$. By $\mid \underline{m}\mid$\, we denot the number $m_1+...+m_n$\,.
The partial derivatives of a function $f(x_1,...,x_m)$\,  in the variables $x_i$\, we denote by
$\partial^{\mid m\mid}/\partial \underline{x}^{\underline{m}}(f(x_1,...,x_m))$\,.
\begin{Notation} Let $X\longrightarrow S$\, be a morphism of schemes. By $\Omega^{(1)}(X/S)$\, we denote the usual sheaf of K\"ahler differentials. The ideal sheaf of the relative diagonal $\Delta_{X/S}\hookrightarrow X\times_SX$\, we denote by $\mathcal I_{X/S}$\,. Thus $\Omega^{(1)}(X/S)\cong \mathcal I_{X/S}/\mathcal I_{X/S}^2.$
\end{Notation}
We collect some basic facts about jet bundles and differential operators. Proofs and more detailed information can be found in \cite{Guenther2}.\\
\begin{facts} If $q: X\longrightarrow S$ is a morphism of finite type of noetherian schemes and $\mathcal E$\, is a  quasi coherent sheaf on $X$, for each $N\in \mathbb N_0\cup\{\mathbb N\}$\,, we denote by $\mathcal J^N(\mathcal E/S)$\,, the $N^{th}$ jet bundle of $-mathcal E$-, relative to $S$, which is a  quasi coherent sheaf on $X$. If $\mathcal E$\, is coherent, so is $\mathcal J^N(\mathcal E/S)$\, for $N\in \mathbb N_0$\,. -- Locally , if $\Spec B\longrightarrow \Spec A$\,  is the restriction of $q$ to Zariski-open subsets and $\mathcal E$\, restricted to the $\Spec B$\, corresponds to the $B$-module $M$, $\mathcal J^N(\mathcal E/S)$\, is  given  by
$$\mathcal J^N(M/A)=B\otimes_AM/I_{B/A}\cdot (B\otimes_AM),$$ where $I_{B/A}$\, is the ideal in $B\otimes_AB$\, which is the kernel of the multiplication map. If $\mathcal E'\subset \mathcal E$\,  is a quasi coherent subsheaf, we denote by $\mathcal J^N(\mathcal E'/S)'$\, the image of the homomorphism $\mathcal J^N(i/S)$\,, where $i: \mathcal E'\hookrightarrow \mathcal E$\, is the inclusion. \\
 We denote the universal derivation $\mathcal E\longrightarrow \mathcal J^N(\mathcal E/S)$\, by $d^N_{\mathcal E/S}.$\\
If $\mathcal E=\mathcal O_X$\,, the jet sheaf  $\mathcal J^N(\mathcal O_X/S)$\, is an $\mathcal O_X$-algebra and we denote by $J^N(X/S):=\Spec_X\mathcal J^N(\mathcal O_X/S)$\,, the associated affine bundle over $X$ with projection $p_X=p_{1,X}: J^N(X/S)\longrightarrow X$\,. There is a second projection $p_{2,X}: J^N(X/S)\longrightarrow X$\, which corresponds to the universal derivation $d^N_{\mathcal O_X/S}$\,.
\end{facts}
\begin{proposition}\mylabel{prop: P4}
 \begin{enumerate}[1]
\item Let $X\longrightarrow S$\, be a morphism of finite type between noetherian schemes. For each $N\in \mathbb N\cup\{\infty\}$\,, let $\mathcal J^{N}(-/S)$\, be the functor from quasi coherent $\mathcal O_X$-modules to quasi coherent $\mathcal J^{ N}(X/S)$-modules sending $\mathcal F$ to $\mathcal J^{ N}(\mathcal F/S)$\,. Then, this functor is right  exact and there is a canonical natural isomorphism $\mathcal J^{ N}(-/S)\stackrel{\cong}\longrightarrow p_2^*(-).$
\item If $\pi:X\longrightarrow S$\, is flat, then $\mathcal J^{\mathbb N}(-/S)$\, is an exact functor.
\item If $\pi: X\longrightarrow S$\, is a smooth morphism of schemes, then for each $N\in \mathbb N$\,, the functor $\mathcal J^N(-/S)$, sending quasi coherent $\mathcal O_X$-modules  to quasi coherent $\mathcal J^N(X/S)$-modules , is exact and equal to $(p^N_2)^*$\,.
\end{enumerate}
\end{proposition}
\begin{proof}(see \cite{Guenther2}[section 3.5, Proposition 3.33, p.19].
\end{proof}
 \begin{definition}\mylabel{def:D46} Let $X\longrightarrow S$\, be an arbitrary morphism of schemes, or more generally of algebraic spaces, and  $\mathcal F_i, i=1,2$\, be  quasi coherent sheaves on $X$. Then, a differential operator of order $\leq N$\, is an $\mathcal O_S$-linear map $D: \mathcal F_1\longrightarrow \mathcal F_2$\, that can be factored as
  $\mathcal F_1\stackrel{d^N_{\mathcal F/S}}\longrightarrow \mathcal J^N(\mathcal F_1/S)$\,
  and an $\mathcal O_X$-linear map $\widetilde{D}: \mathcal J^N(\mathcal F_1/S)\longrightarrow \mathcal F_2$\,. \\
  A differential operator of order $N$\, is a differential operator that is of order $\leq N$\, but not of order $\leq N-1$\,.
 \end{definition}
 Thus, in this situation, there is a 1-1 correspondence between differential operators $\mathcal F_1\longrightarrow \mathcal F_2$\, relative to $S$ and $\mathcal O_X$-linear maps $\mathcal J^N(\mathcal F_1/S)\longrightarrow \mathcal F_2$\,.
\begin{facts} Let $q: X\longrightarrow S$\, be a morphism of finite type of noetherian schemes and $\mathcal F_1,\mathcal F_2$\, be  quasi coherent sheaves on $X$. By $DO^N_{X/S}(\mathcal F_1,\mathcal F_2)$\, we denote the $\mathcal O_S$-module of global differential operators $D: \mathcal F_1\longrightarrow \mathcal F_2$\, relative to $S$, which is by definition the quasi coherent $\mathcal O_X$-module
$$DO^N_{X/S}(\mathcal F_1,\mathcal F_2):=\Hom_{\mathcal O_X}(\mathcal J^N(\mathcal F_1/S), \mathcal F_2).$$
If $\mathcal F_1,\mathcal F_2$\, are coherent, so is $DO^N(\mathcal F_1,\mathcal F_2)$\,.\\
  If $S=\Spec k$\, is a point, we use the simplified notation $DO^N(\mathcal F_1,\mathcal F_2)$\,.
\end{facts}
\begin{facts} Let $q: X\longrightarrow S$\, be a  morphism of finite type between noetherian schemes and $\mathcal E$\, be a coherent sheaf on $X$. Then we denote for each $N\in \mathbb N$\, the standard jet bundle exact sequence by
$$j^N(\mathcal E/S): \quad 0\longrightarrow \mathcal I_{X/S}^N\mathcal E/\mathcal I_{X/S}^{N+1}\cdot \mathcal E\longrightarrow \mathcal J^N(\mathcal E/S)\longrightarrow \mathcal J^{N-1}(\mathcal E/S)\longrightarrow 0.$$ If $\mathcal E$\, is locally free and the morphism $q: X\longrightarrow S$\, is smooth, the natural homomorphism
\begin{gather*}\mathcal E\otimes_{\mathcal O_X}\Omega^{(1)}(X/S)^{\otimes^s N}\cong \mathcal E\otimes_{\mathcal O_X}\mathcal (I_{X/S}/\mathcal I_{X/S}^2)^{\otimes^s N}\\
\longrightarrow  \mathcal I_{X/S}^{N}/\mathcal I_{X/S}^{N+1}\otimes_{\mathcal O_X}\mathcal E\longrightarrow \mathcal I_{X/S}^N\cdot \mathcal E/\mathcal I_{X/S}^{N+1}\cdot\mathcal E
\end{gather*}
is an isomorphism, which follows from the fact, that in this case $\Delta_{X/S}: X\hookrightarrow X\times_SX$\, is a regular embedding.
\end{facts}
\begin{Notation}If $q: X\longrightarrow S$\, is a smooth morphism of finite type of noetherian schemes  and $\mathcal E_1$\, and $\mathcal E_2$\, are locally free sheaves on $X$,  for each $N\in \mathbb N$\, we denote the standard exact sequence of differential operators by
\begin{gather*} do^N_{X/S}(\mathcal E_1,\mathcal E_2):\quad 0\longrightarrow DO^{N_1}_{X/S}(\mathcal E_1,\mathcal E_2)\longrightarrow DO^{N}_{X/S}(\mathcal E_1,\mathcal E_2)\\
\longrightarrow \Hom_{\mathcal O_X}(\mathcal E_1,\mathcal E_2)\otimes_{\mathcal O_X}\mathcal T^1(X/S)^{\otimes^s N}\longrightarrow 0,
\end{gather*} which is $\Hom_{\mathcal O_X}(j^N(\mathcal E_1/S),\mathcal E_2).$
\end{Notation}
\section{Introduction}
 In this note, we want to investigate the question, given a complete  algebraic projective scheme $X/k$ and a coherent sheaves  $\mathcal E_1,\mathcal E_2$\, on $X$, under which conditions does there exist a differential operator of order $N$,  $D: \mathcal E_1\longrightarrow \mathcal E_2$\,. We in particular treat the case where $X=\mathbb P^n_k, n> 1$\, is projective $n$-space over $k$\,.  One result (see \prettyref{prop:P67}), is that for each $n\in \mathbb Z$\, and each locally free sheaf $\mathcal E$\, on $\mathbb P^n_k$\, there exists for $N>>0$\, a global linear partial differential operator $D: \mathcal E\longrightarrow \mathcal E(n)$\, of order $N$. This is in contrast to the case of $\mathcal O_X$-linear homomorphisms $\phi: \mathcal E\longrightarrow \mathcal E(n)$\, where for $n<< 0$\, thanks to Serre duality no global homomorphisms exist. We have in addition that for fixed $N>0$\, and given nontorsion coherent $\mathcal O_X$-modules $\mathcal F_1,\mathcal F_2$\, there exists a global section $D\in \Gamma(X,DO^N(\mathcal F_1,\mathcal F_2(n)))$\, for $n>>0$\, which does not lie in $\Gamma(X, DO^{N-1}(\mathcal F_1,\mathcal F_2(n))$\, (see \prettyref{lem:L21}). This is a consequence of Serre vanishing and the fact, that the inclusion of coherent sheaves $DO^{N-1}(\mathcal F_1,\mathcal F_2)\subsetneq DO^N(\mathcal F_1,\mathcal F_2)$\, is strict, thanks to the result proven in \cite{Guenther2}[section 3.7, Proposition 3.44, pp. 26-29]. \\ We also calculate for fixed locally free $\mathcal E_1,\mathcal E_2$\, the number of global sections $\Gamma(\mathbb P^n_k, DO^N(\mathcal E_1,\mathcal E_2))$\, as $N$ tends to infinity. \\
 Next, we study the behavior of differential operators $D: \mathcal E\longrightarrow \mathcal E$\ on arbitrary smooth projective $X/k$\, , with respect to the Harder-Narhasimhan filtration  (HN-filtration for short) of $\mathcal E$\,.
 Our main result is \prettyref{prop:P60}, which says, that if $X$\, is not uniruled, then $D$ respects the Harder-Narhasimhan-filtration of $\mathcal E$\,. If moreover, $HN^{\bullet}(\Omega^{(1)}(X/k))$\, is the HN-filtration of the cotangent sheaf and the minimal slope $\mu_{min}(\Omega^{(1)}(X/k))>0$\,, then $D$ respects the HN-filtration and the differential operator $gr^iHN(\mathcal E)\longrightarrow gr^iHN(\mathcal E)$\, is $\mathcal O_{X}$-linear (see \prettyref{prop:P60}).\\
 Finally we study the question of global elliptic differential operators on locally free sheaves. We show that they exist on abelian varieties but on smooth projective varieties with $\mu_{min}(\Omega^{(1)}(X/k))>0$\, they do not exist at all. In \cite{Indexsatz} , an algebraic index theorem has been proved, and the result proved here puts an end to speculuations that there could exist an algebraic index theorem equivalent to the Hirzebruch-Riemann-Roch theorem.
\section{Global differential operators on coherent sheaves}
If $X/k$ is a complete algebraic scheme, or , more generally a complete algebraic space  over a field $k$\, and $\mathcal F$  is a coherent sheaf on $X$, the question arises, are there global differential operators $\mathcal F\longrightarrow \mathcal F$\,? To get a feeling for this subtle question, we prove a few lemmas and give some examples.
\begin{lemma}\mylabel{lem:L21} Let $X/k$\, be a projective scheme, $\mathcal F_1,\mathcal F_2$\, be  coherent nontorsion sheaves on $X$ and $\mathcal O_X(1)$\, be an ample invertible sheaf on $X$. Then, for each $N\in \mathbb N$\,  there is $M=M(\mathcal F_1,\mathcal F_2,N)$\, such that for all $m\geq M$\, there is a differential operator of order $N$,  $D:\mathcal F_1\longrightarrow \mathcal F_2(m)$.\\
In particular for $m>>0$\, there always exist non-$\mathcal O_X$-linear operators.
\end{lemma}
\begin{proof} By \cite{Guenther2}[section 3.7, Proposition 3.44, pp. 26-29], for any $N\in \mathbb N$\,, the sheaf
$$DO^{N}(\mathcal F_1,\mathcal F_2):=\ensuremath{\mathcal Hom}_X(\mathcal J^N(\mathcal F_1/k),\mathcal F_2)$$ is stricly larger then the sheaf $DO^{N-1}_X(\mathcal F_1,\mathcal F_2).$
 There is $M= M(\mathcal F_1,\mathcal F_2)$\, such that
 \begin{gather*}DO^{N}(\mathcal F_1,\mathcal F_2)(m)=DO^N(\mathcal F_1,\mathcal F_2(m))\quad \text{and}\\
  DO^{N-1}_{\mathcal O_X}(\mathcal F_1,\mathcal F_2)(m)=DO^{N-1}(\mathcal F_1,\mathcal F_2(m))\quad
     \text{and the quotient sheaf}\\
   \mathcal Q(m,N)= DO^N(\mathcal F_1,\mathcal F_2(m))/(DO^{N-1}(\mathcal F_1,\mathcal F_2(m))\\
   =(DO^N(\mathcal F_1,\mathcal F_2)/DO^{N-1}(\mathcal F_1,\mathcal F_2))(m)
   \end{gather*}  are nonzero,  globally generated for $m\geq M$\, and have no higher cohomology.\\
   The sheaf $\mathcal Q(m.N)$\,  is for $\mathcal F_i, i=1,2$\, locally free  and $X/k$\, smooth the sheaf
   $$ \mathcal Q(m,N):=\ensuremath{\mathcal Hom}_{\mathcal O_X}(\mathcal F,\mathcal F(m))\otimes S^N\mathcal T^1(X/k).$$  Then we get an exact sequence
 \begin{gather*}0\longrightarrow H^0(X, DO^{N-1}(\mathcal F_1,\mathcal F_2(m)))\longrightarrow H^0(X, DO^N(\mathcal F_1,\mathcal F_2(m)))\\
 \longrightarrow H^0(X, Q(m,N))\longrightarrow 0.
 \end{gather*}
 All cohomology groups are nonzero (because the sheaves are nonzero and globally generated) and thus there exists a global differential operator $D_N\in H^0(X,DO^N(\mathcal F,\mathcal F(m)))$\, not contained in $H^0(X,DO^{N-1}(\mathcal F,\mathcal F(m)))$\,.  Putting $N=0$\, and observing that $DO^0(\mathcal F,\mathcal F(m))=\ensuremath{\mathcal Hom}_{\mathcal O_X}(\mathcal F,\mathcal F(m))$\, we get the last claim.
 \end{proof}
 \begin{lemma}\mylabel{lem:L108} For each projective  morphism $f:X\longrightarrow S$\, there exists a locally free $\mathcal O_X$-module $\mathcal E$\, and a global differential operator $\mathcal E\longrightarrow \mathcal E$\, of arbitrary high order relative to $S$.
 \end{lemma}
 \begin{proof}  Let $\mathcal O_X(1)$\, be an $f$-ample invertible sheaf. By the previous proposition, there exists differential operators $D_{12}: \mathcal O_X(m)\longrightarrow \mathcal O_X(n+m)$\, for $n>>0$\, of arbitrary high order. Put $\mathcal E:=\mathcal O_X(m)\oplus \mathcal O_X(n+m)$\, and define $D$ to be given by the matrix of differential operators
 \[ D=
 \begin{pmatrix}  \Id_{\mathcal O_X(m)} & D_{1,2}\\
                  0 & \Id_{\mathcal O_X(m+n)}
                  \end{pmatrix}
                  \]
     Then $D$ is a differential operator on $\mathcal E$\,. The Harder-Narasimhan filtration on $\mathcal E$\, is the filtration $\mathcal HN^{\bullet}\mathcal E$\, with  $\mathcal HN^1(\mathcal E)=\mathcal O_X(n+m)$\, and $gr^2\mathcal HN^{\bullet}(\mathcal E)=\mathcal O_X(m)$\, and  the induced differential operators on the graded pieces are $\mathcal O_X$-linear. By taking direct sums of $\mathcal O_X(n_i)$, $n_i$\, appropriately choosen, we can manage to get the rank of $\mathcal E$\, arbitrarily high.
 \end{proof}
 We have the following
 \begin{proposition}\mylabel{prop:P30102} Let $q: X\longrightarrow S$\, be a proper morphism of noetherian schemes and $D: \mathcal E_1\longrightarrow \mathcal E_2$\, be a differential operator of coherent sheaves relative to $S$ of some order $\leq N$\,. Then, there are $\mathcal O_S$-linear maps
 $$R^iD: R^iq_*\mathcal E_1\longrightarrow R^iq_*\mathcal E_2.$$
 \end{proposition}
 \begin{proof} There is the standard factorization of $D$ as
 $$\mathcal E_1\stackrel{d^N_{\mathcal E_1/S}}\longrightarrow \mathcal J^N(\mathcal E_1/S)\stackrel{\widetilde{D}}\longrightarrow \mathcal E_2.$$
 The first map $d^N_{\mathcal E_1/S}$\, is $\mathcal O_X$-linear if the $\mathcal O_X$-bimodule $\mathcal J^N(X/S)$\, is considered as a coherent  $\mathcal O_X$-module via the second $\mathcal O_X$-module structure. For a  detailed  discussion of this, see \cite{Guenther2}[section 3.5, pp. 16-17, Lemma 3.27].  Hence, we get an $\mathcal O_S$-linear map
 $$R^iq_*\mathcal E_1\stackrel{R^iq_*d^N_{\mathcal E_1/S}}\longrightarrow R^iq_*(\mathcal J^N(\mathcal E_1/S)^{(2)}).$$ Regarding $\mathcal J^N(\mathcal E_1/S)$\, with its first $\mathcal O_X$-module-structure, we get an $\mathcal O_S$-linear map
 $$R^iq_*\widetilde{D}: R^iq_*(\mathcal J^N(\mathcal E_1/S)^{(1)})\longrightarrow R^iq_*\mathcal E_2.$$
 By \cite{Guenther2}[section 3.5, pp. 16-17, Lemma 3.27], we have
 $$\mathcal J^N(\mathcal E_1/S)=p_{1*}(\overline{\mathcal O_S\otimes_{\mathcal O_S}\mathcal E_1)}= p_{2,*}\overline{\mathcal O_X\otimes_{\mathcal O_S}\mathcal E_1},$$
  where $p_1,p_2: J^N(X/S)\longrightarrow X$\, are the two projection morphisms, which are morphisms of $S$-schemes and where $J^N(X/S)=\Spec_X\mathcal J^N(X/S)$.
 Since $q\circ p_1=q\circ p_2$\,  and $p_1$ and $p_2$\, are affine, we have
 \begin{gather*}
 R^iq_*\mathcal J^N(\mathcal E_1/S)^{(2)}=R^i(q\circ p_2)_*\overline{\mathcal O_X\otimes_{\mathcal O_S}\mathcal E_1}\\
 =R^i(q\circ p_1)_*\overline{\mathcal O_X\otimes_{\mathcal O_S}\mathcal E_1} =R^iq_*\mathcal J^N(\mathcal E_1/S)^{(1)}.
 \end{gather*}
 Thus, we can compose $R^iq_*d^N_{\mathcal E_1/S}$\, with the map $R^iq_*{\widetilde{D}}$\, to get the  required $\mathcal O_S$-linear map
 $R^iq_*D: R^iq_*\mathcal E_1\longrightarrow R^iq_*\mathcal E_2.$
\end{proof}
\subsection{Extensions of differential operators}
The basic question is the following: If
$$0\longrightarrow \mathcal E_1\longrightarrow \mathcal E\longrightarrow \mathcal E_2\longrightarrow 0$$
is an exact sequence of coherent sheaves on an $S$-scheme $X/S$\, and $D_1:\mathcal E_1\longrightarrow \mathcal E_1$\, and $D_2:\mathcal E_2\longrightarrow \mathcal E_2$\, are differential operators on $\mathcal E_1,\mathcal E_2$\,, respectively, relative to $S$, can one find a differential operator $D:\mathcal E\longrightarrow \mathcal E$\, on $\mathcal E$\, fitting the exact sequence. We  want to show that this is not always the case.
\begin{proposition}\mylabel{prop:P20} Let $X/S$\, be an integral scheme of finite type and
$$0\longrightarrow \mathcal E_1\longrightarrow \mathcal E\longrightarrow T\longrightarrow 0$$ be an exact sequence, where  $\mathcal E_1,\mathcal E$\, are locally free and $T$ is a torsion sheaf. Given a differential operator $D_1:\mathcal E_1\longrightarrow \mathcal E_1$, there is at most one differential operator $D:\mathcal E\longrightarrow \mathcal E$\,, extending $D_1$\,.
\end{proposition}
\begin{proof} Suppose, $D_1$\, is of order $N$\, and there  are two differential operators $D,D'$\, of order $\leq N'$\, with $N'\geq N$\, on $\mathcal E$\, extending $D_1$\,. Without loss of generality, we may assume that $N=N'$\,. Considering the difference $D-D'$\,, we may assume that $D_1=0$\,. We look for an $\mathcal O_X$-linear homomorpism
$$\widetilde{D}: \mathcal J^N(\mathcal E/k)\longrightarrow \mathcal E.$$ By assumption, $\widetilde{D}$\, restricted to $\mathcal J^N(\mathcal E_1/k)'\subset \mathcal J^N(\mathcal E/k)$\, is the zero homomorphism. But, by the right exactness of the jet module functor, the quotient module is the jet-module $\mathcal J^N(\mathcal E/\mathcal E_1/S)$\,. By \cite{Guenther2}[section 3.5, Lemma 3.30, p.18], this is a torsion sheaf, since $\mathcal E/\mathcal E_1\cong T$\, is so.
Thus $\widetilde{D}$\, factors through $\mathcal J^N(\mathcal E/k)/\mathcal J^N(\mathcal E_1/k)'\longrightarrow \mathcal E$\,. But the first sheaf is a torsion $\mathcal O_X$-module so $\widetilde{D}$\, must be the zero homomorphism.
\end{proof}
\begin{example} Let $(X,\mathcal O_X(H))$\, be a polarized projective scheme with $\mathcal O_X(H)$\, very ample and $H=\text{div}(s), s\in \Gamma(X,\mathcal O_X(H))$\, be a smooth section. Let $n\in \mathbb N$\, be choosen such that there exists a global differential operator $D_{12}: \mathcal O_X\longrightarrow \mathcal O_X(nH)$\,.
Consider the extension of locally free sheaves
$$0\longrightarrow \mathcal O_X\oplus \mathcal O_X(nH)\longrightarrow \mathcal O_X(H)\oplus \mathcal O_X((n+1)H)\longrightarrow \mathcal O_H(H)\oplus \mathcal O_H((n+1)H)\longrightarrow 0.$$ The global  $\mathcal O_X$-($\mathcal O_H$-linear) endomorphisms of the last sheaf contain the endomorphisms in diagonal form and thus
$$h^0(H, \Hom_H(\mathcal O_H(H)\oplus \mathcal O_H((n+1)H, \mathcal O_H(H)\oplus \mathcal O_H((n+1)H)))\geq 2.$$
By the previous proposition, the differential operator $D: \mathcal O_X\oplus \mathcal O_X(nH)\longrightarrow \mathcal O_X\oplus \mathcal O_X(nH)$\, constructed in \prettyref{lem:L108} has at most one extension to an operator $E$\, on $\mathcal O_X(H)\oplus \mathcal O_X((n+1)H)$\, so not every pair $(D, \phi)$\, where $\phi$\, is an $\mathcal O_H$-linear map on the right hand side extends.
\end{example}

\subsection{Differential operators and semistable sheaves}
Let $X/k$\, be a projective  algebraic scheme of dimension $n$ and $\mathcal O_X(H)=\mathcal O_X(1)$\, be an ample invertibel sheaf on $X$. Let $\mathcal E$\, be a $\mu$-semi-stable sheaf on $X$ with respect to the polarization $\mathcal O_X(1)$\,. We want to investigate the question under which circumstances there exists a global differential operator $\mathcal E\longrightarrow \mathcal E$\,. We first give a criterion, when the answer is always negative.
\begin{proposition}\mylabel{prop:P13} Let $X/k, \mathcal E,\, \mathcal O_X(H)$\, be as above and suppose, that for some $m_0\in \mathbb N$\,, there are effective  Cartier divisors $D_i, i=1,...,N$\, with $D_i\cdot H^{n-1}>0$\, and a homomorphism with torsion kernel
$$\bigoplus_{i=1}^{M(m)}\mathcal O_X(D_i)\longrightarrow \Omega^{(1)}(X/k)^{\otimes^s m}\,\,\forall m\geq m_0$$ Then , for every torsion free  coherent semistable sheaf $\mathcal E$,  every global differential operator $D: \mathcal E\longrightarrow \mathcal E$\,  has order $< m_0$\,. In particular if $m_0=1$\,, then every differential operator on $\mathcal E$\, is $\mathcal O_X$-linear. If $\mathcal E$\, is arbitrary torsion free, then each $D$ respects the $HN$-filtration and the differential operators on the graded pieces are $\mathcal O_X$-linear.
\end{proposition}
\begin{proof} Let $\mathcal E$\, be a torsion free semistable sheaf on $X$ and $D: \mathcal E\longrightarrow \mathcal E$\, be a differential operator corresponding to an $\mathcal O_X$-linear map: $\widetilde{D}: \mathcal J^N(\mathcal E/k)\longrightarrow \mathcal E$\, with $N$ minimal  with $N>m_0$\,. From the jet bundle exact sequence  $j^N_{X/S}(\mathcal E_1)$, we get a homomorphism of coherent sheaves
$$(\bigoplus_{i=1}^N\mathcal O_X(D_i))\otimes_{\mathcal O_X} \mathcal E\longrightarrow \Omega^{(1)}(X/k)^{\otimes_s^N}\otimes_{\mathcal O_X}\mathcal E\stackrel{f}\longrightarrow \mathcal E.$$
If $f$ is not the zero homomorphism, there must be a direct summand $\mathcal E(D_i)$\, such that the above homomorphism restricted to $\mathcal E(D_i)$\, gives a nonzero homomorphism $\mathcal E(D_i)\longrightarrow \mathcal E$\,. Then $\mathcal E(D_i)$\, is semistable and
$$\mu(\mathcal E(D_i))=\mu(E)+D_i\cdot H^{n-1}>\mu(\mathcal E).$$
 By \cite{Huybrechts-Lehn}[chapter 1, Proposition 1.2.7, p.11], this homomorphism must be zero. Thus, $\widetilde{D}$\, restricted to $\Omega^{(1)}(X/k)^{\otimes^s N}\otimes \mathcal E$\, is zero. By the same standard exact sequences for the jet modules $j^N_{X/S}(\mathcal E)$, $\widetilde{D}$\, factors over $\mathcal J^{N-1}(\mathcal E/k)$\,. This is in contradiction with the assumed minimality of $N$. Thus we must have had $N\leq m_0$\,.\\
To the last point, if $m_0=1$\, and $\mathcal E$\, is arbitrary torsion free, let $D: \mathcal E\longrightarrow \mathcal E$\, and  the Harder Narhasimhan filtration $HN^{\bullet}(\mathcal E)$ be given. Let $j$ be minimal such that $D(HN^1(\mathcal E))\subset HN^j(\mathcal E)$\, so we get $\overline{D}:gr^1HN(\mathcal E)\longrightarrow gr^jHN(\mathcal E)$\,, since by \cite{Guenther2}[section 3.6 Lemma 3.43, p.24], differential operators restricted to subsheaves and quotient sheaves are again differential operators). Argueing as above, we see that $j$ must be equal to $1$ and by the first part of the proposition, $\overline{D}$\, must be $\mathcal O_X$-linear. So we know that $D(HN^1(\mathcal E))\subseteq HN^1(\mathcal E)$. We get a differential operator $D': \mathcal E/HN^1(\mathcal E)\longrightarrow \mathcal E/HN^1(\mathcal E)$\, and we can continue by induction on the lenght of the $HN$-filtration, the start of the induction given by the first part of the proposition.
\end{proof}
\begin{proposition}\mylabel{prop:P14} Let $X/k$\, be a smooth projective algebraic scheme, $\mathcal O_X(H)$\, be a very ample invertible sheaf on $X$ and $\mathcal E$\, be a coherent sheaf on $X$. Suppose, $\Omega^{(1)}(X/k)$\, is globally generated, e.g., $X$ is an abelian variety. Then each differential operator $D: \mathcal E\longrightarrow \mathcal E$\, respects the Harder-Narasimhan-Filtration $HS^i(\mathcal E)$\,.
\end{proposition}
\begin{proof} Let $D:\mathcal E\longrightarrow \mathcal E$\, and  the Harder-Narhasimhan filtration $HN^{\bullet}(\mathcal E)$\, be given. Let $j$ be minimal such that $D(HN^1(\mathcal E))\subseteq HN^j(\mathcal E)$\,. Then, we get a nonzero differential operator $\overline{D}: HN^1(\mathcal E)\longrightarrow gr^jHN(\mathcal E)$\,.
If $\Omega^{(1)}(X/k)$\, is globally generated, so is each symmetric power. Let  $N\in \mathbb N$\, be minimal such that $\overline{D}$ factors over $\mathcal J^N(\mathcal HN^1(\mathcal E)/k)$. We get homomorphisms of $\mathcal O_X$-modules
$$(\bigoplus \mathcal O_X)\otimes HN^1\mathcal E\twoheadrightarrow \Omega^{(1)}(X/k)^{\otimes^sN}\otimes_{\mathcal O_X}\mathcal E\longrightarrow gr^jHN(\mathcal E),$$
Restricting to each single direct summand $\mathcal O_X\otimes HN^1(\mathcal E)\cong HN^1(\mathcal E)\longrightarrow gr^jHN(\mathcal E)$\, the standard arguement shows, that this map must be zero unless $j=1$. Since this homomorphism is then zero for each direct summand,  so is the homomorphism
$$\Omega^{(1)}(X/k)^{\otimes^s N}\otimes_{\mathcal O_X}HN^1(\mathcal E)\longrightarrow gr^jHN(\mathcal E)$$  But then $\overline{D}$\, factors through $\mathcal J^{N-1}(HN^1(\mathcal E)/k)$\,, a contradiction. Thus, $j=1$\,, we take the quotient differential operator $D_1: \mathcal E/HN^1(\mathcal E)\longrightarrow \mathcal E/HN^1(\mathcal E)$\, and go on by induction.
\end{proof}
\begin{remark}\mylabel{rem:R2412} The same arguement shows that it suffices to assume that $\Omega^{(1)}(X/k)$\, is almost globally generated, i.e. that there is a homomorphism $\bigoplus \mathcal O_X\longrightarrow \Omega^{(1)}(X/k)$\,, with torsion cokernel.
\end{remark}
\begin{proposition}\mylabel{prop:P60} Let $(X,\mathcal O_X(1))$\, be a  smooth polarized projective variety over a field $k$ of characteristic zero and  suppose that $\mu_{min}(\Omega^{(1)}(X/k))\geq 0$\,. If $\mathcal E$\, is an arbitrary torsion free sheaf on $X$, then each differential operator $D:\mathcal E\longrightarrow \mathcal E$\, respects the $HN$-filtration, and  if $\mu_{min}(\Omega^{(1)}(X/k))>0$\,, the differential operator $\overline{D}_i: gr^iHN(\mathcal E)\longrightarrow gr^iHN(\mathcal E)$\, is $\mathcal O_X$-linear.
\end{proposition}
\begin{proof} The arguement is similar to the proof of the previous proposition.  Observe first, that if $\mu_{min}(\Omega^{(1)}(X/k))>0$\, then also $\mu_{min}(\Omega^{(1)}(X/k)^{\otimes^s m})>0$\,. Here we need that the characteristic of $k$\, is zero, because we need that the symmetric tensor product of semi-stable sheaves is again semistable. Now,  let $\mathcal E$\, be torsion free and $D: \mathcal E\longrightarrow \mathcal E$\,be given.  Let again $j$\, be minimal such that $D(HN^1(\mathcal E))\subset HN^j(\mathcal E)$\, so we get $\overline{D}: HN^1(\mathcal E)\longrightarrow gr^jHN(\mathcal E)$\,. Let $N\in \mathbb N$\, be minimal such that $\overline{D}$ factors over $\mathcal J^N(\mathcal HN^1(E)/k)$\,. We then get a nonzero homomorphism
$$\Omega^{(1)}(X/k)^{\otimes^s N}\otimes_{\mathcal O_X}HN^1(\mathcal E)\longrightarrow gr^jHN(\mathcal E).$$
We show by induction that this homomorphism is zero if $j>1$\, and $N>1$\,. Let $HN^{\bullet}(\Omega^{(1)}(X/k)^{\otimes^s N})$\, be the Harder-Narasimhan-filtration. Let $\Omega_{1,N}=HN^1$\, be the maximal destabilizing subsheaf. We get by restriction a homomorphism
$$\Omega_{1,N}\otimes HN^1(\mathcal E)\longrightarrow gr^j\mathcal HN^{\bullet}(\mathcal E).$$
$\Omega_{1,N}\otimes HN^1(\mathcal E)$ is semistable and
\begin{gather*}\mu(\Omega_{1,N}\otimes HN^1(\mathcal E))=\mu(\Omega_{1,N})+\mu(\mathcal HN^1(\mathcal E))\\
>\mu_{min}(\Omega^{(1)}(X/k)^{\otimes^s N})+\mu(HN^1(\mathcal E))>\mu_j(\mathcal E),
\end{gather*} so by \cite{Huybrechts-Lehn}[chapter 1, Proposition 1.2.7, p.11], this homomorphism is zero. At this point, we also need that the characteristic is zero since we need the semi stability  of the tensor product. Suppose that for some $n\in \mathbb N$\, the restricted homomorphism
$$HN^n(\Omega^{(1)}(X/k)^{\otimes^s N})\otimes HN^1(\mathcal E)\longrightarrow gr^jHN(\mathcal E)$$ is nonzero. We then get a homomorphsim
$$\Omega_{n+1,N}\otimes \mathcal HN^1(\mathcal E):=gr^{n+1}HN^{\bullet}(\Omega^{(1)}(X/k)^{\otimes^s N})\otimes HN^1(\mathcal E)\longrightarrow gr^jHN(\mathcal E)$$
and the same as the previous arguement shows, since $\mu(\Omega_{n+1,N})>\mu_{min}>0$\, that this map must be zero if $j>1$. Thus, the entire map $\Omega^{(1)}(X/k)^{\otimes^s N}\otimes HN^1(\mathcal E)\longrightarrow gr^jHN(\mathcal E)$\, is zero. Since $N$ was choosen minimal, we get by the $N^{th}$ exact sequence of jet bundles a contradiction. Thus $j=1$\,. We get a quotient differential operator $D_1: \mathcal E/HN^1(\mathcal E)\longrightarrow \mathcal E/HN^1(\mathcal E)$\, and we argue by induction on the lenght of the HN-filtration.\\
 To the last point, if $\mathcal E$\, is semistable and a differential operator $D:\mathcal E\longrightarrow \mathcal E$\, is given, choose again $N$ minimal such that $D$ factors over $\mathcal J^N(\mathcal E/k)$\, and show, as in the first part of this proof, that the homomorphism
$$\Omega^{(1)}(X/k)^{\otimes^s N}\otimes \mathcal E\longrightarrow \mathcal E$$
is the zero map if $N>1$\,.
\end{proof}
\begin{remark}\mylabel{rem:R200} By \cite{Miyaoka}[Lecture III, 2.14 Theorem, p. 67], the condition that \\
 $\mu_{min}(\Omega^{(1)}(X/k))\geq 0$\, is equivalent to the uniruledness of $X$.
\end{remark}
\subsection{Vector bundles with global differential operators on projective $n$-space}
\begin{proposition}\mylabel{prop:P67} Let $X=\mathbb P_k^n$\, be projective $n$-space with $n\geq 2$. Then for each  pair of locally free sheaves $\mathcal F_i, i= 1,2,$ there exists an $N=N(\mathcal F_i)$\, such that for each $N\geq N(\mathcal F_i)$ there exist   operators $D: \mathcal F_1\longrightarrow \mathcal F_2$\,  of order $\geq N$\,.
\end{proposition}
\begin{proof} The tangent sheaf $\mathcal T^1(\mathbb P^n_k/k)$\, is ample . For each $N\in \mathbb N$, we consider the exact sequence
$$j^N(\mathcal F_1): 0\longrightarrow \Omega^{(1)}(X/k)^{\otimes^s N}\otimes_{\mathcal O_X}\mathcal F_1\longrightarrow \mathcal J^N(\mathcal F_1/k)\longrightarrow \mathcal J^{N-1}(\mathcal F_1/k)\longrightarrow 0.$$ Taking duals and tensoring with $\mathcal F_2$\, we get exact sequences
\begin{gather*}do^N(\mathcal F_1,\mathcal F_2): 0\longrightarrow DO^{N-1}(\mathcal F_1,\mathcal F_2)\stackrel{i_N}\longrightarrow DO^N(\mathcal F_1,\mathcal F_2)\\
\stackrel{\sigma^N}
\longrightarrow \mathcal T^1(\mathbb P^n_k/k)^{\otimes^s N}\otimes Hom_{\mathcal O_X}(\mathcal F_1,\mathcal F_2)\longrightarrow 0.
\end{gather*}
We have $DO^0(\mathcal F_1,\mathcal F_2)\cong Hom_{\mathcal O_X}(\mathcal F_1,\mathcal F_2)$\,. The claim is that for $N>>0$\,
$$H^0(\mathbb P^n_k, DO^N(\mathcal F_1,\mathcal F_2))\supsetneq H^0(\mathbb P^n_k, DO^{N-1}(\mathcal F_1,\mathcal F_2)\supsetneq H^0(\mathbb P^n_k, Hom_{\mathcal O}(\mathcal F_1,\mathcal F_2)).$$
By \cite{Lazarsfeld}[II, chapter 6.1.B, Theorem 6.1.10, p.11] for each coherent sheaf $\mathcal E$\, on $\mathbb P^n_k$\,  and for all $m\geq m(\mathcal E)$\, we have $H^i(\mathbb P^n_k, S^m\mathcal T^1(\mathbb P^n_k)\otimes \mathcal E)=0\,\,\forall i>0$\, and the sheaf $S^m\mathcal T^1(\mathbb P^n_k)\otimes \mathcal E$\, is globally generated. Putting $\mathcal E=Hom_{\mathcal O_X}(\mathcal F_1,\mathcal F_2)$ and $i=1$\, we get
$$H^1(\mathbb P^n_k, S^m\mathcal T^1(\mathbb P^n_k/k)\otimes Hom_{\mathcal O_X}(\mathcal F_1,\mathcal F_2))=0\,\, \forall m\geq m(\mathcal E).$$
Writing out the long exact cohomology sequences for the exact sequence $do^N(\mathcal F_1,\mathcal F_2), N\geq m(\mathcal E)$\,, we get
\begin{gather*} 0\longrightarrow H^0(\mathbb P^n_k, DO^{N-1}(\mathcal F_1,\mathcal F_2))\longrightarrow H^0(\mathbb P^n_k, DO^N(\mathcal F_1,\mathcal F_2))\\
\longrightarrow H^0(\mathbb P^n_k, S^N\mathcal T^1(\mathbb P^n_k/k)\otimes Hom_{\mathcal O_X}(\mathcal F_1,\mathcal F_2))\\
\longrightarrow H^1(\mathbb P^n_k, DO^{N-1}(\mathcal F_1,\mathcal F_2)))\longrightarrow  H^1(\mathbb P^n_k, DO^N(\mathcal F_1,\mathcal F_2))\\
\longrightarrow H^1(\mathbb P^n_k, S^N\mathcal T^1(\mathbb P^n_k/k)\otimes Hom_{\mathcal O_X}(\mathcal F_1,\mathcal F_2))=0.
\end{gather*}
 The $k$-linear maps $H^1(\mathbb P^n_k, i_N)$\, are for $N>m(\mathcal F_i)$\, surjective. Thus, there must be an $N(\mathcal F_i)\geq m(\mathcal F_i)$\, such that $H^1(\mathbb P^n_k,i_N)$\, are isomorphisms for $N\geq N(\mathcal F_i)$\,. Looking into the long exact cohomology sequence, we see that for such $N$, the maps
 $$H^0(\mathbb P^n_k, DO^N(\mathcal F_1,\mathcal F_2))\longrightarrow H^0(\mathbb P^n_k, S^N\mathcal T^1(\mathbb P^n_k/k)\otimes_{\mathcal O_X}Hom_{\mathcal O_X}(\mathcal F_1,\mathcal F_2))\longrightarrow 0$$ are surjective.  The $k$-vector space dimension $h^0(\mathbb P^n_k, S^N\mathcal T^1(\mathbb P^n_k/k))$\, is always greater than zero for $N>>0$\,, because by \cite{Lazarsfeld}[II, chapter 6.1.B, Theorem 6.1.10, p.11] the sheaf $S^N\mathcal T^1(\mathbb P^n_k/k)\otimes Hom_{\mathcal O_X}(\mathcal F_1,\mathcal F_2)$\, is globally generated for $N>>0$\,. So there are elements in $H^0(\mathbb P^n_k, DO^N(\mathcal F_1,\mathcal F_2))$\, that do not lie in $H^0(\mathbb P^n_k, DO^{N-1}(\mathcal F_1,\mathcal F_2))$\,. So for $N>>0$\,, we have strict inclusions
 $$H^0(\mathbb P^n_k, \Hom_{\mathcal O_X}(\mathcal F_1,\mathcal F_2))\subsetneq H^0(\mathbb P^n_k, DO^{N-1}(\mathcal F_1,\mathcal F_2))\subsetneq H^0(\mathbb P^n_k, DO^{N}(\mathcal F_1,\mathcal F_2)).$$
\end{proof}
\begin{remark}\mylabel{rem:R100} By global generation  of the sheaf $DO^N(\mathcal F_1,\mathcal F_2)$\, after a sufficiently high ample twist, on gets immediately differential operators $\mathcal E\longrightarrow \mathcal E(n)$\, for $n>>0$\,. This result implies, that for each $n>0$\, there exist differential operatos of arbitrary high degree $\mathcal E\longrightarrow \mathcal E(-n)$\,.
\end{remark}
 Using the ampleness of $\mathcal T^1(\mathbb P^n/k)$ we  now want to  estimate the growth of $DO^N(\mathcal F_1,\mathcal F_2)$\, as $N$\, tends to infinity.\\
 Let $\mathcal E_1$\, and $\mathcal E_2$\, be locally free sheaves on $\mathbb P^n_k$\,, $n\geq 2$\, be given. By the ampleness of $\mathcal T^1_{\mathbb P^n_k/k}$\,, we know that for some $N\in \mathbb N_0$\, and all $d\geq N$, we have that $H^0(\mathbb P^n_k, S^d\mathcal T^1(\mathbb P^n/k)\otimes \Hom_{\mathcal O}(\mathcal E_1,\mathcal E_2))$\, is globally generated and all higher cohomology groups vanish. As we have shown, there is $M\in \mathbb N$\,, that only depends on $\mathcal E_1,\mathcal E_2$\, such that the sequence
 \begin{gather*}0\longrightarrow H^0(\mathbb P^n_k, DO^{d-1}(\mathcal E_1,\mathcal E_2))\longrightarrow H^0(\mathbb P^n_k, DO^d(\mathcal E_1,\mathcal E_2))\\
 \longrightarrow H^0(\mathbb P^n_k, S^d\mathcal T^1\otimes \Hom_{\mathcal O_{\mathbb P^n_k}}(\mathcal E_1,\mathcal E_2))\longrightarrow 0
 \end{gather*}
 is exact for all $d\geq M$\,. For $d>>M$\, we can thus write for $h^0(\mathbb P^n_k, DO^d(\mathcal E_1,\mathcal E_2))$\,
 \begin{gather*}h^0(\mathbb P^n_k, DO^d(\mathcal E_1,\mathcal E_2))= h^0(\mathbb P^n_k, DO^M(\mathcal F_1,\mathcal F_2))\\
 + \sum_{M+1\leq k\leq d}h^0(\mathbb P^n_k, S^k\mathcal T^1(\mathbb P^n_k/k)\otimes \Hom_{\mathcal O_{\mathbb P^n}}(\mathcal E_1,\mathcal E_2))=\\
  h^0(\mathbb P^n_k, DO^M(\mathcal F_1,\mathcal F_2))+\sum_{M+1\leq k\leq d}\chi(\mathbb P^n_k,S^k\mathcal T^1(\mathbb P^n_k/k)\otimes \Hom_{\mathcal O}(\mathcal E_1,\mathcal E_2)).
 \end{gather*}
 If we tensor the symmetric power of the Euler sequence ($\mathbb P^n_k=\mathbb P(V)$)
 $$0\longrightarrow S^{k-1}V\otimes \mathcal O_{\mathbb P}(-1)\longrightarrow S^k(V)\otimes \mathcal O_{\mathbb P}\longrightarrow S^k\mathcal T^1(\mathbb P^n/k)(-k)\longrightarrow 0,\quad\text{or}$$
 $$0\longrightarrow S^{k-1}V\otimes \mathcal O(k-1)\longrightarrow S^k(V)\otimes \mathcal O(k)\longrightarrow S^k\mathcal T^1(\mathbb P^n_k/k)\longrightarrow 0$$
 (see \cite{Huybrechts-Lehn}[chapter 1.4, p. 19]) with $\Hom_{\mathcal O}(\mathcal E_1,\mathcal E_2)$\, and use the additivity of the Euler characteristic, we get
 \begin{gather*}\chi(\mathbb P^n_k, S^k\mathcal T^1(\mathbb P^n_k/k)\otimes_{\mathcal O}\Hom_{\mathcal O}(\mathcal E_1,\mathcal E_2))=\chi(\mathbb P^n_k, S^k(V)\otimes \mathcal O(k)\otimes \Hom_{\mathcal O}(\mathcal E_1,\mathcal E_2))\\
 - \chi(\mathbb P^n_k, S^{k-1}V\otimes \mathcal O(k-1)\otimes \Hom_{\mathcal O}(\mathcal E_1,\mathcal E_2))=\\
 \binom{n+k}{k}\cdot \chi(\mathbb P^n_k, \Hom_{\mathcal O}(\mathcal E_1,\mathcal E_2(k)))
 - \binom{n+k-1}{k-1}\cdot \chi(\mathbb P^n, \Hom_{\mathcal O}(\mathcal E_1,\mathcal E_2(k-1))).
 \end{gather*}
  Summing over $M+1\leq k\leq d$\, we get for $d>>M$\,
 \begin{gather*}h^0(\mathbb P^n_k, DO^d(\mathcal E_1,\mathcal E_2))=h^0(\mathbb P^n_k, DO^M(\mathcal E_1,\mathcal E_2))\\
 +\sum_{M+1\leq k\leq d}(\binom{n+k}{k}\cdot \chi(\mathbb P^n_k, \Hom_{\mathcal O}(\mathcal E_1,\mathcal E_2(k)))\\
 -\binom{n+k-1}{k-1}\cdot\chi(\mathbb P^n_k, \Hom_{\mathcal O}(\mathcal E_1,\mathcal E_2(k-1))))\\
 =
 h^0(\mathbb P^n_k, DO^M(\mathcal E_1,\mathcal E_2))\\
 +\binom{n+d}{d}\cdot \chi(\mathbb P^n_k, \Hom_{\mathcal O}(\mathcal E_1,\mathcal E_2(d)))-\binom{n+M}{M}\cdot\chi(\mathbb P^n_k, \Hom_{\mathcal O}(\mathcal E_1,\mathcal E_2(M))).
 \end{gather*}
 We thus have proved the following
  \begin{proposition}\mylabel{prop:P2612} Let $\mathcal E_1,\mathcal E_2$ be locally free sheaves on $\mathbb P^n_k, n\geq 2$\, . Let $F(N): = h^0(\mathbb P^n_k, DO^N(\mathcal E_1,\mathcal E_2)).$ There is a polynomial of degree $2n$, $P(N) =P_{(\mathcal E_1,\mathcal E_2)}(N)$\, such that for $N>>0$\,
 $$F(N)= P(N).$$
 The polynomial $P(N)$\, is  explicitely given by
 \begin{gather*}P(N):= \binom{n+N}{N}\cdot \chi(\mathbb P^n_k, \Hom_{\mathcal O}(\mathcal E_1,\mathcal E_2)(N))+\\
 h^0(\mathbb P^n_k, DO^M(\mathcal E_1,\mathcal E_2)) - \binom{n+M}{M}\cdot \chi(\mathbb P^n_k, \Hom_{\mathcal O}(\mathcal E_1,\mathcal E_2)(M)),
 \end{gather*}
 where $M=M(\mathcal E_1,\mathcal E_2)$\, is a fixed natural number.
 \end{proposition}
 \begin{proof}
 \end{proof}
 For the special case where $\mathcal E_1=\mathcal O(m_1)$\, and $\mathcal E_2=\mathcal O(m_2)$\,, the formula then reads,
 \begin{gather*}h^0(\mathbb P^n_k, DO^d(\mathcal O(m_1),\mathcal O(m_2)))= h^0(\mathbb P_k^n, DO^M(\mathcal O(m_1), \mathcal O(m_2)))\\
 + \binom{n+d}{d}\cdot \chi(\mathbb P^n, \mathcal O(m_2-m_1+d))-\binom{n+M}{M}\cdot \chi(\mathbb P^n, \mathcal O(m_2-m_1+M)).
 \end{gather*}
 \subsection{Elliptic operators in algebraic geometry}
\begin{definition}\mylabel{def:D13113}(Definition of the symbol of a differential operator)\\
Let $X/k$\, be a smooth variety and $D: \mathcal E_1\longrightarrow \mathcal E_2$\, be a differential operator of order $N\in \mathbb N$\,. Consider the standard exact sequence
\begin{gather*}do^N(\mathcal E_1,\mathcal E_2): 0\longrightarrow DO^{N-1}_{(X/k)}(\mathcal E_1,\mathcal E_2)\longrightarrow DO^N_{(X/k}(\mathcal E_1,\mathcal E_2)\\
\stackrel{\sigma_N}\longrightarrow Hom_{\mathcal O_X}(\mathcal E_1,\mathcal E_2)\otimes_{\mathcal O_X}S^N\mathcal T^1(X/k).
\end{gather*}
The element $\sigma^N(D)\in Hom_{\mathcal O_X}(\mathcal E_1,\mathcal E_2)\otimes_{\mathcal O_X}S^N\mathcal T^1(X/k)$\, is called the symbol of the differential operator $D$.
\end{definition}
Recall, that if $\mathcal F$\, is a coherent sheaf on $X$ and $F: =\Spec_X\text{Sym}^{\bullet}(\mathcal F)$\, with projection $\pi: F\longrightarrow X$\, is the associated affine bundle, for each $N\in \mathbb N$\,, there is a canonical homomorphism
$c_{N,\mathcal F}:\quad\pi^*S^N\mathcal F\longrightarrow \mathcal O_F$\,.  Zariski locally, if $X=\Spec A$\, and $\mathcal F$\, corresponds to the $A$-module $M$\,, we have
$$\Gamma(\Spec A, \mathcal O_F)=\text{Sym}^{\bullet}M\quad  \text{and}\quad \pi^*S^N\mathcal F\mid_{\Spec A}=S^NM\otimes_A\text{Sym}^{\bullet}M$$ and the required map is simply the tensor multiplication map
$S^NM\otimes_A\text{Sym}^{\bullet}M\longrightarrow \text{Sym}^{\bullet}M.$\\
In our case, $\mathcal F=\mathcal T^1(X/k)$\, and $ T^{1*}(X/k):=\Spec_X\text{Sym}^{\bullet}\mathcal T^1_{X/k}$\, is the classical cotangent bundle of $X$ with projection $\pi_X: T^{1*}(X/k)\longrightarrow X$\,. Put $T'(X/S):=T^{1,*}(X/S)\backslash s_0(X)$\, where $s_0: X\hookrightarrow T^{1,*}(X/k)$\, is the zero section with same projection $\pi_X: T'(X/k)\longrightarrow X$\,.\\
 We have the following classical
\begin{definition}\mylabel{def:D2512}(Definition of an elliptic operator in algebraic geometry)\\
 With notation as  just introduced, let $X/k$\, be a smooth scheme of finite type over the base field $k$\, , $\mathcal E_1,\mathcal E_2$\, be locally free sheaves on $X$ and $D: \mathcal E_1\longrightarrow \mathcal E_2$\, be a differential operator of order $N$. The operator $D$ is called elliptic, if
 $$c_{N,\mathcal T^1}\circ \pi_X^*(\sigma^N(D)): \pi_X^*\mathcal E_1\longrightarrow \pi_X^*\mathcal E_2$$ is an isomorphism of locally free sheaves on $T'(X/S)$\,.
\end{definition}
This is just an adoptation of the classical definition of an elliptic operator in differential geometry to the setting of locally free sheaves. Observe that $\mathcal E_1$\, and $\mathcal E_2$\, must then  have the same rank.
\begin{proposition}\mylabel{prop:P13114} Let $X/k$\, be a smooth complete variety such that\\
 $\mu_{min}(\Omega^{(1)}(X/k))>0$\, with respect to some polarization $H$\,. Then, there is no locally free sheaf $\mathcal E$\, plus an elliptic differential operator $D: \mathcal E\longrightarrow \mathcal E$\, on $X$.
\end{proposition}
\begin{proof} The proof is relatively straight forward. Let $\mathcal E_1$\, be the maximal destabilizing subsheaf. By \prettyref{prop:P60} , $D$ respects the Harder Narasimhan-filtration and $D|_{\mathcal E_1}$\, is $\mathcal O_X$-linear.  The sheaves $\mathcal E_1$\, and $\mathcal E/\mathcal E_1$\, are torsion free and are locally free outside a codimension $2$ subset of $X$. So there exists a Zariski-open $\Spec A\subset X$\, such that $\mathcal E$\, and $\mathcal E_1$\, and $\mathcal E/\mathcal E_1$\,  are free on $X$. Furthermore, by shrinking $\Spec A$\,, we may assume that $\mathcal T^1(X/k)$\, is free on $\Spec A$\,. Choose a trivialization
\begin{gather*}\mathcal E|_{\Spec A}\cong A^{\oplus n}\cong A^{\oplus k}\oplus A^{\oplus l} \,\,\text{with}\,\,\\
\mathcal E_1|_{\Spec A}\cong A^{\oplus k}\,\,\text{and}\,\, \mathcal E/\mathcal E_1|_{\Spec A}\cong A^{\oplus l}.
\end{gather*}
The symbol $\sigma(D)$\, is given by an $n\times n$-matrix of $N^{th}$-order differential operators on $\Spec A$\,. But  the $(k\times k)$-submatrix that corresponds to $\sigma(D)|_{\mathcal E_1}$\, is the zero matix since $D$ restricted to $\mathcal E_1$\, is $\mathcal O_X$-linear. By definition of ellipticity, the symbol $\sigma(D)$\, can never be a fibrewise isomorphism outside the zero section.
\end{proof}
\begin{remark}This just says, that for "most" varieties, e.g., if $\Omega^{(1)}(X/k)$\, is ample, elliptic operators on locally free sheaves do not exist. So there is no hope to prove the existence of an algebraic Atiyah-Singer-Index theorem that is equivalent to the Hirzebruch-Riemann-Roch theorem.
\end{remark}
In order to give an example of an ellitic operator on a smooth complete variety, we prove
\begin{proposition}\mylabel{prop:P26113} Let $A/\mathbb C$\, be an abelian variety, or, more generally a complex torus. Then, there exist elliptic operators $\mathcal O_A\longrightarrow \mathcal O_A$\, of arbitrary high order.
\end{proposition}
\begin{proof} Let $A=\mathbb C^n/\Lambda$\, with $\Lambda$\, generated by the vectors $\lambda_j+i\mu_j, j=1,...,2n$\,. Then, if $D$ is a differential operator on $\mathbb C^n$\, with constant coefficents, then $D$ is obviously translation invariant, since $d^1z_i=d^1(z_i+c)$\, for all $c\in \mathbb C$\, and thus $D$ descends to a differential operator $D_A: \mathcal O_A\longrightarrow \mathcal O_A$\,.\\
Now, on $\mathbb C^n$\, on can construct for arbitrary $N>>0$\, elliptic operators with constant coefficients.
\end{proof}

redpajama_set_name: RedPajamaArXiv


This card is a little too busy for me. I think it's that striped background that is the problem.
The Noel is from the "Christmas Stockings" Thinlits dies. I cut the letters off from the line they were attached to. I had used the red glitter paper to do the letters.
The stockings were cut using the small stocking die and the Dazzling Diamonds glitter paper.
I set the letters in place with Dimensionals. They sit on a circle and a scalloped circle which are taped directly onto the decorative paper. That busy paper came from the "Presents & Pinecones" Designer Series Paper pack.
I added a strip of "Candy Cane Lane" Washi tape and tied a couple of pieces of Candy Cane Lane Baker's Twine around the card. And to finish it off I added a couple enamel dots.
A very simple, quick and busy card!!!


text:
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The devastating impact of Bt Cotton in Haryana.
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Re: Re: British Charity Funding Religious Extremism
Hate as a history lesson
Re: BEST WIISHES ON DIWALI FESTIWAL FROM JHARKHAND...
free vaccine camp in india
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Dear Doctors,
Please compare the situation to India where numerous deaths fail to rouse the authorities. Rather the person(s) raising questions are rebuked. As I had told a sympathetic doctor, "It is unfortunate that unless a very big tragedy happens, no change will take place in India".
Why have we become inured to the deaths and disability amongst children?
Recently I had paid a visit to a very big hospice in my city who were holding a free vaccine camp. I came to know that they were giving the Hep-B and the Typhoid vaccine. The free camp was being held to make the patients aware about other vaccines available and also to drive home the message that vaccines are the child's "birth right". The impression was given to me that the Hep-B vaccine is "mandatory" for both children and adults.
It is a pity that the doctor whom I was interviewing was not left alone but chaperoned by the Business Development Executive of the hospital.
When I started probing further I found that the deaths from the measles vaccine in India was brushed aside as a rumour. However the hospital was ready with adrenaline, oxygen, IV fluids and resuscitation kits in case of an emergency. Despite the local IAP chapter warning the hospitals not to involve in NGOs in vaccination camps, this hospital was taking the help of three NGOs to increase their outreach.
The business development executive was very proud of the fact that the vaccine camp had been widely advertised and they were expecting to vaccinate 750 children on a single day. He talked about getting government permission in the next camp so as to include vaccines given under the universal immunisation program.
To my question about vaccine storage they replied that the vaccines were being brought in by the manufacturers and not stored in the hospital.
I was distracted by the presence of the business executive and did not want the doctor to get into any kind of problem therefore I restricted my questions.
I have my reservations about the "free" part. As the vaccine camp was also being called a health check up camp would fees be charged and medicines prescribed?
Jagannath.
Teenager's death spurs vaccination questions, fears
http://www.kyivpost .com/nation/ 30691
Violations in vaccine testing procedures has many pointing the blame at Ukraine's Health Ministry
The death of a teenager from vaccination last spring is scaring Ukrainians and making them refuse immunizations. The trend is expected to increase as investigators keep uncovering violations in vaccine testing procedures, selecting people who need immunization and other bad practices. "The Health Ministry made fatal mistakes and violated laws," said Victor Korzh, head of the parliamentary commission investigating the case.
Anton Tyshchenko, a 17-year old from Kramatorsk, died from a measles vaccine in May this year. The serum was developed in India and apparently wasn't tested in Ukraine, according to the findings of two investigation commissions. They also found the boy did not actually need a vaccination at all because he had already been immunized twice.
Tyshchenko was vaccinated in school on May 12. A few hours later, he started feeling sick and developed a fever. He was hospitalized the same night, but died eight hours after the vaccine was injected. "There is a direct cause-effect relation between Anton Tyshchenko's death and his vaccination for measles and rubella," Prosecutor General Oleksandr Medvedko said in his interview with Fakty daily last month.
Tyshchenko was vaccinated with one of the nine million doses of Indian serum provided by UNICEF as humanitarian aid to Ukraine - part of a joint immunization campaign with the Health Ministry. The campaign has been put on hold until the investigation of this case is over.
Investigators said that they suspected at least six violations in the process of immunization. Medvedko said one of them was immunizing in the spring, when children's health is weak after the winter. Immediately after the immunization, Tyshchenko was sent to hike over five kilometers to the nearest clinic for a compulsory lung X-ray.
A parliamentary commission that conducted a parallel investigation said the serum used for Tyshchenko was not even registered or tested in Ukraine. However, 207 cases of complications have already been found, according to their reports. "There were numerous outrages during the additional vaccination campaign," said Korzh. The vaccine itself was banned from use in India, its country of origin, and was not used in any country within the European Union, Korzh said.
WHO and UNICEF, however, insisted there was nothing wrong with the vaccine. "The measles and rubella vaccine used in Ukraine is pre-qualified by WHO and produced in accordance with the highest international standards by the Serum Institute of India, the largest producer of measles and rubella vaccine globally," wrote Maryanna Zaichykova, a communications assistant at UNICEF Ukraine, in response to questions from the Kyiv Post. Zaichykova said that the septic shock that caused the teenager's death was "unrelated to immunization. "
The Rada investigative committee also discovered that the scale of immunization suggested by the Health Ministry is largely overstated. The Ministry expected to apply vaccine to nine million people aged 16-29, but experts said only 20 percent of these people need the jab. "For those people who have immunity, such immunization is not only harmful and dangerous for their health, but can also cause drastic consequences, " Korzh said. Tyshchenko was one of them because he had already received measles vaccine twice in previous years.
As more gruesome details about mass vaccination come out into public light, more Ukrainians are turning away from vaccinations. Some data suggests that the number of parents who refuse to vaccinate their children has grown two or three-fold this year, and even more in Kyiv.
Vyacheslav Kostylev, head of the League of Civil Rights Protection, a non-government organization, said the number of people who ask for legal help on how to refuse vaccinating their children is growing. Ukraine's law is contradictory: on the one hand, parents are required to present vaccination certificates for their children when they start school or kindergarten. On the other hand, the right to education is guaranteed by the Constitution and vaccination is the parents' free choice.
Kostylev, whose kids have not been vaccinated, avoided the conflict by sending them to a private kindergarten where the vaccination rules are not so strict. He said other parents prefer simply to pay for a fake certificate to avoid headaches. "They mostly buy certificates, because it is very hard to fight [with legal mechanisms], " he said.
The Health Ministry keeps insisting the nation needs mass immunization. "Refusal to vaccinate is a threat to many people's lives," said Mykola Prodanchuk, deputy health minister, who signed documents allowing the Indian vaccine into Ukraine. He said the number of people who require immunization is currently high enough to cause an outbreak of measles.
WHO is also pushing to resume a mass immunization campaign in Ukraine. "The global health partners urge the government to reaffirm its commitment to the WHO strategy to eliminate measles and rubella in Europe by 2010," its statement said.
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McDaniel's 28 points not enough as Michigan State ends PKI tournament 0-2
Kamaria McDaniel passes the ball through a lane of defenders during Michigan State's 109-44 victory over Florida A&M on November 17, 2022. Photo Credit: Sarah Smith/WDBM
Luca Melloni, Sports Editor
PORTLAND, Ore. – The Michigan State women's basketball team fell to No. 18 Oregon in the Phil Knight Invitational consolation game. Despite MSU guard Kamaria McDaniel scoring a game-high 28 points and seven assists, the Ducks played a strong match and earned an 86-78 win over the Spartans.
The all-time series between MSU and Oregon is now 3-1 after Sunday's game.
"Playing hard is one thing, competing is another. I thought this team competed today," head coach Suzy Merchant said. "We have a lot of new faces, and we're trying to figure ourselves out a little bit."
The Ducks improve their season record to 5-1, while the Spartans sit at 6-2 as they travel back to East Lansing.
McDaniel played a team-high 37 minutes and was influential in trying to get the Spartans back in the game. She also had three three-pointers to go along with her impressive individual performance.
"I'm trying to visualize myself being where I want to be and coach has instilled so much confidence in me," McDaniel said. "I'm going to continue putting the work in and continue being the best I can be."
McDaniel was the go-to point guard for the Spartans for the majority of the game.
The Spartans were close to taking the lead in the fourth quarter when sophomore guard Matilda Ekh tied the game at 68-68 with a fastbreak layup. Ekh returned to the lineup after missing the Nov. 24 game against No. 5 Iowa State due to her playing with the Swedish National team.
She flew directly to Portland on Friday after playing with the Swedish team and recorded 14 points against the Ducks.
Oregon guard Ahlise Hurst extended the Ducks' lead to double-digits with four minutes remaining in the fourth quarter. By that point, Michigan State did not have enough firepower to catch up.
"It was a good performance all the way," Oregon head coach Kelly Graves said. "Michigan State is a good team, man. Coach Suzy [Merchant] is one of my favorite coaches."
The Ducks went on to close the game with a series of free throws and a layup by freshman forward Grace VanSlooten. Senior guard Taya Hanson made the final two free throws of the game, solidifying the 86-78 victory.
The Spartans ended their PKI journey 0-2 against two ranked opponents.
"Our one-on-one accountability defensively has to continue to get better," Merchant said. "I learned a lot about our team and a lot about each one of these kids."
The Spartans return to action to face Georgia Tech in East Lansing. Tip-off at the Breslin Center is scheduled for 8 p.m. on Thursday, Dec. 1.
Luca is a senior journalism student and the sports editorial assistant at Impact 89FM. He is the men's and women's soccer beat reporter where he spent the last three seasons covering the two programs, as well as the men's basketball beat writer for the Impact. He is the host of the DeMartin Download podcast. Outside of Impact, Luca is the executive producer for Focal Point News at Michigan State University. Luca is also a news producer at WSYM FOX 47 in Lansing.
The State – 01/30/23
Exposure – Empowering Women in Law – 1/29/23
Coming Soon: More Than Coffee
All Impact 89FM | WDBM-FM Picks Reader Picks Sort: Newest


ALice Cooper describes how Jesus saved him from a premature 'rockers' death from alcoholism.
The Jihadist attack on Paris is actually an expression of the ancient resistance to God by fallen spiritual beings and their human followers.
Does Islam promote peace as say so many moderate muslims and many Western leaders. A broad review of the evidence..
Islamic jihad is about an intolerant and violent propagation of the Islamic faith and subjugation of resistant populations.
The world doesn't want Christian peace-makers, it wants Christian peace-keepers.


Tom Stienstra Nov. 29, 2018 Updated: Feb. 26, 2019 10:46 a.m.
The rain and wind this week has been pure sweetness for the lakes nestled across the foothills of the greater Bay Area.
Sweet, that is, as in the infusion of cool, fresh water, stirred by wind, that has refreshed dozens of lakes across the region that provide fishing and boating.
Between the Department of Fish and Wildlife, lake managers and park districts, 15 lakes in the greater Bay Area have been stocked with rainbow trout this month, crowned by the most fish at Los Vaqueros, Lake Chabot and Del Valle.
In addition, even Lake Merced North in San Francisco, left fallow for most of the year, was stocked twice this month, including this week.
Amid cool weather, occasional rain and the typical holiday commitments, many often overlook the late-fall and early-winter season. But it is the change of weather that jump-starts outdoor recreation at many lakes.
The spotlight, as in past years, is at 10 lakes in Alameda and Contra Costa counties that are receiving bonus trophy-size rainbow trout courtesy of purchases from daily fishing permits.
On the other end of the spectrum, the San Francisco Peninsula is again the black hole of fishing and boating this winter, with no boating or fish stocks at any lake in San Mateo County. In fact, south of Lake Merced in San Francisco, the nearest lake (without crossing a bridge) that is stocked for fishing is little Sandy Wool Lake at Levin Park in Milpitas. Sandy Wool is 50 miles away and across a span of about 3.5 million people who are blocked from seven lakes in their respective vicinities.
Lake Chabot, Castro Valley: Stocked twice in November, including this week, with 2,400 pounds of trout. Catch rates high from shore until storm runoff with muddy water, deeper into winter, puts the ki-bosh on fishing. This is the site of Bay Area lake-record bass, a 19-pounder; 87 percent full, 318 surface acres.
Del Valle Res., Livermore: Stocked three times in November, including this week, with 3,000 pounds of trout. The lake level is down, best estimate is about 60 percent full, and the boat ramp and dock are still functional, with the best fishing often in Swallow Bay; 750 acres, 5 miles long, 16 miles of shoreline.
Don Castro, Hayward hills: Stocked this week with 750 pounds of trout; 23 acres.
Quarry Lakes/Horseshoe Lake, Newark: Stocked twice in November, including last week, with 2,000 pounds of trout. Known for good shoreline fishing, landmark exchange program to prohibit use of lead weights; lakes complex is 350 acres.
Shadow Cliffs Lake, Pleasanton: Stocked last week with 1,000 pounds of trout. Former water quarry known for good water clarity even during heavy storm activity, and occasional giant bass, catfish and trout; 80 acres.
Trout plant updates: Department of Fish and Wildlife: Regional headquarters, 707-944-5581, https://nrm.dfg.ca.gov/FishPlants.
Contra Costa Water District: Los Vaqueros Marina, 925-371-2628; watershed information, www.ccwater.com/losvaqueros.
Marin Watershed: Fishing hotline, Sky Oaks Ranger Station, 415-945-1194, www.marinwater.org.
Rules for lakes that permit boating often can restrict boating to rentals and limits on gas engines, electric motors, speed, and type and size of craft, and some may require hand-launch boats only. Mussel inspections are required at most lakes that allow recreational boating of any type.
Note: Lake and marina managers, not an independent organization, verify and report these fishing records; over the years, The Chronicle has logged and tracked them.
2-hook rig: Tie a small snap swivel on your line. From the snap swivel, use 4-pound leader and about 18 inches of line to a No. 8 hook. Then add a separate line with 12 inches of leader to a second hook. On the bottom hook with the longer leader, with a worm threader, skewer a nightcrawler on the hook and line so it lies straight, and then on the top hook with the shorter leader, add a "Power Nugget" or "Mice Trail" for bait.
Troll: Tie a barrel swivel on your line and then add a 4- or 6-pound fluorocarbon leader, about 36 inches, to a snap swivel. For spoons, use silver-black or rainbow-colored Needlefish, Humdinger or similar. For plugs, use gold-black jointed Rapala or similar. Get speed and depth exactly right, often best accomplished with downrigger.
Lake Temescal, Oakland hills: Stocked twice in November, including this week, with 1,750 pounds of trout; 15 acres.
Boating: Del Valle, Chabot, Aquatic Park, Bethany, Elizabeth, Merritt, Quarry, Shadow Cliffs. No boating: Anza, Cull Canyon, Don Castro, Jordan Pond, Shinn Pond, Temescal. No access: Upper San Leandro (hiking by permit from EBMUD only), San Antonio Reservoir.
Contra Loma, Antioch hills: Stocked twice in November, including this week, with 1,750 pounds of trout. Local spot for shoreline baitdunkers; 70 acres.
Heather Farm Pond, Walnut Creek: DFW stocked twice in November, including last week; small lake in city park.
Hidden Lakes Park, Martinez: DFW stocked last week; small lake in city park.
Lafayette Res., Lafayette: DFW stocked twice in November, including last week; 67 percent full, 126 acres.
Los Vaqueros Res., Livermore: Stocked twice in November, including last week. When wind isn't howling, one of the best fishing lakes in California. Rich aquatic food chain and consistent trout stocks provide good fishing by boat and shore in South Cove, Cowboy Cove and other spots. Chance for big striped bass, catfish; 93 percent full, 1,500 acres, 62-degree surface temperature.
San Pablo Res., El Sobrante: 73 percent full, 800 surface acres. Closed for winter.
Boating: Contra Loma, Lafayette, Los Vaqueros, San Pablo. No boating: Briones. No access: Antioch Municipal Reservoir, Clifton Court Forebay.
Bon Tempe, Fairfax: DFW stocked this week and last week. Shoreline fishing ignites after a stock, then after three days, tapers off. The only realistic chance for trout fishing in Marin County; 82 percent full.
No boating: Alpine, Bon Tempe, Lagunitas, Kent, Phoenix, Nicasio, Stafford and Soulejule.
Lake Berryessa, Napa: No trout stocks this month. Berryessa provides trolling for holdovers from past plants, often at the mouth of Skier's Cove and the Narrows; 77 percent full, 20,700 acres.
Boating: Lake Berryessa, Lake Hennessey (Conn Dam). No access: Lake Curry, Milliken and Rector.
Lake Merced North: DFW stocked twice in November, including this week. San Francisco's backyard fishing hole is back on the map. Any type of boat can put you over the top, baitfishing along the edge of the tules just before they break into open shoreline; 650 acres.
Boating: Lake Merced North, Lake Merced South, Stow Lake.
Zilch: No stocks, boating or access to any of the reservoirs.
No boating: Waterdog Lake. No access: Upper Crystal Springs, Lower Crystal Springs, San Andreas Lake, Pilarcitos Lake.
Sandy Wool Lake, Milpitas: DFW stocked once this month.
Boating: As of this week, Calero (36 percent full), Lexington (28), Stevens Creek (19), plus little Shoreline Sailing Lake, are open to boating. Anderson and Coyote are closed for the season. All others are closed indefinitely to boating: Almaden, Arastradero, Campbell Perc Ponds, Chesbro, Cottonwood, Coyote, Cunningham, Grant, Guadalupe, Sandy Wool, Uvas and Vasona. No access: Boronda (Palo Alto residents only), Calaveras, Elsman, Felt.
Boating: Chabot Reservoir, Lake Solano. No access: Lake Madigan.
Ralphine Lake, Santa Rosa: DFW stocked three times in November, including this week.
Boating: Spring, Ralphine. No access: Laguna Lake.
Tom Stienstra is The San Francisco Chronicle's outdoors writer. His Outdoor Report can be heard at 7:35 a.m., 9:35 a.m. and 12:35 p.m. Saturdays on KCBS (740 and 106.9). Email: tstienstra@sfchronicle.com Twitter: @StienstraTom.


text:
NBA, union expected to agree to Dec. 22 start…
NBA, union expected to agree to Dec. 22 start to season, according to reports
The NBA's board of governors and players' association will reportedly hold separate meetings on Thursday that are expected to end with an agreement on starting the 2020-21 season on Dec. 22 and playing a reduced 72-game schedule. (Photo by Kevin C. Cox/Getty Images)
By Kyle Goon | kgoon@scng.com |
PUBLISHED: November 4, 2020 at 5:39 p.m. | UPDATED: November 4, 2020 at 5:40 p.m.
With hundreds of millions of dollars on the line, the NBA and the players' union are poised to approve a Dec. 22 start to the 2020-21 season as soon as Thursday, multiple media outlets reported.
The NBA has spent the last few weeks pitching a short turnaround between seasons, with a 72-game schedule that would include games on Christmas Day. ESPN and The New York Times reported that the league's board of governors and the National Basketball Players' Association were expected to vote on the proposed changes to the Collective Bargaining Agreement on Thursday after weeks of negotiations, and both bodies are expected to approve it.
At stake is an estimated $500 million to $1 billion for a league that is already expecting severe revenue shortfalls next year, due to the COVID-19 pandemic. With many arenas situated in cities that have yet to lift mass gathering restrictions, the NBA is on track to lose a significant chunk of its revenue through gate receipts, which has been reported as high as 40 percent.
The league is hoping to make up for those losses in part with a shorter break between seasons, one that will thrust the Lakers and Miami Heat into a new season just 72 days after Game 6 of the NBA Finals.
While it is likely to add stress to an already unusual offseason that will host free agency likely just days before players have to report to their markets (expected to be Dec. 1), the financial implications are widely seen to be too great to ignore.
Several Lakers, including veterans Danny Green and Jared Dudley, publicly expressed support for a later date. But as one agent told The Athletic of the early start: "We have to."
The presumed agreement would sidestep a nuclear option of ending the CBA and the league's revenue split with the players. The NBA and NBPA already have extended the deadline for an agreement four times, the last of these extensions landing on Friday. ESPN reported earlier this week that Commissioner Adam Silver told team owners that the league was running out of time to start the season in December if negotiations dragged on longer.
ESPN reported that the expedited timeline will mean the NBA's trade moratorium would be lifted after the changes to the CBA, followed by the NBA Draft on Nov. 18. The New York Times reported that free agency could begin two or three days following the draft.
One of the major factors that still has to be settled is the amount of player salary held in escrow. ESPN reported the league will hold an estimated 18 percent over the next two years to smooth out the revenue losses. The NBA typically holds 10 percent of player salary in escrow until the revenue split between players and owners is determined at the end of the season.
ESPN's Bobby Marks estimated $720 million to $800 million would be held in escrow under the presumed model for next season.
Lakers could face a season-long battle with complacency
Warriors knock off uninspired Lakers, who blow 19-point lead
Lakers prepare for schedule difficulty to ramp up
Dynasty in the making? Lakers see the pieces to pick up where the Warriors left off
While the title-winning Lakers will have to thrust quickly back into the fray with a roster of veterans, many other teams have had more normal offseasons, including the eight non-bubble teams that have not played since March 11, and non-playoff teams which were eliminated from the Orlando bubble by Aug. 15.
A fast-track offseason will also allow the NBA to begin getting back in sync with its traditional calendar year and finish before the start of the Tokyo Olympics in 2021, both insulating the league from more television competition (ratings dipped dramatically this postseason) and getting players and personnel back to their regular four-month offseason.
It's unclear how the NBA will schedule games and travel or play in markets that still restrict the number of people at gatherings. COVID cases have been rising nationally over the last month, and other sports leagues – particularly the NFL and college football – have struggled with playing games with regular travel in home markets due to positive tests.
Kyle Goon | Lakers reporter
Kyle Goon is a Maryland native who kept traveling west in search of stories. Kyle has been a Lakers beat reporter since 2018, and was one of a few reporters inside the NBA bubble in 2020. He previously worked at The Salt Lake Tribune covering the Utah Jazz and Utah Utes. He is a University of Maryland alum.
kgoon@scng.com
Follow Kyle Goon	@kylegoon
More in Lakers


Brent is a 21 year veteran of the industry and currently serves as the Sr. Director of Enterprise Fraud, Security, & Investigations at Ally Bank. In this role he is responsible for addressing fraud risk through establishment of the bank's fraud program, including fraud prevention and detection, intelligence and reconnaissance, organized ring investigations, and deployment of the machine-learning, behavioral fraud risk engine. Prior to this Brent served in various Audit, Information Security, IT Process & Delivery, Risk and Consulting roles across top US banks and external consulting firms. Starting in 2011 he contributed to building a global fraud penetration testing ("red team") service where he developed and executed external fraud control tests at several banks across US, Canada, and the UK. In this role he successfully identified and (legally) compromised hundreds of controls at client banks to demonstrate weaknesses across Consumer and Commercial banking businesses and their supporting channels, systems and processes. Following his career breaking into banks around the globe, he joined Ally to assist with building their Consumer Bank Information Security Program which, once established, led to a new role developing the Fraud Risk Programs for multiple business lines within Ally.


Our Psychotronic Summer series is taking us back to 1988 this week as we mark the 30th anniversary of Robert Zemeckis's WHO FRAMED ROGER RABBIT!
From its origins as a novel whose plot was changed drastically when adapted to film, to its innovative and techniques used to blend live action and animated characters, we discuss all aspects of this ground-breaking and beloved film.
Then we ask the big question: does it belong in the Psychotronic Hall of Fame?


The tales of a Louisiana expat in the cold, cold north. Told in a casual, conversational style, with musings, observations, some humor, some pictures, and huge dose of humility.
About me: I'm a 50 year old recent transplant to Finland. I had a mid-life transition to getting married again, leaving my (adult) children behind and moving to the cold, cold north after having lived most of my life in warm places like the Caribbean, Central American, Texas, Louisiana and Southern California. I'm enjoying this adventure quite a bit!


The economy, while still by no means great, is improving slowly but surely and as such, new opportunities are springing up all across the country. But where do you go and why?
Based on data from the U.S. Department of Labor and Adecco's own internal data on job growth and demand, we compiled a list of the top ten cities in America that are hiring RIGHT NOW. Oh yeah, you heard me–cool places to live with companies that are actively looking for new members. Believe it.
Would you relocate to any of these places for a job? Sound off in the comments!
I am willing to move to any city as long as it is a good opportunity and good pay.
I wish to work in one of the best Oil organizations that are why I choose UAE; there are plenty of organizations which offer great jobs in oil industry.


Beauty Icons: The Supremes | Top Beauty Blog in the Philippines covering Makeup Reviews, Swatches, Skincare and More!
"Great hair. Great Makeup. Great style — and of course, great voice. That is what The Supremes is all about."
Woe the Supremes they sure are icon.
Such great style. Thanks for sharing.
I didn't know them, but their hairstyles are so cool!