Global algebraic linear differential operators

TL;DR

This paper studies the existence and growth of global algebraic differential operators on projective schemes, especially projective space, and explores the non-existence of algebraic elliptic operators on most smooth varieties.

Contribution

It characterizes when global differential operators exist, proves their dimension grows polynomially with order, and introduces the concept of algebraic elliptic operators with non-existence results.

Findings

Global differential operators exist for large N on projective space.

Dimension of global operators grows polynomially with degree 2n.

Algebraic elliptic operators generally do not exist on most smooth varieties.

Abstract

In this note, we want to investigate the question, given a projective algebraic scheme X/k and coherent sheaves F, E on X, when do global differential operators of order N greater than zero, between E and F exist. We investigate in particular the case of projective n-space and prove that for large N always global differential operators between E and F exist. We also show that the dimension of global operators of order N grows like a polynomial in N of degree 2n. Finally, we define algebraic elliptic operators in the classical sense and show that for "most" algebraic smooth complete varieties, they do not exist on locally free sheaves.

Equations

$${\mathcal{J}}^N(M/A)=B{\otimes }_{A}M/I_{B/A}\cdot (B{\otimes }_{A}M),$$

$$D{O}_{X/S}^N({\mathcal{F}}_1,{\mathcal{F}}_2):={\mathrm{Hom}}_{{\mathcal{O}}_X}({\mathcal{J}}^N({\mathcal{F}}_1/S),{\mathcal{F}}_2).$$

$$j^N(\mathcal{E}/S):0⟶{\mathcal{I}}_{X/S}^N\mathcal{E}/{\mathcal{I}}_{X/S}^{N+1}\cdot \mathcal{E}⟶{\mathcal{J}}^N(\mathcal{E}/S)⟶{\mathcal{J}}^{N-1}(\mathcal{E}/S)⟶0.$$

$$\mathcal{E}{\otimes }_{{\mathcal{O}}_X}{\Omega }^{(1)}(X/S{)}^{{\otimes }^{s}N}\cong \mathcal{E}{\otimes }_{{\mathcal{O}}_X}(I_{X/S}/{\mathcal{I}}_{X/S}^2{)}^{{\otimes }^{s}N}$$

$$⟶{\mathcal{I}}_{X/S}^N/{\mathcal{I}}_{X/S}^{N+1}{\otimes }_{{\mathcal{O}}_X}\mathcal{E}⟶{\mathcal{I}}_{X/S}^N\cdot \mathcal{E}/{\mathcal{I}}_{X/S}^{N+1}\cdot \mathcal{E}$$

$$d{o}_{X/S}^N({\mathcal{E}}_1,{\mathcal{E}}_2):0⟶D{O}_{X/S}^{N_1}({\mathcal{E}}_1,{\mathcal{E}}_2)⟶D{O}_{X/S}^N({\mathcal{E}}_1,{\mathcal{E}}_2)$$

$$⟶{\mathrm{Hom}}_{{\mathcal{O}}_X}({\mathcal{E}}_1,{\mathcal{E}}_2){\otimes }_{{\mathcal{O}}_X}{\mathcal{T}}^1(X/S{)}^{{\otimes }^{s}N}⟶0,$$

$$D{O}^N({\mathcal{F}}_1,{\mathcal{F}}_2):=\mathcal{H}o{m}_X({\mathcal{J}}^N({\mathcal{F}}_1/k),{\mathcal{F}}_2)$$

$$D{O}^N({\mathcal{F}}_1,{\mathcal{F}}_2)(m)=D{O}^N({\mathcal{F}}_1,{\mathcal{F}}_2(m))\text{and}$$

$$D{O}_{{\mathcal{O}}_X}^{N-1}({\mathcal{F}}_1,{\mathcal{F}}_2)(m)=D{O}^{N-1}({\mathcal{F}}_1,{\mathcal{F}}_2(m))\text{and the quotient sheaf}$$

$$\mathcal{Q}(m,N)=D{O}^N({\mathcal{F}}_1,{\mathcal{F}}_2(m))/(D{O}^{N-1}({\mathcal{F}}_1,{\mathcal{F}}_2(m))$$

$$=(D{O}^N({\mathcal{F}}_1,{\mathcal{F}}_2)/D{O}^{N-1}({\mathcal{F}}_1,{\mathcal{F}}_2))(m)$$

$$\mathcal{Q}(m,N):=\mathcal{H}o{m}_{{\mathcal{O}}_X}(\mathcal{F},\mathcal{F}(m))\otimes S^N{\mathcal{T}}^1(X/k).$$

$$0⟶H^0(X,D{O}^{N-1}({\mathcal{F}}_1,{\mathcal{F}}_2(m)))⟶H^0(X,D{O}^N({\mathcal{F}}_1,{\mathcal{F}}_2(m)))$$

$$⟶H^0(X,Q(m,N))⟶0.$$

$$D=\left(\begin{array}{cc}{\mathrm{Id}}_{{\mathcal{O}}_X(m)}& D_{1,2}\\ 0& {\mathrm{Id}}_{{\mathcal{O}}_X(m+n)}\end{array}\right)$$

$$R^{i}D:R^i{q}_{\ast }{\mathcal{E}}_1⟶R^i{q}_{\ast }{\mathcal{E}}_2.$$

$${\mathcal{E}}_1\stackrel{d_{{\mathcal{E}}_1/S}^N}{⟶}{\mathcal{J}}^N({\mathcal{E}}_1/S)\stackrel{\tilde{D}}{⟶}{\mathcal{E}}_2.$$

$$R^i{q}_{\ast }{\mathcal{E}}_1\stackrel{R^i{q}_{\ast }{d}_{{\mathcal{E}}_1/S}^N}{⟶}{R}^i{q}_{\ast }({\mathcal{J}}^N({\mathcal{E}}_1/S{)}^{(2)}).$$

$$R^i{q}_{\ast }\tilde{D}:R^i{q}_{\ast }({\mathcal{J}}^N({\mathcal{E}}_1/S{)}^{(1)})⟶R^i{q}_{\ast }{\mathcal{E}}_2.$$

$${\mathcal{J}}^N({\mathcal{E}}_1/S)=p_{1\ast }(\overline{{\mathcal{O}}_S{\otimes }_{{\mathcal{O}}_S}{\mathcal{E}}_1)}=p_{2,\ast }\overline{{\mathcal{O}}_X{\otimes }_{{\mathcal{O}}_S}{\mathcal{E}}_1},$$

$$R^i{q}_{\ast }{\mathcal{J}}^N({\mathcal{E}}_1/S{)}^{(2)}=R^i(q\circ p_2{)}_{\ast }\overline{{\mathcal{O}}_X{\otimes }_{{\mathcal{O}}_S}{\mathcal{E}}_1}$$

$$=R^i(q\circ p_1{)}_{\ast }\overline{{\mathcal{O}}_X{\otimes }_{{\mathcal{O}}_S}{\mathcal{E}}_1}=R^i{q}_{\ast }{\mathcal{J}}^N({\mathcal{E}}_1/S{)}^{(1)}.$$

$$0⟶{\mathcal{E}}_1⟶\mathcal{E}⟶{\mathcal{E}}_2⟶0$$

$$0⟶{\mathcal{E}}_1⟶\mathcal{E}⟶T⟶0$$

$$\tilde{D}:{\mathcal{J}}^N(\mathcal{E}/k)⟶\mathcal{E}.$$

$$0⟶{\mathcal{O}}_X\oplus {\mathcal{O}}_X(nH)⟶{\mathcal{O}}_X(H)\oplus {\mathcal{O}}_X((n+1)H)⟶{\mathcal{O}}_H(H)\oplus {\mathcal{O}}_H((n+1)H)⟶0.$$

$$h^0(H,{\mathrm{Hom}}_H({\mathcal{O}}_H(H)\oplus {\mathcal{O}}_H((n+1)H,{\mathcal{O}}_H(H)\oplus {\mathcal{O}}_H((n+1)H)))\ge 2.$$

$$\underset{i=1}{\overset{M(m)}{⨁}}{\mathcal{O}}_X(D_i)⟶{\Omega }^{(1)}(X/k{)}^{{\otimes }^{s}m}\forall m\ge m_0$$

$$(\underset{i=1}{\overset{N}{⨁}}{\mathcal{O}}_X(D_i)){\otimes }_{{\mathcal{O}}_X}\mathcal{E}⟶{\Omega }^{(1)}(X/k{)}^{{\otimes }_s^N}{\otimes }_{{\mathcal{O}}_X}\mathcal{E}\stackrel{f}{⟶}\mathcal{E}.$$

$$\mu (\mathcal{E}(D_i))=\mu (E)+D_i\cdot H^{n-1}>\mu (\mathcal{E}).$$

$$(⨁{\mathcal{O}}_X)\otimes H{N}^1\mathcal{E}\twoheadrightarrow {\Omega }^{(1)}(X/k{)}^{{\otimes }^{s}N}{\otimes }_{{\mathcal{O}}_X}\mathcal{E}⟶g{r}^{j}HN(\mathcal{E}),$$

$${\Omega }^{(1)}(X/k{)}^{{\otimes }^{s}N}{\otimes }_{{\mathcal{O}}_X}H{N}^1(\mathcal{E})⟶g{r}^{j}HN(\mathcal{E})$$

$${\Omega }^{(1)}(X/k{)}^{{\otimes }^{s}N}{\otimes }_{{\mathcal{O}}_X}H{N}^1(\mathcal{E})⟶g{r}^{j}HN(\mathcal{E}).$$

Full text

After remembering the necessary foundational material, we prove one result in this direction, namely that on $\mathbb{P}^{n}_{k},\,\,n>1$, for each locally free sheaf $\mathcal{E}$  and each $n\in\mathbb{Z}$ , there exist global differential operators $\mathcal{E}\longrightarrow\mathcal{E}(n)$  of order $N$ , if $N\geq N(n)$  is sufficiently large. Also, we calculate the order of growth of differential operators of order $\leq N$ , as $N$  tends to infinity. As a final result, we prove, that, with the standard definition, for ”most” projective algebraic varieties, algebraic elliptic operators on a locally free sheaf do not exist.

Stefan Günther

1. 1 Notation and Conventions and Basic Definitions

2. 2 Introduction

3. 3 Global differential operators on coherent sheaves

4. 3.1 Extensions of differential operators

5. 3.2 Differential operators and semistable sheaves

6. 3.3 Vector bundles with global differential operators on projective $n$-space

7. 3.4 Elliptic operators in algebraic geometry

1 Notation and Conventions and Basic Definitions

Convention 1

By $\mathbb{N}$  we denote the natural numbers, by $\mathbb{N}_{0}$  the set of nonnegative integers.

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 $\partial^{\mid m\mid}/\partial\underline{x}^{\underline{m}}(f(x_{1},...,x_{m}))$ .

Notation 1

Let $X\longrightarrow S$  be a morphism of schemes. By $\Omega^{(1)}(X/S)$  we denote the usual sheaf of Kähler differentials. The ideal sheaf of the relative diagonal $\Delta_{X/S}\hookrightarrow X\times_{S}X$  we denote by $\mathcal{I}_{X/S}$ . Thus $\Omega^{(1)}(X/S)\cong\mathcal{I}_{X/S}/\mathcal{I}_{X/S}^{2}.$

We collect some basic facts about jet bundles and differential operators. Proofs and more detailed information can be found in [1].

Facts 1.1

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 $-mathcalE$-, 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 $\operatorname{Spec}B\longrightarrow\operatorname{Spec}A$  is the restriction of $q$ to Zariski-open subsets and $\mathcal{E}$  restricted to the $\operatorname{Spec}B$  corresponds to the $B$-module $M$, $\mathcal{J}^{N}(\mathcal{E}/S)$  is given by

[TABLE]

*where $I_{B/A}$  is the ideal in $B\otimes_{A}B$  which is the kernel of the multiplication map. If $\mathcal{E}^{\prime}\subset\mathcal{E}$  is a quasi coherent subsheaf, we denote by $\mathcal{J}^{N}(\mathcal{E}^{\prime}/S)^{\prime}$  the image of the homomorphism $\mathcal{J}^{N}(i/S)$ , where $i:\mathcal{E}^{\prime}\hookrightarrow\mathcal{E}$  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 $J^{N}(X/S):=\operatorname{Spec}_{X}\mathcal{J}^{N}(\mathcal{O}_{X}/S)$ , 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 $d^{N}_{\mathcal{O}_{X}/S}$ .*

Proposition 1.2

1. 1

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 $\mathcal{J}^{N}(-/S)\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}p_{2}^{*}(-).$ 2. 2

If $\pi:X\longrightarrow S$  is flat, then $\mathcal{J}^{\mathbb{N}}(-/S)$  is an exact functor. 3. 3

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 $(p^{N}_{2})^{*}$ .

Proof.

(see [1][section 3.5, Proposition 3.33, p.19]. ∎

Definition 1.3

*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 $\mathcal{F}_{1}\stackrel{{\scriptstyle d^{N}_{\mathcal{F}/S}}}{{\longrightarrow}}\mathcal{J}^{N}(\mathcal{F}_{1}/S)$  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 $\leq N-1$ .*

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 $\mathcal{J}^{N}(\mathcal{F}_{1}/S)\longrightarrow\mathcal{F}_{2}$ .

Facts 1.4

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

[TABLE]

*If $\mathcal{F}_{1},\mathcal{F}_{2}$  are coherent, so is $DO^{N}(\mathcal{F}_{1},\mathcal{F}_{2})$ .

If $S=\operatorname{Spec}k$  is a point, we use the simplified notation $DO^{N}(\mathcal{F}_{1},\mathcal{F}_{2})$ .*

Facts 1.5

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

[TABLE]

If $\mathcal{E}$  is locally free and the morphism $q:X\longrightarrow S$  is smooth, the natural homomorphism

[TABLE]

is an isomorphism, which follows from the fact, that in this case $\Delta_{X/S}:X\hookrightarrow X\times_{S}X$  is a regular embedding.

Notation 2

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

[TABLE]

which is $\operatorname{Hom}_{\mathcal{O}_{X}}(j^{N}(\mathcal{E}_{1}/S),\mathcal{E}_{2}).$

2 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 Proposition 3.11), 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 Lemma 3.1). 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 [1][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 Proposition 3.9, 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 $gr^{i}HN(\mathcal{E})\longrightarrow gr^{i}HN(\mathcal{E})$  is $\mathcal{O}_{X}$-linear (see Proposition 3.9).

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 [5] , 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.

3 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.

Lemma 3.1

*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.*

Proof.

By [1][section 3.7, Proposition 3.44, pp. 26-29], for any $N\in\mathbb{N}$ , the sheaf

[TABLE]

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

[TABLE]

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

[TABLE]

Then we get an exact sequence

[TABLE]

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 $DO^{0}(\mathcal{F},\mathcal{F}(m))=\mathcal{H}om_{\mathcal{O}_{X}}(\mathcal{F},\mathcal{F}(m))$  we get the last claim. ∎

Lemma 3.2

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 $S$.

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

[TABLE]

Then $D$ is a differential operator on $\mathcal{E}$ . The Harder-Narasimhan filtration on $\mathcal{E}$  is the filtration $\mathcal{H}N^{\bullet}\mathcal{E}$  with $\mathcal{H}N^{1}(\mathcal{E})=\mathcal{O}_{X}(n+m)$  and $gr^{2}\mathcal{H}N^{\bullet}(\mathcal{E})=\mathcal{O}_{X}(m)$  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. ∎

We have the following

Proposition 3.3

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

[TABLE]

Proof.

There is the standard factorization of $D$ as

[TABLE]

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 [1][section 3.5, pp. 16-17, Lemma 3.27]. Hence, we get an $\mathcal{O}_{S}$-linear map

[TABLE]

Regarding $\mathcal{J}^{N}(\mathcal{E}_{1}/S)$  with its first $\mathcal{O}_{X}$-module-structure, we get an $\mathcal{O}_{S}$-linear map

[TABLE]

By [1][section 3.5, pp. 16-17, Lemma 3.27], we have

[TABLE]

where $p_{1},p_{2}:J^{N}(X/S)\longrightarrow X$  are the two projection morphisms, which are morphisms of $S$-schemes and where $J^{N}(X/S)=\operatorname{Spec}_{X}\mathcal{J}^{N}(X/S)$. Since $q\circ p_{1}=q\circ p_{2}$  and $p_{1}$ and $p_{2}$  are affine, we have

[TABLE]

Thus, we can compose $R^{i}q_{*}d^{N}_{\mathcal{E}_{1}/S}$  with the map $R^{i}q_{*}{\widetilde{D}}$  to get the required $\mathcal{O}_{S}$-linear map $R^{i}q_{*}D:R^{i}q_{*}\mathcal{E}_{1}\longrightarrow R^{i}q_{*}\mathcal{E}_{2}.$ ∎

3.1 Extensions of differential operators

The basic question is the following: If

[TABLE]

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.

Proposition 3.4

Let $X/S$  be an integral scheme of finite type and

[TABLE]

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 $D_{1}$ .

Proof.

Suppose, $D_{1}$  is of order $N$  and there are two differential operators $D,D^{\prime}$  of order $\leq N^{\prime}$  with $N^{\prime}\geq N$  on $\mathcal{E}$  extending $D_{1}$ . Without loss of generality, we may assume that $N=N^{\prime}$ . Considering the difference $D-D^{\prime}$ , we may assume that $D_{1}=0$ . We look for an $\mathcal{O}_{X}$-linear homomorpism

[TABLE]

By assumption, $\widetilde{D}$  restricted to $\mathcal{J}^{N}(\mathcal{E}_{1}/k)^{\prime}\subset\mathcal{J}^{N}(\mathcal{E}/k)$  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 [1][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 $\mathcal{J}^{N}(\mathcal{E}/k)/\mathcal{J}^{N}(\mathcal{E}_{1}/k)^{\prime}\longrightarrow\mathcal{E}$ . But the first sheaf is a torsion $\mathcal{O}_{X}$-module so $\widetilde{D}$  must be the zero homomorphism. ∎

Example 3.5

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

[TABLE]

The global $\mathcal{O}_{X}$-($\mathcal{O}_{H}$-linear) endomorphisms of the last sheaf contain the endomorphisms in diagonal form and thus

[TABLE]

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 Lemma 3.2 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.

$\square$

3.2 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.

Proposition 3.6

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

[TABLE]

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.

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

[TABLE]

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

[TABLE]

By [3][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 $\overline{D}:gr^{1}HN(\mathcal{E})\longrightarrow gr^{j}HN(\mathcal{E})$ , since by [1][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 $D^{\prime}:\mathcal{E}/HN^{1}(\mathcal{E})\longrightarrow\mathcal{E}/HN^{1}(\mathcal{E})$  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. ∎

Proposition 3.7

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 $HS^{i}(\mathcal{E})$ .

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 $\overline{D}:HN^{1}(\mathcal{E})\longrightarrow gr^{j}HN(\mathcal{E})$ . 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 $\mathcal{J}^{N}(\mathcal{H}N^{1}(\mathcal{E})/k)$. We get homomorphisms of $\mathcal{O}_{X}$-modules

[TABLE]

Restricting to each single direct summand $\mathcal{O}_{X}\otimes HN^{1}(\mathcal{E})\cong HN^{1}(\mathcal{E})\longrightarrow gr^{j}HN(\mathcal{E})$  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

[TABLE]

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. ∎

Remark 3.8

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.

Proposition 3.9

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 $\overline{D}_{i}:gr^{i}HN(\mathcal{E})\longrightarrow gr^{i}HN(\mathcal{E})$  is $\mathcal{O}_{X}$-linear.

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 $\overline{D}:HN^{1}(\mathcal{E})\longrightarrow gr^{j}HN(\mathcal{E})$ . Let $N\in\mathbb{N}$  be minimal such that $\overline{D}$ factors over $\mathcal{J}^{N}(\mathcal{H}N^{1}(E)/k)$ . We then get a nonzero homomorphism

[TABLE]

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

[TABLE]

$\Omega_{1,N}\otimes HN^{1}(\mathcal{E})$ is semistable and

[TABLE]

so by [3][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

[TABLE]

is nonzero. We then get a homomorphsim

[TABLE]

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 $\Omega^{(1)}(X/k)^{\otimes^{s}N}\otimes HN^{1}(\mathcal{E})\longrightarrow gr^{j}HN(\mathcal{E})$  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

[TABLE]

is the zero map if $N>1$ . ∎

Remark 3.10

*By [4][Lecture III, 2.14 Theorem, p. 67], the condition that

$\mu_{min}(\Omega^{(1)}(X/k))\geq 0$  is equivalent to the uniruledness of $X$.*

3.3 Vector bundles with global differential operators on projective $n$-space

Proposition 3.11

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 $\geq N$ .

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

[TABLE]

Taking duals and tensoring with $\mathcal{F}_{2}$  we get exact sequences

[TABLE]

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 $N>>0$

[TABLE]

By [2][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

[TABLE]

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

[TABLE]

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

[TABLE]

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 [2][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

[TABLE]

∎

Remark 3.12

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 $\mathcal{E}\longrightarrow\mathcal{E}(-n)$ .

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 $H^{0}(\mathbb{P}^{n}_{k},S^{d}\mathcal{T}^{1}(\mathbb{P}^{n}/k)\otimes\operatorname{Hom}_{\mathcal{O}}(\mathcal{E}_{1},\mathcal{E}_{2}))$  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

[TABLE]

is exact for all $d\geq M$ . For $d>>M$  we can thus write for $h^{0}(\mathbb{P}^{n}_{k},DO^{d}(\mathcal{E}_{1},\mathcal{E}_{2}))$

[TABLE]

If we tensor the symmetric power of the Euler sequence ($\mathbb{P}^{n}_{k}=\mathbb{P}(V)$)

[TABLE]

[TABLE]

(see [3][chapter 1.4, p. 19]) with $\operatorname{Hom}_{\mathcal{O}}(\mathcal{E}_{1},\mathcal{E}_{2})$  and use the additivity of the Euler characteristic, we get

[TABLE]

Summing over $M+1\leq k\leq d$  we get for $d>>M$

[TABLE]

We thus have proved the following

Proposition 3.13

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 $N>>0$

[TABLE]

The polynomial $P(N)$  is explicitely given by

[TABLE]

where $M=M(\mathcal{E}_{1},\mathcal{E}_{2})$  is a fixed natural number.

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,

[TABLE]

3.4 Elliptic operators in algebraic geometry

Definition 3.14

*(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*

[TABLE]

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 $D$.

Recall, that if $\mathcal{F}$  is a coherent sheaf on $X$ and $F:=\operatorname{Spec}_{X}\text{Sym}^{\bullet}(\mathcal{F})$  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 $X=\operatorname{Spec}A$  and $\mathcal{F}$  corresponds to the $A$-module $M$ , we have

[TABLE]

and the required map is simply the tensor multiplication map $S^{N}M\otimes_{A}\text{Sym}^{\bullet}M\longrightarrow\text{Sym}^{\bullet}M.$

In our case, $\mathcal{F}=\mathcal{T}^{1}(X/k)$  and $T^{1*}(X/k):=\operatorname{Spec}_{X}\text{Sym}^{\bullet}\mathcal{T}^{1}_{X/k}$  is the classical cotangent bundle of $X$ with projection $\pi_{X}:T^{1*}(X/k)\longrightarrow X$ . Put $T^{\prime}(X/S):=T^{1,*}(X/S)\backslash s_{0}(X)$  where $s_{0}:X\hookrightarrow T^{1,*}(X/k)$  is the zero section with same projection $\pi_{X}:T^{\prime}(X/k)\longrightarrow X$ .

We have the following classical

Definition 3.15

*(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*

[TABLE]

is an isomorphism of locally free sheaves on $T^{\prime}(X/S)$ .

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.

Proposition 3.16

*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 $X$.*

Proof.

The proof is relatively straight forward. Let $\mathcal{E}_{1}$  be the maximal destabilizing subsheaf. By Proposition 3.9 , $D$ 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 $\operatorname{Spec}A\subset X$  such that $\mathcal{E}$  and $\mathcal{E}_{1}$  and $\mathcal{E}/\mathcal{E}_{1}$  are free on $X$. Furthermore, by shrinking $\operatorname{Spec}A$ , we may assume that $\mathcal{T}^{1}(X/k)$  is free on $\operatorname{Spec}A$ . Choose a trivialization

[TABLE]

The symbol $\sigma(D)$  is given by an $n\times n$-matrix of $N^{th}$-order differential operators on $\operatorname{Spec}A$ . 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. ∎

Remark 3.17

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.

In order to give an example of an ellitic operator on a smooth complete variety, we prove

Proposition 3.18

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.

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 $d^{1}z_{i}=d^{1}(z_{i}+c)$  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. ∎