Algebraic Atiyah-Singer index theorem

TL;DR

This paper develops an algebraic weak version of the Atiyah-Singer index theorem and illustrates it through examples involving elliptic differential operators on smooth projective schemes over a field.

Contribution

It introduces an algebraic approach to the Atiyah-Singer index theorem and applies it to specific elliptic operators derived from the Atiyah class.

Findings

Computed examples of elliptic operators on smooth projective schemes

Established an algebraic framework for the index theorem

Connected Atiyah class to index computations

Abstract

The aim of this work is to give an algebraic weak version of the Atiyah-Singer index theorem. We compute then a few small examples with the elliptic differential operator of order $\leq 1$ coming from the Atiyah class in $\mathrm{Ext}^1_{\mathcal{O}_X}(\mathcal{O}_X,\Omega^1_{X/k})$, where $X \longrightarrow \mathrm{Spec}(k)$ is a smooth projective scheme over a perfect field $k$.

Equations

$$X\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${\mathcal{P}}_{X/S}^n={\Delta }_{X/S}^{-1}({\mathcal{O}}_{X{\times }_{S}X})/I_f^{n+1}.$$

$${\mathcal{P}}_{X/S}^{\infty }=\underset{n}{\underleftarrow{\lim }}{\mathcal{P}}_{X/S}^n.$$

$$d_{X/S}^n:{\mathcal{O}}_X⟶{\mathcal{P}}_{X/S}^n.$$

$${\mathcal{P}}_{X/S}^n(\mathcal{F})={\mathcal{P}}_{X/S}^n{\otimes }_{{\mathcal{O}}_X}\mathcal{F}.$$

$$\mathcal{F}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${\mathrm{Diff}}_{X/S}(\mathcal{F},\mathcal{G})=\bigcup _n{\mathrm{Diff}}_{X/S}^n(\mathcal{F},\mathcal{G}).$$

$${\mathrm{Hom}}_{{\mathcal{O}}_X}({\mathcal{P}}_{X/S}^n(\mathcal{F}),\mathcal{G})\stackrel{\cong }{⟶}{\mathrm{Diff}}_{X/S}^n(\mathcal{F},\mathcal{G}).$$

$$R^q{f}_{\ast }(\mathcal{F})=a_{Zar}(U\mapsto H_{Zar}^q(f^{-1}U,\mathcal{F}))$$

$${\Gamma }_{ab}(X,\mathcal{F})\stackrel{defn}{=}{\mathrm{Hom}}_{\underline{Ab}}(\mathbb{Z},\mathcal{F})⟶{\mathrm{Hom}}_{\underline{Ab}}(\mathbb{Z},\mathcal{G})\stackrel{defn}{=}{\Gamma }_{ab}(\mathbb{Z},\mathcal{G}),$$

$$\begin{gathered} {\Gamma }_{mod}(X,{\mathcal{P}}_{X/S}^n{\otimes }_{{\mathcal{O}}_X}\mathcal{F})={\mathrm{Hom}}_{{\mathcal{O}}_X-Mod}({\mathcal{O}}_X,{\mathcal{P}}_{X/S}^n{\otimes }_{{\mathcal{O}}_X}\mathcal{F})⟶{\mathrm{Hom}}_{{\mathcal{O}}_X-Mod}({\mathcal{O}}_X,\mathcal{G}) \\ ={\Gamma }_{mod}(X,\mathcal{G}). \end{gathered}$$

$$D(X):{\Gamma }_{mod}(X,{\mathcal{P}}_{X/k}^n{\otimes }_{{\mathcal{O}}_X}\mathcal{F})⟶{\Gamma }_{mod}(X,\mathcal{G})$$

$$D(X):{\Gamma }_{mod}(X,{\mathcal{P}}_{X/A}^n{\otimes }_{{\mathcal{O}}_X}\mathcal{F})⟶{\Gamma }_{mod}(X,\mathcal{G})$$

$$Ind(D)\stackrel{defn}{=}di{m}_k(Ker(D(X))-di{m}_k(Coker(D(X)).$$

$$\mathbb{V}(\mathcal{E})\stackrel{defn}{=}\mathrm{Spec}(Sy{m}_{{\mathcal{O}}_X}(\mathcal{E})).$$

$${\mathrm{Hom}}_{Sch/X}(Y,\mathbb{V}(\mathcal{E}))\stackrel{\cong }{⟶}{\mathrm{Hom}}_{{\mathcal{O}}_X-Alg}(Sy{m}_{{\mathcal{O}}_X}(\mathcal{E}),f_{\ast }{\mathcal{O}}_Y)\stackrel{\cong }{⟶}{\mathrm{Hom}}_{{\mathcal{O}}_X-Mod}(\mathcal{E},f_{\ast }{\mathcal{O}}_Y).$$

$$K_0(\mathbb{V}(\mathcal{E}))\stackrel{defn}{=}K(Coh(\mathbb{V}(\mathcal{E})))$$

$$K_0(X)\stackrel{s_{0\ast }}{⟶}{K}_0(\mathbb{V}(\mathcal{E}))\stackrel{j^{\ast }}{⟶}{K}_0(\mathbb{V}(\mathcal{E})-s_0(X))⟶0.$$

$$0⟶\mathcal{F}\stackrel{D}{⟶}\mathcal{G}⟶0.$$

$$0⟶{\mathcal{P}}_{X/S}^n{\otimes }_{{\mathcal{O}}_X}\mathcal{F}⟶\mathcal{G}⟶0.$$

$$0⟶{\pi }^{\ast }({\mathcal{P}}_{X/S}^n{\otimes }_{{\mathcal{O}}_X}\mathcal{F})⟶{\pi }^{\ast }\mathcal{G}⟶0.$$

$${\pi }^{\ast }({\mathcal{P}}_{X/S}^n)\stackrel{\cong }{⟶}{\mathcal{P}}_{\mathbb{V}(\mathcal{E})/X}^n.$$

$$\sigma (D):{\mathcal{P}}_{\mathbb{V}(\mathcal{E})/X}^n{\otimes }_{{\mathcal{O}}_{\mathbb{V}(\mathcal{E})}}{\pi }^{\ast }{\mathcal{F}}_{|\mathbb{V}(\mathcal{E})-s_0(X)}\stackrel{\cong }{⟶}{\pi }^{\ast }{\mathcal{G}}_{|\mathbb{V}(\mathcal{E})-s_0(X)},$$

$$K^0(X)\stackrel{defn}{=}K(Vect(X)).$$

$$f^{\ast }:K^0(X)⟶K^0(Y),[\mathcal{E}]\mapsto [f^{\ast }\mathcal{E}].$$

$$f_{\ast }:K_0(Y)⟶K_0(X),\mathcal{F}\mapsto \sum _i(-1{)}^i[R^i{f}_{\ast }\mathcal{F}],$$

$$0⟶{\mathcal{F}}^{\prime }⟶\mathcal{F}⟶{\mathcal{F}}^{\prime \prime }⟶0$$

$$\cdots ⟶R^i{f}_{\ast }({\mathcal{F}}^{\prime })⟶R^i{f}_{\ast }(\mathcal{F})⟶R^i{f}_{\ast }({\mathcal{F}}^{\prime \prime })⟶R^{i+1}{f}_{\ast }({\mathcal{F}}^{\prime })⟶\cdots$$

$$E_2^{p,q}=R^p{g}_{\ast }(R^q{f}_{\ast }\mathcal{F})\Rightarrow R^{p+q}(g\circ f{)}_{\ast }\mathcal{F}.$$

$$K^0(X)\stackrel{\delta }{⟶}{K}_0(X)$$

Full text

We follow Grothendieck [EGA4, §16.8] to recall briefly the notion of differential operators. Let $f:X\longrightarrow S$ be a morphism of schemes. Consider the Cartesian square

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The diagonal $\Delta_{X/S}:X\longrightarrow X\times_{S}X$ is an immersion. One defines the $n$-th normal invariant of $\Delta_{X/S}$ as

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It is clear that $\{\mathcal{P}^{n}_{X/S}\}_{n}$ form a projective system. One defines

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The first projection $p:X\times_{S}X\longrightarrow X$ induces an $\mathcal{O}_{X}$-algebra structure on $\mathcal{P}^{n}_{X/S}$ and the second projection $q:X\times_{S}X\rightarrow X$ induces a morphism

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If $\mathcal{F}\in\mathcal{O}_{X}-Mod$ is an $\mathcal{O}_{X}$-module, one defines

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Let $\mathcal{F},\mathcal{G}\in\mathcal{O}_{X}-Mod$ be two $\mathcal{O}_{X}$-modules. Let $D:\mathcal{F}\longrightarrow\mathcal{G}$ be a morphism of the underlying abelian sheaves. $D$ is called a differential operator of order $\leq n$ relative $S$, if there exists a morphism of $\mathcal{O}_{X}$-modules $u:\mathcal{P}^{n}_{X/S}(\mathcal{F})\rightarrow\mathcal{G}$, such that the following diagram commutes:

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The morphism $D:\mathcal{F}\rightarrow\mathcal{G}$ between the underlying abelian sheaves is not $\mathcal{O}_{X}$-linear, but it is $f^{-1}\mathcal{O}_{S}$-linear. Let us denote by $\mathrm{Diff}^{n}_{X/S}(\mathcal{F},\mathcal{G})$ the set of all differential operator of order $\leq n$ between $\mathcal{F}$ and $\mathcal{G}$. One defines

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It follows from the definition [EGA4, 16.8.3.1] that one has an isomorphism of abelian groups

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If $f:X\rightarrow S$ is locally of finite type, then $\mathcal{P}^{n}_{X/S}$ is a quasi-coherent $\mathcal{O}_{X}$-module of finite type (cf. [EGA4, Prop. 16.3.9]). If $f:X\rightarrow S$ is locally of finite presentation, then $\mathcal{P}^{n}_{X/S}$ is a quasi-coherent $\mathcal{O}_{X}$-module of finite presentation (cf. [EGA4, Cor. 16.4.22]). Consequently, if $\mathcal{F}$ is a quasi-coherent $\mathcal{O}_{X}$-module of finite type resp. locally of finite presentation and $f:X\rightarrow S$ is locally of finite type resp. locally of finite presentation, then $\mathcal{P}^{n}_{X/S}(\mathcal{F})=\mathcal{P}^{n}_{X/S}\otimes_{\mathcal{O}_{X}}\mathcal{F}$ is also a quasi-coherent $\mathcal{O}_{X}$-module of finite type resp. locally of finite presentation. If $S$ is locally Noetherian and $f:X\rightarrow S$ is proper, then by [EGA3, Thm. 3.2.1] the higher direct image sheaf

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is coherent for any $\mathcal{F}\in Coh(X)$ and $\forall q\geq 0$. Consequently, if $X\longrightarrow\mathrm{Spec}(A)$ is proper over a Noetherian ring $A$, then for all coherent sheaf $\mathcal{F}\in Coh(X)$ the Zariski cohomology groups $H^{q}_{Zar}(X,\mathcal{F})$ are $A$-modules of finite type $\forall q\geq 0$. A differential operator $D:\mathcal{F}\longrightarrow\mathcal{G}$ induces a homomorphism of abelian groups

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where we write $\underline{Ab}$ for the category of abelian sheaves on $X$ and $\mathbb{Z}$ is the terminal object in $\underline{Ab}$. In general, we can not say anything about the kernel or cokernel of this homomorphism. However, $D$ induces also a homomorphism of $\Gamma_{mod}(X,\mathcal{O}_{X})$-modules

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This motivates us to give the following definition:

1. (1)

Let $X\longrightarrow\mathrm{Spec}(k)$ be a $k$-scheme of finite type, where $k$ is a field and $D:\mathcal{F}\rightarrow\mathcal{G}$ be a differential operator between two $\mathcal{O}_{X}$-modules of order $\leq n$. $D$ is called a Fredholm operator, if the $k$-linear morphism between $k$-vector spaces

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has finite-dimensional kernel and cokernel. 2. (2)

Let $X\rightarrow\mathrm{Spec}(A)$ be a scheme of finite type over a Noetherian ring $A$ and $D:\mathcal{F}\rightarrow\mathcal{G}$ be a differential operator between two $\mathcal{O}_{X}$-modules. $D$ is called a Fredholm operator, if the kernel and cokernel of the $A$-module morphism

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are $A$-modules of finite type.

For a proper $A$-scheme $X$, where $A$ is a Noetherian ring, any differential operator $D:\mathcal{F}\longrightarrow\mathcal{G}$ between coherent $\mathcal{O}_{X}$-modules is a Fredholm operator.

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Let $X$ be any scheme and $QsCoh(X)$ be the category of quasi-coherent $\mathcal{O}_{X}$-modules. Following Grothendieck [EGA2, §1.7] we define a vector bundle associated to a quasi-coherent sheaf $\mathcal{E}\in QsCoh(X)$ to be the $X$-scheme

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The structure morphism $\pi:\mathbb{V}(\mathcal{E})\rightarrow X$ is an affine morphism. For any morphism $f:Y\rightarrow X$, we have

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We call $s_{0}:X\rightarrow\mathbb{V}(\mathcal{E})$ the [math]-section, if it corresponds to the [math]-homomorphism $\mathcal{E}\rightarrow\mathcal{O}_{X}$. Now we assume $\mathcal{E}$ is an $\mathcal{O}_{X}$-module of finite type. Then $\pi:\mathbb{V}(\mathcal{E})\rightarrow X$ is a morphism of finite type [EGA2, Prop. 1.7.11 (ii)]. So if $X$ is a Noetherian scheme, then $\mathbb{V}(\mathcal{E})$ is also a Noetherian scheme. Consider the locally small abelian category $Coh(\mathbb{V}(\mathcal{E}))$ of coherent sheaves on $\mathbb{V}(\mathcal{E})$. We let

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to be the Grothendieck group of $\mathbb{V}(\mathcal{E})$. By [EGA2, Prop. 1.7.15] the morphism $s_{0}:X\longrightarrow\mathbb{V}(\mathcal{E})$ is a closed immersion. Let $j:\mathbb{V}(\mathcal{E})-s_{0}(X)\longrightarrow\mathbb{V}(\mathcal{E})$ be the complement open immersion. By [SGA6, §IX, Prop. 1.1] one has a localization exact sequence

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If $f:X\rightarrow S$ is the structure morphism of $X$ over a base scheme $S$, then any differential operator $D\in\mathrm{Diff}^{n}_{X/S}(\mathcal{F},\mathcal{G})$ of order $\leq n$ between $\mathcal{O}_{X}$-modules defines a two terms complex of the underlying abelian sheaves

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Or equivalently, a two terms complex of $\mathcal{O}_{X}$-modules

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We obtain then a two terms complex of $\mathcal{O}_{\mathbb{V}(\mathcal{E})}$-modules

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Remark that if $S$ is Noetherian and $f:X\longrightarrow S$ is of finite type, then $X$ is also Noetherian, so $\pi^{*}(\mathcal{F})$, $\pi^{*}(\mathcal{P}^{n}_{X/S})$ and $\pi^{*}(\mathcal{G})$ are coherent $\mathcal{O}_{\mathbb{V}(\mathcal{E})}$-modules, if $\mathcal{F}$ and $\mathcal{G}$ are coherent. Moreover, by [EGA4, Prop. 16.4.5] there is a canonical isomorphism

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The isomorphism $\sigma(D)$ is called the symbol of $D$.

By the localization exact sequence 1, an elliptic operator $D$ of order $\leq n$ with respect to an $\mathcal{O}_{X}$-module of finite type $\mathcal{E}$ defines a class $[\mathcal{P}^{n}_{\mathbb{V}(\mathcal{E})/X}]\cdot[\pi^{*}\mathcal{F}]-[\pi^{*}\mathcal{G}]\in K_{0}(\mathbb{V}(\mathcal{E}))$, which becomes [math] after restricting to the complement of the [math]-section. If $f:X\rightarrow S$ a morphism locally of finite type, then the sheaf of relative differentials $\Omega^{1}_{X/S}=I_{f}/I_{f}^{2}$ is an $\mathcal{O}_{X}$-module of finite type.

Let $X$ be a Noetherian scheme. Denote by $Vect(X)$ the locally small exact category of locally free sheaves of finite rank on $X$. We let

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For an arbitrary morphism of Noetherian schemes $f:Y\rightarrow X$ one has a functorial pullback homomorphism

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For a proper morphism $f:Y\longrightarrow X$ between Noetherian schemes there is a pushforward homomorphism

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which is well-defined as an exact sequence of coherent $\mathcal{O}_{Y}$-modules

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gives rise to an exact sequence of coherent $\mathcal{O}_{X}$-modules

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The functoriality of the pushforward follows easily from the Leray-Grothendieck spectral sequence

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If $X$ is a regular Noetherian scheme, then the canonical homomorphism induced by the obvious exact embedding of categories

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is an isomorphism, since every coherent sheaf on a regular scheme has a finite locally free resolution.

Let now $S$ be an arbitrary scheme. We denote by $SH(S)$ the stable motivic homotopy category together with the formalism of six functors $(f^{*},f_{*},f_{!},f^{!},\wedge,\underline{\mathrm{Hom}})$ as in [Ay08], [CD12] and [Hoy14, Appendix C]. Let $\mathbf{E}$ be a motivic ring spectrum parameterized by schemes. We will assume $\mathbf{E}$ is stable by base change, i.e. for any morphism of schemes $f:X\longrightarrow Y$ there is an isomorphism of spectra in $SH(X)$

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Let $S$ be a scheme and $\mathbf{E}_{S}\in SH(S)$ be a motivic ring spectrum. The unit $\varphi_{S}:\mathbb{S}^{0}\rightarrow\mathbf{E}_{S}$ gives rise to a class

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We have a tower of $S$-schemes given by the obvious embeddings

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Let $\mathbb{P}^{\infty}_{S}=colim_{n}\mathbb{P}^{n}_{S}$, which is an object in the pointed unstable motivic homotopy category of Morel-Voevodsky $Ho_{\mathbb{A}^{1},\bullet}(S)$ [MV99]. Let $i:\mathbb{P}^{1}_{S}\longrightarrow\mathbb{P}^{\infty}_{S}$ be the obvious map in $Ho_{\mathbb{A}^{1},\bullet}(S)$, which gives rise to a map $\Sigma^{\infty}(i):\Sigma^{\infty}(\mathbb{P}^{1}_{S})_{+}\longrightarrow\Sigma^{\infty}(\mathbb{P}^{\infty}_{S})_{+}$. $\mathbf{E}_{S}$ is called oriented, if there is a class

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such that $i^{*}(c_{S})=[\varphi_{S}]$. We say $\mathbf{E}$ is oriented, if $\mathbf{E}_{S}$ is oriented for any scheme $S$ and for any morphism $f:T\rightarrow S$ one has $f^{*}(c_{S})=c_{T}$. If $S$ is regular, by [MV99, §4, 1.15, 3.7] there is a canonical isomorphism $B\mathbb{G}_{m}\cong\mathbb{P}^{\infty}_{S}$ in $Ho_{\mathbb{A}^{1},\bullet}(S)$, which gives to a canonical isomorphism

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For an oriented motivic ring spectrum $\mathbf{E}$, one can define the first Chern class as

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Assume $S$ is a regular scheme. Let $X\longrightarrow S$ be a smooth $S$-scheme. For a vector bundle $\mathbb{V}(\mathcal{E})$ associated to a locally free $\mathcal{O}_{X}$-module $\mathcal{E}$ of finite rank $r$, there is an isomorphism (cf. [NSO09, Thm. 2.11])

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where $\mathbb{P}(\mathcal{E})=\mathrm{Proj}(Sym_{\mathcal{O}_{X}}(\mathcal{E}))\stackrel{{\scriptstyle p}}{{\longrightarrow}}X$ is the projective bundle associated to $\mathcal{E}$ [EGA2, §4]. So $\mathbf{E}^{*,*}(\mathbb{P}(\mathcal{E}))$ is a free module over $\mathbf{E}^{*,*}(X)$ with the basis

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So for a vector bundle $\mathbb{V}(\mathcal{E})$ of rank $r$ one can define the higher Chern classes

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where one puts $c_{0}(\mathbb{V}(\mathcal{E}))=1$ and $c_{i}(\mathbb{V}(\mathcal{E}))=0$ for $i\notin[0,r]$. Recall that one has a homotopy cofiber sequence [MV99, §3]

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where $Th_{X}(\mathbb{V}(\mathcal{E}))$ denotes the Thom space of the vector bundle $\mathbb{V}(\mathcal{E})$. The homotopy cofiber sequence induces a long exact sequence

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The projective bundle formula tells us that $i^{*}$ is a split epimorphism, so $\mathbf{E}^{*,*}(Th_{X}(\mathbb{V}(\mathcal{E})))$ is a free module of rank $1$ over $\mathbf{E}^{*,*}(X)$, which is just $Ker(i^{*})$. The Thom class is the unique class in $\mathbf{E}^{2r,r}(Th_{X}(\mathbb{V}(\mathcal{E})))$

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where $p:\mathbb{P}(\mathcal{E}\oplus\mathcal{O}_{X})\rightarrow X$ is the structure morphism. Let $s:X\longrightarrow\mathbb{P}(\mathcal{E}\oplus\mathcal{O}_{X})$ be its section. The Thom isomorphism is given by

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Its inverse is the composition

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which sends

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If $\xi$ is the universal quotient bundle on $\mathbb{P}(\mathcal{E}\oplus\mathcal{O}_{X})$, i.e. there is an exact sequence

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Then by Whitney sum formula we have

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Now let $KGL_{S}\in SH(S)$ be the motivic ring spectrum representing the algebraic $K$-theory constructed in [Rio10], [CD12, §13]. If $S$ is regular and $X\in Sm/S$ is a smooth $S$-scheme, then one has a natural isomorphism

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$KGL$ is an oriented motivic ring spectrum. Indeed, by Bott periodicity one has an isomorphism

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The Bott element $\beta$ is the image of $1$ under this isomorphism and $\beta\in KGL^{-2,-1}(S)$. The orientation of $KGL$ is given by

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For any line bundle $L\in Pic(S)$, its first Chern class is

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For an arbitrary morphism $f:T\rightarrow S$ of regular schemes one has a commutative diagram [CD12, §13.1]

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Now assume $\mathcal{E}$ is a locally free sheaf of finite rank on a smooth scheme $X$ over a regular base $S$. We have a homotopy cofiber sequence (cf. [MV99, §3])

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where $Th_{X}(\mathbb{V}(\mathcal{E}))$ denotes the Thom space of the vector bundle $\mathbb{V}(\mathcal{E})$. This homotopy cofiber sequence gives rise to an exact sequence

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The natural isomorphism 2 gives rise to a commutative diagram with exact rows (cf. [CD12, §13.4, (K6a)]), where we abbreviate $[-,-]$ for $\mathrm{Hom}_{SH(S)}(-,-)$:

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The bottom row is the exact sequence 1. For a Noetherian scheme $X$ and an $\mathcal{O}_{X}$-module of finite type $\mathcal{E}$ the functor

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is exact, since $\pi$ is an affine morphism, which implies that $R^{i}\pi_{*}\mathcal{F}=0,\forall i>0,\forall\mathcal{F}\in Coh(\mathbb{V}(\mathcal{E}))$. So we still can define the pushforward

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Hence, one may try to define the topological index by applying the homomorphism

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then composing with $f_{*}\pi_{*}$. However, $\pi:\mathbb{V}(\mathcal{E})\rightarrow X$ is not necessary projective, so we may run into difficulties, when we apply later the Grothendieck-Riemann-Roch theorem. Now we restrict ourselves to the situation where $f:X\rightarrow\mathrm{Spec}(k)$ is a smooth proper scheme over a field $k$. $\Omega^{1}_{X/k}$ is a locally free $\mathcal{O}_{X}$-module of finite rank. Let $D:\mathcal{F}\rightarrow\mathcal{G}$ be an elliptic operator of order $\leq n$. Let $\pi:\mathbb{V}(\Omega^{1}_{X/k})\longrightarrow X$ denotes the vector bundles associated to $\Omega^{1}_{X/k}$. The symbol $\sigma(D)$ defines then an element

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which lies in the image of the homomorphism

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Let us denote by $[\sigma(D)]\in K_{0}(Th_{X}(\mathbb{V}(\Omega^{1}_{X/k})))$ a class, which is mapped to

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such that its image in $K_{0}(\mathbb{V}(\Omega^{1}_{X/k})-s_{0}(X))$ is trivial. We call this class $[\sigma(D)]$ a symbol class associated to the symbol $\sigma(D)$.

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where

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is the inverse Thom isomorphism of algebraic $K$-theory and

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If $S$ is any scheme, let $H\mathbb{Q}$ denote the Beilinson rational motivic cohomology ring spectrum in $SH(S)$ constructed in [Rio10], [CD12, Defn. 14.1.2]. Let

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be the total Chern character, which is an isomorphism of rational spectra, if $S=\mathrm{Spec}(k)$, where $k$ is a perfect field [Rio10, Defn. 6.2.3.9, Rem. 6.2.3.10]. One has the following result due to Riou:

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As explained in [CD12, §13.7] $f_{\star}$ induces the usual pushforward $f_{*}$ on $K$-theory. Now we can compute

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where $\int_{X}$ means that we take the pushforward on motivic cohomology

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Now let $D\in\mathrm{Diff}^{n}_{X/A}(\mathcal{F},\mathcal{G})$ be an arbitrary differential operator of order $\leq n$ on a projective scheme $X\rightarrow\mathrm{Spec}(A)$ over a Noetherian ring $A$ and $\mathcal{F},\mathcal{G}\in Coh(X)$. As $D$ gives rise to an $\mathcal{O}_{X}$-module homomorphism

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we may take the twisting

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which in turn gives us a differential operator of order $\leq n$

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We call $D(m)$ a twisting of $D$.

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and a number $m_{0}$, such that for all $q>0$ and all $b\geq m_{0}$

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We take $N_{0}=max(n_{0},m_{0})$ to be the maximal of $n_{0}$ and $m_{0}$. We have trivially that for all $N\geq N_{0}$

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The proposition follows now easily from the Hirzebruch-Riemann-Roch theorem

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∎

Now we are ready to state the following algebraic weak form of the Atiyah-Singer index theorem

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[TABLE]

which means that we have to show there is a commutative diagram

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By construction, we have a commutative diagram

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So it remains to see that we have a commutative diagram

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But this is quite obvious, since the closed immersion $s:X\longrightarrow\mathbb{P}(\Omega^{1}_{X/k}\oplus\mathcal{O}_{X})$ is the composition of the [math]-section $s_{0}:X\longrightarrow\mathbb{V}(\Omega^{1}_{X/k})$ and $\mathbb{V}(\Omega^{1}_{X/k})\longrightarrow\mathbb{P}(\Omega^{1}_{X/k}\oplus\mathcal{O}_{X})$ (cf. [EGA2, Prop. 8.3.2]). ∎

Finally, we will give in the last part of this paper a few small examples. For this purpose, we always restrict now ourselves to the case of a smooth projective $k$-scheme $X\longrightarrow\mathrm{Spec}(k)$ of dimension $\mathrm{dim}(X)=d$, where $k$ is a perfect field. Let us begin with the following lemma:

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Therefore,

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Consider the Euler sequence

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One has

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The lemma follows easily. ∎

Consider the Atiyah class in $\mathrm{Ext}^{1}_{\mathcal{O}_{X}}(\mathcal{O}_{X},\Omega^{1}_{X/k})$

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The morphism $u:\mathcal{P}^{1}_{X/k}\longrightarrow\mathcal{O}_{X}$ defines a differential operator of order $\leq 1$

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For each point $P\in\mathbb{V}(\Omega^{1}_{X/k})$ we have by [EGA4, Cor. 16.4.12]

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The morphism $\pi^{*}(u):\mathcal{P}^{1}_{\mathbb{V}(\Omega^{1}_{X/k})/X}\longrightarrow\mathcal{O}_{\mathbb{V}(\Omega^{1}_{X/k})}$ becomes obviously an isomorphism at each point $P\in\mathbb{V}(\Omega^{1}_{X/k})-s_{0}(X)$. ∎

From now on we always consider the elliptic differential operator $D_{u}$. We write $H=c_{1}(\mathcal{O}_{X}(1))$ and for a curve $X$

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For $N>>0$ one has

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for any number $n\geq 0$, where we write $H=c_{1}(\mathcal{O}_{C}(1))$ for the hyperplane section. As $g_{C}=1$, the canonical divisor $K_{C}=0$ is trivial. So we have $Td(T_{C/k})=1$ and hence

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This implies easily that $Ind(D_{u}(N))=N\cdot deg(H)=4N$ for $N>>0$. ∎

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We take two embeddings $i_{1}:\mathbb{P}^{1}\longrightarrow\mathbb{P}^{M}$ and $i_{2}:X\longrightarrow\mathbb{P}^{M}$, such that they are compatible with $f$

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Let $N_{0}>>0$ be a big integer number. By [EGA3, Cor. 7.9.13, Seconde partie] $f_{*}(\Omega^{1}_{X/K}(N_{0}))$ is a locally free $\mathcal{O}_{\mathbb{P}^{1}}$-module of rank $\chi(X_{\eta},\Omega^{1}_{X_{\eta},\eta}(N_{0}))$. By flat base change for Zariski cohomology of coherent sheaves it is enough to compute $\chi(X_{\bar{\eta}},\Omega^{1}_{X_{\bar{\eta}}}(N_{0}))$, where $X_{\bar{\eta}}$ is the base change of $X_{\eta}$ to an algebraic closure $\bar{\eta}$. $X_{\bar{\eta}}$ must be an elliptic curve. For an elliptic curve $(E,P_{0})$, we know that $\Omega^{1}_{E}\cong\mathcal{O}_{E}$ and $\mathcal{L}(3P_{0})$ is very ample, i.e. $\mathcal{L}(3P_{0})\cong\mathcal{O}_{E}(1)$. The last isomorphism means simply that we can embed

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By Riemann-Roch theorem for curves we also have $dim\,H^{0}(E,\mathcal{L}(nP_{0}))=n$. Now we can apply the theorem of Grothendieck for the decomposition of vector bundles on $\mathbb{P}^{1}$ and we have

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Let $N_{0}^{\prime}>>0$ be another big integer number. We have

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So

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If this dimension is already odd, then there is nothing to prove. So we assume

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Let $N_{0}^{\prime\prime}>>0$ be another big integer number. After twisting by $\mathcal{O}(N_{0}^{\prime\prime})$ we have

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Now we take simply $N_{0}\equiv 1\mod 2$, $N_{0}^{\prime}\equiv 1\mod 2$ and $N_{0}^{\prime\prime}\equiv 0\mod 2$. We obtain

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By projection formula we have for any number $n>0$ an isomorphism

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This implies

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where the first equality follows easily from the adjunction

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Therefore, we can conclude that there exists a number $N=N_{0}+N_{0}^{\prime}+N_{0}^{\prime\prime}>>0$ such that $Ind(D_{u}(N))\equiv 1\mod 2$, which finishes the proof. ∎

1. (1)

If $deg(X)=4$, then for a big number $N>>0$ one has

[TABLE] 2. (2)

If $deg(X)=6$, then for a big number $N>>0$ one has

[TABLE]

[TABLE]

For a smooth projective $K3$-surface one has $K=0$ and $K^{2}=0$. $\Omega^{1}_{X/k}$ is the dual bundle of $T_{X/k}$. So we have by lemma 10

[TABLE]

For any number $n\geq 0$ one has

[TABLE]

If $deg(X)=4$ then $(d_{1},d_{2})=(2,2)$ and if $deg(X)=6$ then $(d_{1},d_{2})=(2,3)$. The result follows now easily, since $deg(X)=\int_{X}H^{2}$ and $\int_{X}c_{2}=24$. ∎