数理解析研究所講究録1090 巻 1999 年 24-25

24

Recent results on the Index Theorem
Pierre Schapira
December 1997

1

Review on elliptic pairs [4]Let $X$ be a complex manifold of dinension and : $T^{*}Xarrow X$ its cotangent, the dualizing complex,
bundle. We shall make use of the sheaves
the structural sheaf,
the sheaf of linear holomorphic partial differential$X,$
operators on
the sheaf on $T^{*}X$ of microdifferential operators ([3]).
If $F$ is an -constructible sheaf, its microlocal Euler class $\mu eu(F)$ is a Lacycle supported by $SS(F)$ , the micro-support of $F$ (see [2]):
$n$

$\pi$

$\omega_{X}\simeq \mathbb{C}_{X}[2n]$$D_{X}$$\mathcal{O}_{X}$$\mathcal{E}_{X}$$\mathbb{R}$$\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{n}_{\mathrm{b}}^{\sigma \mathrm{i}\mathrm{a}\mathrm{n}}$$\mu eu(F)\in H_{ss()}^{0}F(T^{*}X;\pi-1\omega_{X})$If At is a coherent
class supported by char

$D_{X}-\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{u}\mathrm{l}\mathrm{e}$$(\mathcal{M})$, its microlocal Euler class is a cohomology, the characteristic vaniety of$\mathcal{M}$$\mu eu(\mathcal{M})\in H_{\mathrm{C}ha}0(T^{*}X;\pi-r(\mathcal{M})\omega_{X})1$

The pair$(\mathcal{M},F)$is ehptic ifchar$(\mathcal{M})\cap SS(F)\subset T_{X}^{*}X$

Under this ellipticity hypothes.is, the microlocal convolution$\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{d}\mathrm{u}\mathrm{c}\mathrm{t}*_{\mu}$:$H_{ch}^{0*}ar(\mathcal{M})(TX;\pi^{-}\omega \mathrm{x})1\otimes H_{s}^{0_{S(}*}F)(\tau X;.\pi^{-}\omega \mathrm{x}1)arrow H_{Char(}^{0}\mathcal{M})+Ss(p)(\tau*X;\pi^{-}\omega \mathrm{x})1$

is well defined, and the microlocal Euler class of the elliptic pairobtained as:$(\mathcal{M},F)$is$\mu eu(\mathcal{M}, F)=\mu eu(\mathcal{M})*_{\mu}\mu eu(F)$its restriction to the zero-section. Hence, ifOne denote by eu$(F)$suppis compact, we get:$(\mathcal{M},F)$eu$suw(\mathcal{M})\cap$$(\mathcal{M},F)\in H_{c}^{0}(X;\omega_{X})$One of the main results of [4] asserts that the complex
finitecohomology and its Euler-Poincar\’e index
calculated by the formula:has
may be$\mathrm{R}\mathrm{H}\mathrm{o}\mathrm{m}_{D}(\mathcal{M}\otimes \mathrm{F},\mathcal{O}_{\mathrm{X}})$$\mathrm{d}\mathrm{i}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}\prime \mathrm{a}\mathrm{l}$$\chi(X, M,F)=\int_{X}eu(\mathcal{M}, F)$$\chi(X, \mathcal{M}, F)$

25.
The renainding problem is to calculate the microlocal Euler classis a coherent nodAssune
is endowed with a good filtration Thenule over the sheaf
and its Chern character is welI defined. Let $Td_{X}(Tx)$denote the Todd class. The following conjecture is nade in [ :$\mathrm{o}\mathrm{f}J\backslash 4$$gr(\mathcal{M})$$\mathcal{M}$$\mathcal{O}_{T^{*}X}$$4|$$\mu eu(shm)=[Ch(gr(J\vee t))\cup\pi^{-1}Tdx(TX)\iota 2n$2.

Chern character of -modules [1]$\mathcal{E}$Let
be a unitary algebra over a field of characteristic zero. It is a wellknown result that the Chern caracter of a finitely generated projective A-nodule
may be calcuIated using
cyclic homology. This construction has been
generalized by Bressler-Nest-Tsygan to perfect sheaves of modules over sheaves
algebras. (There is also a recent and elegant construction due toof rings of{ is a coherent .-module, its Chern character
B. KelIer.) Hence,
.
$A$

$k$

$\mathrm{n}\mathrm{e}_{\epsilon}\sigma \mathrm{a}\mathrm{t}\mathrm{i}\mathrm{V}\mathrm{e}$$k_{X}$$\mathrm{i}\mathrm{f}/\vee$$ch(\mathcal{M})$$\mathcal{E}_{\mathrm{Y}}$$\mathrm{b}\mathrm{e}\mathrm{I}\mathrm{o}\mathrm{n}\mathrm{g}_{\mathrm{S}\mathrm{t}}\mathrm{o}\oplus_{j=1}^{n}H_{Chr\prime}^{2\mathrm{j}}(a(\triangleright l)\tau\tau*X,\mathbb{C}*\mathrm{x})$Theorem 2.1 [1] Assume$\mathcal{M}$is endowed with a good filtration. Then:$ch(M\rangle$$=$$ch(gr(\mathcal{M}))\cup\pi^{-1}Td\mathrm{x}(TX)$$ch(\mathcal{M})^{2n}$$=$$\mu eu(\mathcal{M})$

Of course, this result in particular implies the conjecture of ,Jn fact, the theorem of (loc.cit.) is more general and is stated in the framework of ’‘quantized defornation algebras”.
$[4|$3Comments

fromIt is interesting to notice that the Todd class appears wheninmodules to -nodules, and that the index theorem, when formulated terms
of $D$-nodules (and Chen character of such modules) do not use Todd classes.

$\prime D-$$\mathrm{p}\mathrm{a}\mathrm{s}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{o}\sigma$$\mathcal{O}$

References[1] P. Bressler, R. Nest and B. Tsygan A Riemann-Roch type formla for the microlocal
Euler class Inter. Math. Research Notices 20 pp 1033-1044 (1997)
der mathe[2] M. Kashiwara and P. Schapira, Sheaves on manifolds,
matischen Wissenschaften, no. 292, Springer, 1990.
[3] M. Sato, T. Kawai and M. Kashiwara Hyperfunctions and pseudodifferential equations in LN in Math 287, Springer-Verlag pp. 265-529 (1973).
[4] P. Schapira and J-P.
ers Index theorem for eltiptc pairs, Ast\’erisq.u.e224,

Grundl.ehren

1994.

Schnei.d

Institut de Math\’ematiques; Analyse Alg\’ebrique; Universit\’eCase 82.; 4, place Jus.sieu;
Paris Cedex 05
email: schapira@nath.jussieu.fr$\mathrm{F}arrow 7..5252$$\mathrm{P}\neg$ierre et Marie Curie;