## Index theorem for elliptic pairs.(English)Zbl 0856.58037

Astérisque. 224. Paris: Société Mathématique de France, 113 p. (1994).
The booklet comprises three parts whose purpose is to establish an index theorem for elliptic differential systems on complex manifolds in the setting of algebraic analysis. Most of the results of the first wo papers have been announced previously [C. R. Acad. Sci., Paris, Sér. I 311, No. 2, 83-86 (1990; Zbl 0718.32011) and ibid. 312, No. 1, 81-84 (1991; Zbl 0724.32006)]. The index theorem in some sense generalizes the relative index theorem of L. Boutet de Monvel and B. Malgrange [Ann. Sci. Éc. Norm. Supér., IV. Sér. 23, No. 1, 151-192 (1990; Zbl 0705.58047)] and the index theorem of M. Kashiwara [cf. M. Kashiwara and the first author, ‘Sheaves on manifolds’, Springer, Berlin (1990; Zbl 0709.18001) ([KS] for reference)].
In the first paper [Astérisque 224, 5-60 (1994; see below)], the authors introduce the notion of an elliptic pair, $$({\mathcal M}, F)$$, on a complex analytic manifold $$X$$. This consists of a coherent module $$\mathcal M$$ over the ring $${\mathcal D}_X$$ of differential operators on $$X$$, and of an $$\mathbb{R}$$-constructible object $$F$$ on $$X$$, satisfying the transversality condition $\text{char}({\mathcal M})\cap SS(F)\subset T^*_X X,$ where $$\text{char}({\mathcal M})$$ and $$SS(F)$$ denote the characteristic variety of $$\mathcal M$$ and the micro-support of $$F$$, resp., as defined in [KS]; $$T^*_X X$$ is the image of the zero section in the cotangent bundle of $$X$$. More generally, they consider $$f$$-elliptic pairs for a morphism $$f: X\to Y$$ with $$\text{chr}({\mathcal M})$$ replaced by $$\text{chr}_f({\mathcal M})$$. If $$f$$ is proper on $$\text{supp}({\mathcal M})\cap \text{supp}(F)$$ and $$\mathcal M$$ allows a good filtration, they prove that
– the direct image $$\underline f_!({\mathcal M}\otimes F)$$ has $${\mathcal D}_Y$$-coherent cohomology (finiteness);
– the duality morphism $$\underline f_!(D' F\otimes \underline D_Y {\mathcal M})\to \underline D_Y \underline f_!({\mathcal M}\otimes F)$$ is in fact an isomorphism (where $$D'$$ and $$\underline D$$ denote the dualizing functors);
– there is a Künneth formula;
– direct images commute with microlocalization.
These results are also proved in the relative case where $$X$$ and $$Y$$ are relative analytic manifolds over an analytic manifold $$S$$, i.e., there are analytic submersions from $$X$$ resp. $$Y$$ onto $$S$$ commuting with $$f$$. As special cases they contain Grauert’s theorem on direct images of coherent $${\mathcal O}$$-modules as well as Kashiwara’s theorem on direct images of coherent $${\mathcal D}$$-modules (also in the non-proper case), and the corresponding duality results.
In the second paper [Astérisque 224, 61-98 (1994; see below)], the authors introduce for an elliptic (or $$f$$-elliptic) pair $$({\mathcal M}, F)$$ the microlocal Euler class $$\mu\text{eu}({\mathcal M}, F)$$ in $$H^0_\Lambda(T^* X; \pi^{- 1} \omega_X)$$ (with $$\Lambda= \Lambda_0+ \Lambda_1= \text{char}({\mathcal M})+ SS(F)$$, $$\pi: T^* X\to X$$ the projection, and $$\omega_X$$ the dualizing complex on $$X$$). Generalizing corresponding results for constructible sheaves of [KS] they prove $\mu\text{eu}({\mathcal M}, F)= \mu\text{eu}({\mathcal M}, \mathbb{C}_X) *_\mu \mu\text{eu}(\Omega_X, F),$ where $$*_\mu: H^0_{\Lambda_0}(T^* X; \pi^{- 1} \omega_X)\times H^0_{\Lambda_1}(T^* X; \pi^{- 1} \omega_X)\to H^0_\Lambda(T^* X; \pi^{- 1} \omega_X)$$ is induced by integration along the fibers of $$T^* X\times_X T^* X\to T^* X$$ (pointwise addition), and, in the $$f$$-elliptic case, $\mu\text{eu}(\underline f_!({\mathcal M}\otimes F))= f_\mu \mu\text{eu}({\mathcal M}, F)$ with a morphism $$f_\mu$$ also induced by integration. Based on these results the authors prove the cohomological index formula $\chi(R\Gamma(X; F\otimes {\mathcal M} \otimes^L_{{\mathcal D}_X} {\mathcal O}_X))= \int_{T^* X} \mu\text{eu} ({\mathcal M}, \mathbb{C}_X)\cup \mu \text{eu}(\Omega_X, F)$ for the Euler-Poincaré-characteristic of the complex $$R\Gamma(X; F\otimes {\mathcal M}\times^L_{{\mathcal D}_X} {\mathcal O}_X)$$ (representing the solutions of the elliptic differential system) which has finite-dimensional cohomology for an elliptic pair $$({\mathcal M}, F)$$ with compact support. Finally, they state some conjectures concerning a microlocal Chern character (confirmed in special cases) which would link the cohomology index theorem with the $$K$$-theoretical index theorem of Atiyah-Singer and its relative version of Boutet de Monvel-Malgrange.
The third paper (by the second named author) [Astérisque 224, 99-113 (1994; see below)] provides an appropriate extension of Houzel’s finiteness theorem [see C. Houzel, Math. Ann. 205, 13-54 (1973; Zbl 0264.32012)] and its application in C. Houzel and the first named author [C. R. Acad. Sci., Paris, Sér. I 298, 461-464 (1984; Zbl 0582.14004)] which is used at a crucial point in the proof of coherence of the cohomology for elliptic pairs.

### MSC:

 58J20 Index theory and related fixed-point theorems on manifolds 32C38 Sheaves of differential operators and their modules, $$D$$-modules 58-02 Research exposition (monographs, survey articles) pertaining to global analysis 58J15 Relations of PDEs on manifolds with hyperfunctions 58J10 Differential complexes 58J40 Pseudodifferential and Fourier integral operators on manifolds 14F40 de Rham cohomology and algebraic geometry