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FJ-LMI Seminar
Seminar information archive ～04/30｜Next seminar｜Future seminars 05/01～
| Organizer(s) | Toshiyuki Kobayashi, Michael Pevzner |
|---|
2025/04/14
17:00-18:00 Room #Main Lecture Hall (Graduate School of Math. Sci. Bldg.)
FJ-LMI Distinguished Lecture
Pierre SCHAPIRA (IMJ - Sorbonne University)
Microlocal sheaf theory and elliptic pairs (英語)
https://fj-lmi.cnrs.fr/wp-content/uploads/2025/03/Tokyo25Colloq.pdf
FJ-LMI Distinguished Lecture
Pierre SCHAPIRA (IMJ - Sorbonne University)
Microlocal sheaf theory and elliptic pairs (英語)
[ Abstract ]
On a complex manifold $X$, an elliptic pair $(\mathcal{M},G)$ is the data of a coherent $\mathcal{D}_X$-module $\mathcal{M}$ and an $\mathbb R$-constructible sheaf $G$ with the property that the characteristic variety $\operatorname{char}(\mathcal{M})$ and the micro-support $\mathrm{SS}(G)$ do not intersect outside the zero-section of $T^*X$. We prove a regularity result which generalizes the classical case of hyperfunction solutions of elliptic systems and a finiteness theorem when assuming that the support of the pair is compact.
Then we introduce the microlocal Euler class of $\mathcal{M}$ and that of $G$ and calculate the Euler-Poincar\'e index of the complex of holomorphic solutions of the pair as the integral over $T^*X$ of the cup product of these two characteristic classes. This construction gives a new approach to the Riemann--Roch or the Atiyah--Singer theorems.
I will start by briefly recalling all necessary notions of microlocal sheaf theory and $\mathcal{D}$-module theory.
[ Reference URL ]On a complex manifold $X$, an elliptic pair $(\mathcal{M},G)$ is the data of a coherent $\mathcal{D}_X$-module $\mathcal{M}$ and an $\mathbb R$-constructible sheaf $G$ with the property that the characteristic variety $\operatorname{char}(\mathcal{M})$ and the micro-support $\mathrm{SS}(G)$ do not intersect outside the zero-section of $T^*X$. We prove a regularity result which generalizes the classical case of hyperfunction solutions of elliptic systems and a finiteness theorem when assuming that the support of the pair is compact.
Then we introduce the microlocal Euler class of $\mathcal{M}$ and that of $G$ and calculate the Euler-Poincar\'e index of the complex of holomorphic solutions of the pair as the integral over $T^*X$ of the cup product of these two characteristic classes. This construction gives a new approach to the Riemann--Roch or the Atiyah--Singer theorems.
I will start by briefly recalling all necessary notions of microlocal sheaf theory and $\mathcal{D}$-module theory.
https://fj-lmi.cnrs.fr/wp-content/uploads/2025/03/Tokyo25Colloq.pdf
