University of Georgia

Mathematics Department Colloquium

Pierre Schapira

University of Paris VI

October 5, 1999 (Tuesday)

Integral transforms in the language of sheaves and D-modules

Roughly speaking, an integral transform is a correspondence between two spaces $X$ and $Y$, through an ``incidence relation'' $S\subset X\times Y$. First, we shall discuss several examples of such geometrical transforms: linear duality, real and complex projective duality, flag correspondences, etc. Next, we shall discuss the ``quantization'' of such correspondences. If $X$ and $Y$ are real manifolds, one associates to a function $\phi$ on $X$, a function $\hat{\phi}$ on $Y$ by a formula like $$\hat{\phi}(y)=\int_X\phi(x)k(x,y)dx$$ where $k(x,y)dy$ is a kernel associated with $S$ (e.g.: a Dirac measure on $S$). This transform is the composition of three operations: inverse image, product and direct image (integration). Such operations exist in various contexts that we shall describe: set theory, sheaves, constructible functions, functions and distributions, D-modules. Indeed, systems of linear differential operators arise naturally in such transformations, and it appears convenient to treat integral transform in the complex domain with the language of systems of linear PDE, that is, with D-module theory. We shall give a general ``adjunction formula'' and show how it allows one to recover, unify, and generalize classical results of Gelfand, Penrose and many others.