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Integral transforms with exponential kernels and Laplace transform

Authors: Masaki Kashiwara and Pierre Schapira
Journal: J. Amer. Math. Soc. 10 (1997), 939-972
MSC (1991): Primary 32C38, 14F10, 44A10
MathSciNet review: 1447834
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Abstract: Let $X \underset {f}{\longleftarrow } Z \underset {g}{\longrightarrow } Y$ be a correspondence of complex manifolds. We study integral transforms associated to kernels $\exp (\varphi )$, with $\varphi$ meromorphic on $Z$, acting on formal or moderate cohomologies. Our main application is the Laplace transform. In this case, $X$ is the projective compactification of the vector space $V \simeq \mathbb {C}^n$, $Y$ is its dual space, $Z=X\times Y$ and $\varphi (z,w) =\langle z,w \rangle$. We obtain the isomorphisms: \begin{align*} &F \otimes ^W \mathcal {O}_V \simeq F^\wedge [n] \otimes ^W \mathcal {O}_{V^*},\quad \operatorname {THom}(F,\mathcal {O}_V) \simeq \operatorname {THom}(F^\wedge [n],\mathcal {O}_{V^*}) \end{align*} where $F$ is a conic and $\mathbb {R}$-constructible sheaf on $V$ and $F^\wedge$ is its Fourier-Sato transform. Some applications are discussed.

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Additional Information

Masaki Kashiwara
Affiliation: RIMS, Kyoto University, Kyoto 606-01, Japan
MR Author ID: 98845

Pierre Schapira
Affiliation: Institut de Mathématiques, Université Paris VI, Case 82, 4 pl Jussieu, 75252 Paris, France

Received by editor(s): September 17, 1996
Received by editor(s) in revised form: May 23, 1997
Article copyright: © Copyright 1997 American Mathematical Society