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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
DOI: https://doi.org/10.1090/S0894-0347-97-00245-2
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 $V \simeq {\Bbb C}^n$, $Y$ is its dual space, $Z=X\times Y$ and $\varphi(z,w) =\langle z,w \rangle$. We obtain the isomorphisms:

\begin{align*}&F \mathop\otimes\limits^W{\cal {O}}_V \simeq F^\wedge[n] \mathop\otimes\limits^W{\cal {O}}_{V^*},\quad \operatorname{ THom}(F,{\cal {O}}_V) \simeq \operatorname{ THom}(F^\wedge[n],{\cal {O}}_{V^*}) \end{align*}

where $F$ is a conic and ${\Bbb R}$-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

Pierre Schapira
Affiliation: Institut de Mathématiques, Université Paris VI, Case 82, 4 pl Jussieu, 75252 Paris, France
Email: schapira@math.jussieu.fr

DOI: https://doi.org/10.1090/S0894-0347-97-00245-2
Received by editor(s): September 17, 1996
Received by editor(s) in revised form: May 23, 1997
Article copyright: © Copyright 1997 American Mathematical Society

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