Integral transforms with exponential kernels and Laplace transform

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.