Hodge-Gaussian maps

Abstract

Let $X$ be a compact Kähler manifold, and let $L$ be a line bundle on $X.$ Define $I_k(L)$ to be the kernel of the multiplication map $Sy{m}^k{H}^0(L)\to H^0(L^k).$ For all $h\le k,$ we define a map $\rho :I_k(L)\to Hom(H^{p,q}(L^{-h}),H^{p+1,q-1}(L^{k-h})).$ When $L=K_X$ is the canonical bundle, the map $\rho$ computes a second fundamental form associated to the deformations of $X.$ If $X=C$ is a curve, then $\rho$ is a lifting of the Wahl map $I_2(L)\to H^0(L^2\otimes K_C^2).$ We also show how to generalize the construction of $\rho$ to the cases of harmonic bundles and of couples of vector bundles.

26 pages, LaTeX