Hodge-Gaussian Maps

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 Sym^k H^0(L) -> H^0(L^k). For all h <= k, we define a map rho: I_k(L) -> 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 ρ is a lifting of the Wahl map I_2(L) -> H^0(L^2 ⊗ K_C^2). We also show how to generalize the construction of ρ to the cases of harmonic bundles and of couples of vector bundles.