Hodge-Gaussian maps

Let $X$ be a compact K\"{a}hler manifold, and let $L$ be a line bundle on $X.$ Define $I_k(L)$ to be the kernel of the multiplication map $Sym^k H^0 (L) \to H^0 (L^k).$ For all $h \leq k,$ 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.