Let X be a compact Kahler manifold, and let L be a line bundle on X.
Define Ik(L) to be the kernel of the multiplication map For all h we define a map When L is the canonical bundle, the map p computes a second fundamental form associated to the deformations of X.
If X = C is a curve, then p is a lifting of the Wahl map 12 (L) ->
KC2).
We also show how to generalize the construction of p to the cases of harmonic bundles and of couples of vector bundles.
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