The infinitesimal Torelli problem for zero sets of sections of vector bundles (Q1088739)

Let X be a compact complex manifold of dimension \(n\) and consider an exact sequence \(0\to {\mathcal G}\to {\mathcal F}\to \Omega^ 1_ X\to 0.\) Fix an integer p, \(0<p\leq n\). The main result asserts that, assuming some cohomological conditions concerning the above data, the canonical map \(H^ 1(X,\Theta_ X)\to Hom(H^{n-p}(X,\Omega^ p_ X),H^{n- p+1}(X,\Omega_ X^{p-1}))\) is injective. By Griffiths, it follows then that the infinitesimal Torelli theorem holds for X. Using this, the author obtains a general result concerning the infinitesimal Torelli problem for zero sets of sections of vector bundles (in rank one, this extends a result of Green). In particular from this one deduces that the infinitesimal Torelli theorem holds for complete intersections in projective spaces (with two precise exceptions), generalizing known results of Griffiths, Peters, Usui.