Type: Article

Some results on deformations of sections of vector bundles

Castorena, Abel; Pirola, Gian P.;

Abstract:

Let E be a vector bundle on a smooth complex projective variety X. We study the family of sections st? H0(E? Lt) where Lt? Pic0(X) is a family of topologically trivial line bundle and L0= OX, that is, we study deformations of s= s0. By applying the approximation theorem of Artin (Invent Math 5:277–291, 1968) we give a transversality condition that generalizes the semi-regularity of an effective Cartier divisor. Moreover, we obtain another proof of the Severi–Kodaira–Spencer theorem (Bloch In Invent Math 17:51–66, 1972). We apply our results to give a lower bound to the continuous rank of a vector bundle as defined by Miguel Barja (Duke Math J 164(3):541–568, 2015) and a proof of a piece of the generic vanishing theorems (Green and Lazarsfeld, Invent Math 90:389–407, 1987) and (Green and Lazarsfeld, J Am Math Soc 4:87–103, 1991) for the canonical bundle. We extend also to higher dimension a result given in (Mendes-Lopes et al. In Geo Topol 17:1205:1223, 2013) on the base locus of the paracanonical base locus for surfaces