Some results on deformations of sections of vector bundles

Abstract

Let $E$ be a vector bundle on a smooth complex projective variety $X$. We study the family of sections $s_t\in H^0(E\otimes L_t)$ where $L_t\in Pi{c}^0(X)$ is a family of topologically trivial line bundle and $L_0={\mathcal{O}}_X,$ that is, we study deformations of $s=s_0$. By applying the approximation theorem of Artin [2] 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 [4]. We apply our results to give a lower bound to the continuous rank of a vector bundle as defined by Miguel Barja [3] and a proof of a piece of the generic vanishing theorems [6] and [7] for the canonical bundle. We extend also to higher dimension a result given in [8] on the base locus of the paracanonical base locus for surfaces.

12 pages. An extra hypothesis is added to the results in last section. Final version will appear in Collectanea Mathematica

References in corpus (3)

• Strong generic vanishing and a higher dimensional Castelnuovo-de Franchis inequality
• Generalized Clifford-Severi Inequality and the Volume of Irregular Varieties
• Continuous families of divisors, paracanonical systems and a new inequality for varieties of maximal Albanese dimension