Some results on deformations of sections of vector bundles

TL;DR

This paper investigates deformations of sections of vector bundles on complex projective varieties, providing new transversality conditions, alternative proofs of classical theorems, and applications to bounds on continuous ranks and generic vanishing theorems.

Contribution

It introduces a generalized transversality condition for deformations, offers a new proof of the Severi-Kodaira-Spencer theorem, and applies these results to bounds on continuous ranks and vanishing theorems.

Findings

Established a transversality condition generalizing semi-regularity.

Provided a new proof of the Severi-Kodaira-Spencer theorem.

Extended results on base loci to higher-dimensional varieties.

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 $L_t\in Pic^0(X)$ 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.