Gaps for geometric genera

TL;DR

This paper studies the possible geometric genera of subvarieties in smooth projective varieties, identifying gaps and extending previous results from surfaces to higher dimensions using new methods.

Contribution

It generalizes the understanding of gaps in geometric genera from surfaces to arbitrary-dimensional smooth projective varieties with a novel approach.

Findings

Identification of gap intervals for geometric genera in higher dimensions

Finiteness of the set of gaps in arbitrary dimensions

Asymptotic bounds for the set of gaps

Abstract

We investigate the possible values for geometric genera of subvarieties in a smooth projective variety. Values which are not attained are called gaps. For curves on a very general surface in $\mathbb{P}^3$, the initial gap interval was found by Xu (see [7] in References), and the next one in our previous paper (see [4] in References), where also the finiteness of the set of gaps was established and an asymptotic upper bound of this set was found. In the present paper we extend some of these results to smooth projective varieties of arbitrary dimension using a different approach.