Positivity of the diagonal

TL;DR

This paper investigates how the positivity properties of the diagonal cycle on a projective variety reflect the variety's geometric features, establishing new links between diagonal positivity and classical geometric invariants.

Contribution

It provides a comprehensive analysis of the conditions under which the diagonal is big, nef, or rigid, and relates these to the vanishing of Hodge groups and classification of low-dimensional varieties.

Findings

Diagonal being big implies vanishing of certain Hodge groups.

Classification results for low-dimensional varieties with nef and big diagonals.

Criteria for the diagonal to be nef, big, or rigid based on geometric properties.

Abstract

We study how the geometry of a projective variety $X$ is reflected in the positivity properties of the diagonal $\Delta_X$ considered as a cycle on $X \times X$. We analyze when the diagonal is big, when it is nef, and when it is rigid. In each case, we give several implications for the geometric properties of $X$. For example, when the diagonal is big, we prove that the Hodge groups $H^{k,0}(X)$ vanish for $k>0$. We also classify varieties of low dimension where the diagonal is nef and big.