Mumford Tate groups and the Hodge conjecture

TL;DR

This paper investigates the Hodge conjecture for singular varieties, proving it for certain normal crossing varieties with constant Mumford-Tate groups and expressing it algebraically for more singular cases.

Contribution

It establishes the conjecture for specific singular varieties and introduces methods to construct relevant families with constant Mumford-Tate groups.

Findings

Proved the Hodge conjecture for simple normal crossing varieties in certain families.

Demonstrated the conjecture can be expressed algebraically for varieties with worse singularities.

Provided techniques to produce families with constant Mumford-Tate groups.

Abstract

In this article we study the (cohomological) Hodge conjecture for singular varieties. We prove the conjecture for simple normal crossing varieties that can be embedded in a family where the Mumford-Tate group remains constant. We show how to produce such families. Furthermore, we show for varieties with worse singularities the conjecture can be expressed solely in terms of the algebraic classes.