Hodge Spaces for Real Toric Varieties

TL;DR

This paper introduces Z/2Z Hodge spaces for real toric varieties, providing methods to compute them using polyhedral duality, especially for smooth fans, and establishing conditions for maximality of the associated real toric variety.

Contribution

It defines Z/2Z Hodge spaces for fans and develops computational techniques using polyhedral duality, including complete determination for smooth fans and criteria for maximality.

Findings

Computed Hodge spaces for reflexive polytope fans

Established conditions for maximality of real toric varieties

Demonstrated complete determination of H_{pq} for smooth fans

Abstract

We define the Z/2Z Hodge spaces H_{pq}(\Sigma) of a fan \Sigma. If \Sigma is the normal fan of a reflexive polytope \Delta then we use polyhedral duality to compute the Z/2Z Hodge Spaces of \Sigma. In particular, if the cones of dimension at most e in the face fan \Sigma^* of \Delta are smooth then we compute H_{pq}(\Sigma) for p<e-1. If \Sigma^* is a smooth fan then we completely determine the spaces H_{pq}(\Sigma) and we show the toric variety X associated to \Sigma is maximal, meaning that the sum of the Z/2Z Betti numbers of X(R) is equal to the sum of the Z/2Z Betti numbers of X(C).