Galois closure and Lagrangian varieties

TL;DR

This paper introduces a novel method using Galois closures of rational maps to construct complex varieties with non-trivial cup product kernels, and applies it to produce new Lagrangian surfaces with specific geometric properties.

Contribution

It develops a new approach based on Galois closures for constructing varieties with particular cohomological features and applies it to generate and analyze Lagrangian surfaces.

Findings

Constructed new Lagrangian surfaces in their Albanese varieties.

Computed Chern invariants of these surfaces.

Proved these surfaces are not fibred over higher genus curves.

Abstract

We use Galois closures of finite rational maps between complex projective varieties to introduce a new method for producing varieties such that the holomorphic part of the cup product map has non-trivial kernel. We then apply our result to the two-dimensional case and we construct a new family of surfaces which are Lagrangian in their Albanese variety. Moreover, we analyze these surfaces computing their Chern invariants, and proving that they are not fibred over curves of genus greater than one.