# Cohomologies of locally conformally symplectic manifolds and solvmanifolds

## TL;DR

This paper explores the cohomological properties of locally conformally symplectic manifolds, introducing new lcs cohomologies, and investigates their applications to solvmanifolds and Oeljeklaus-Toma manifolds, including dualities and Lefschetz conditions.

## Contribution

It introduces lcs cohomologies, studies their properties, and applies these to specific classes of manifolds, establishing new results on their geometric and arithmetic structures.

## Key findings

- Oeljeklaus-Toma manifolds with one complex place satisfy the Mostow property
- The study establishes dualities and Hard Lefschetz conditions for lcs cohomologies
- Inoue surface of type S^0 is included in the class satisfying these properties

## Abstract

We study the Morse-Novikov cohomology and its almost-symplectic counterpart on manifolds admitting locally conformally symplectic structures. More precisely, we introduce lcs cohomologies and we study elliptic Hodge theory, dualities, Hard Lefschetz Condition. We consider solvmanifolds and Oeljeklaus-Toma manifolds. In particular, we prove that Oeljeklaus-Toma manifolds with precisely one complex place, and under an additional arithmetic condition, satisfy the Mostow property. This holds in particular for the Inoue surface of type $S^0$.