# Topology of Lagrangian fibrations and Hodge theory of hyper-K\"ahler manifolds

## TL;DR

This paper proves a new correspondence between the topology of Lagrangian fibrations and the Hodge theory of hyper-K"ahler manifolds, establishing a compact analog of the P=W conjecture and exploring its implications.

## Contribution

It establishes a compact analog of the P=W conjecture for Lagrangian fibrations on hyper-K"ahler manifolds, linking perverse and Hodge numbers.

## Key findings

- Perverse numbers match Hodge numbers for holomorphic symplectic varieties with Lagrangian fibrations.
- The perverse filtration is multiplicative under cup product.
- Applications include insights into the topology of fibrations and refined Gopakumar-Vafa invariants.

## Abstract

We establish a compact analog of the P = W conjecture. For a holomorphic symplectic variety with a Lagrangian fibration, we show that the perverse numbers associated with the fibration match perfectly with the Hodge numbers of the total space. This builds a new connection between the topology of Lagrangian fibrations and the Hodge theory of hyper-K\"ahler manifolds. We present two applications of our result, one on the topology of the base and fibers of a Lagrangian fibration, the other on the refined Gopakumar-Vafa invariants of a K3 surface. Furthermore, we show that the perverse filtration associated with a Lagrangian fibration is multiplicative under cup product.