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Topology of Lagrangian fibrations and Hodge theory of hyper-K\"ahler manifolds
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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.
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Cited by 2 Pith papers
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P=W for Lagrangian fibrations and degenerations of hyper-K\"ahler manifolds
For every Lagrangian fibration of a projective hyper-Kähler manifold, the perverse filtration equals the monodromy weight filtration of an associated type III degeneration.
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Torus fibers and the weight filtration
A single real torus inside a log Calabi-Yau complement (or a Calabi-Yau degeneration) computes the odd weight filtration, yielding P=W type results for rational surfaces and K3 surfaces.
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