Topology of Lagrangian fibrations and Hodge theory of hyper-Kähler manifolds. (English) Zbl 1490.14019

Shen, Junliang; Yin, Qizheng

Duke Math. J. 171, No. 1, 209-241 (2022).

In the paper, a compact analogue of the \(P=W\) conjecture is proved, namely that, given an irreducible holomorphic symplectic variety equipped with a Lagrangian fibration, its Hodge and perverse numbers coincide. Given \(\pi\colon X\rightarrow Y\) a proper morphism with \(X\) a nonsingular algebraic variety, the perverse \(t\)-structure on the constructible derived category \(D_c^b(Y)\) induces a filtration on the cohomology \(H^*(X,\mathbb{Q})\): \[ P_{0} H^*(X,\mathbb{Q})\subseteq P_{1} H^*(X,\mathbb{Q})\subseteq\cdots \subseteq H^*(X,\mathbb{Q}). \] The perverse numbers are then given by the dimensions of the graded pieces of \(H^*(X,\mathbb{Q})\) associated to this filtration: \[ ^p h^{i,j}(X):= \dim\left( P_i H^{i+j}(X,\mathbb{Q})/P_{i-1} H^{i+j}(X,\mathbb{Q})\right). \] In the case where \(X\) is an irreducible holomorphic symplectic variety and \(\pi\) is a Lagrangian fibration, the authors show in Theorem 0.2 that Hodge and perverse numbers coincide, i.e., that \[ ^p h^{i,j}(X)=h^{i,j}(X). \] As an application, it is shown in Theorem 0.4.b that the intersection cohomology of the base of a Lagrangian fibration is always isomorphic to the cohomology of \(\mathbb{P}^n\). This gives a positive answer to a cohomological version of the well known conjecture which predicts that the base \(B\) is isomorphic to a projective space. The equality between Hodge and perverse numbers can also be used to study the restriction of cohomological classes on \(X\) to a nonsingular fiber \(X_b\): Denoting by \(\eta\) the relative ample class, it is proven in Theorem 0.4.a that the image of the restriction map \(H^d(X,\mathbb{Q})\rightarrow H^d(X_b,\mathbb{Q})\) is spanned by \(\eta^k|_{X_b}\) if \(d=2k\) is even and it is zero if \(d\) is odd. An alternative and more direct proof of this last statement is provided by Voisin in Appendix B. As an additional application, the authors describe in Theorem 0.5 the role played by perverse numbers in the construction of curve counting invariants on \(S\times \mathbb{C}\), where \(S\) is a \(K3\) surface.

Reviewer: Mauro Varesco (Bonn)

MSC:

14C30	Transcendental methods, Hodge theory (algebro-geometric aspects)
14J42	Holomorphic symplectic varieties, hyper-Kähler varieties
53C26	Hyper-Kähler and quaternionic Kähler geometry, “special” geometry
14D06	Fibrations, degenerations in algebraic geometry
14J28	\(K3\) surfaces and Enriques surfaces
14F45	Topological properties in algebraic geometry
14N35	Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)

Keywords:

curve counting; Hodge theory; hyper-Kähler manifolds; Lagrangian fibrations; perverse sheaves

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