Remarks on degenerations of hyper-K\"ahler manifolds

  title={Remarks on degenerations of hyper-K\"ahler manifolds},
  author={J. Koll'ar and R. Laza and Giulia Sacc{\`a} and C. Voisin},
  journal={arXiv: Algebraic Geometry},
Using the Minimal Model Program, any degeneration of K-trivial varieties can be arranged to be in a Kulikov type form, i.e. with trivial relative canonical divisor and mild singularities. In the hyper-K\"ahler setting, we can then deduce a finiteness statement for monodromy acting on $H^2$, once one knows that one component of the central fiber is not uniruled. Independently of this, using deep results from the geometry of hyper-K\"ahler manifolds, we prove that a finite monodromy projective… Expand
The LLV decomposition of hyper-Kaehler cohomology
Looijenga-Lunts and Verbitsky showed that the cohomology of a compact hyper-Kaehler manifold $X$ admits a natural action by the Lie algebra $\mathfrak{so} (4, b_2(X)-2)$, generalizing the HardExpand
Deformation principle and Andr\'e motives of projective hyperk\"ahler manifolds
Let $X_1$ and $X_2$ be deformation equivalent projective hyperkahler manifolds. We prove that the Andre motive of $X_1$ is abelian if and only if the Andre motive of $X_2$ is abelian. Applying thisExpand
Lagrangian fibrations
We review the theory of Lagrangian fibrations of hyperkähler manifolds as initiated by Matsushita [Mat99, Mat01, Mat05]. We also discuss more recent work of Shen–Yin [SY18] and Harder–Li–Shen–YinExpand
Birational geometry of the intermediate Jacobian fibration of a cubic fourfold
We show that the intermediate Jacobian fibration associated to any smooth cubic fourfold $X$ admits a hyper-K\"ahler compactification $J(X)$ with a regular Lagrangian fibration $J \to \mathbb P^5$.Expand
The geometry of degenerations of Hilbert schemes of points
Given a strict simple degeneration $f \colon X\to C$ the first three authors previously constructed a degeneration $I^n_{X/C} \to C$ of the relative degree $n$ Hilbert scheme of $0$-dimensionalExpand
GIT versus Baily-Borel compactification for quartic K3 surfaces
Looijenga has introduced new compactifications of locally symmetric varieties that give a complete understanding of the period map from the GIT moduli space of plane sextics to the Baily-BorelExpand
GIT versus Baily-Borel compactification for K3's which are quartic surfaces or double covers of quadrics
Looijenga has introduced new compactifications of locally symmetric varieties that give a complete understanding of the period map from the GIT moduli space of plane sextics to the Baily-BorelExpand
Constructing local models for Lagrangian torus fibrations
We give a construction of Lagrangian torus fibrations with controlled discriminant locus on certain affine varieties. In particular, we apply our construction in the following ways. We find aExpand
Calabi-Yau Structures, Spherical Functors, and Shifted Symplectic Structures
A categorical formalism is introduced for studying various features of the symplectic geometry of Lefschetz fibrations and the algebraic geometry of Tyurin degenerations. This approach is informed byExpand
Extending holomorphic forms from the regular locus of a complex space to a resolution of singularities
We investigate under what conditions holomorphic forms defined on the regular locus of a reduced complex space extend to holomorphic (or logarithmic) forms on a resolution of singularities. We give aExpand


Large Complex Structure Limits of K3 Surfaces
Motivated by the picture of mirror symmetry suggested by Strominger, Yau and Zaslow, we made a conjecture concerning the Gromov-Hausdorff limits of Calabi-Yau n-folds (with Ricci-flat K\"ahlerExpand
Lagrangian fibrations on hyperk\"ahler manifolds - On a question of Beauville
Let X be a compact hyperk\"ahler manifold containing a complex torus L as a Lagrangian subvariety. Beauville posed the question whether X admits a Lagrangian fibration with fibre L. We show that thisExpand
The classical Weil-Petersson metric on the Teichmuller space of compact Riemann surfaces is a Kahler metric, which is complete only in the case of elliptic curves [Wo]. It has a naturalExpand
The essential skeleton of a degeneration of algebraic varieties
In this paper, we explore the connections between the Minimal Model Program and the theory of Berkovich spaces. Let $k$ be a field of characteristic zero and let $X$ be a smooth and projectiveExpand
A hyper-K\"ahler compactification of the Intermediate Jacobian fibration associated to a cubic fourfold
For a general cubic fourfold, it was observed by Donagi and Markman that the relative intermediate Jacobian fibration associated to the family of its hyperplane sections carries a natural holomorphicExpand
The dual complex of Calabi–Yau pairs
A log Calabi–Yau pair consists of a proper variety X and a divisor D on it such that $$K_X+D$$KX+D is numerically trivial. A folklore conjecture predicts that the dual complex of D is homeomorphic toExpand
Symplectic singularities
We introduce in this paper a particular class of rational singularities, which we call symplectic, and classify the simplest ones. Our motivation comes from the analogy between rational GorensteinExpand
Homological mirror symmetry and torus fibrations
In this paper we discuss two major conjectures in Mirror Symmetry: Strominger-Yau-Zaslow conjecture about torus fibrations, and the homological mirror conjecture (about an equivalence of the FukayaExpand
A global Torelli theorem for hyperkahler manifolds
A mapping class group of an oriented manifold is a quotient of its diffeomorphism group by the isotopies. We compute a mapping class group of a hypekahler manifold $M$, showing that it isExpand
On the Periods of Certain Rational Integrals: II
In this section we want to re-prove the results of ?? 4 and 8 using sheaf cohomology. One reason for doing this is to clarify the discussion in those paragraphs and, in particular, to show howExpand