Mapping class group and global Torelli theorem for hyperkahler manifolds: an erratum

OG6 型簇的强全局 Torelli 定理

A Strong Global Torelli theorem for OG6 type varieties

本文证明 OG6 型超凯勒流形由其次积分上同调及其 Hodge 结构和 Beauville--Bogomolov--Fujiki 形式唯一决定，并由此推出双亚纯等价即双全纯等价。

Torelli-type theorems for gravitational instantons with quadratic volume growth

We prove Torelli-type uniqueness theorems for both ALG$^*$ gravitational instantons and ALG gravitational instantons which are of order $2$. That is, the periods uniquely characterize these types of gravitational instantons up to diffeomorphism. We define a period mapping $\mathscr{P}$, which we show is surjective in the ALG cases, and open in the ALG$^*$ cases. We also construct some new degenerations of hyperkähler metrics on the K3 surface which exhibit bubbling of ALG$^*$ gravitational instantons.

hyperkähler流形的光滑Birman-Hilden理论

A smooth Birman-Hilden theory for hyperkähler manifolds

本文为容许hyperkähler结构的流形证明了Birman-Hilden定理的光滑类比，借此探究具有非平凡基本群的4维流形的光滑映射类群，并应用于Enriques曲面的Nielsen实现。

Twisted cubics on cubic fourfolds and stability conditions

We give an interpretation of the Fano variety of lines on a cubic fourfold and of the hyperkahler eightfold, constructed by Lehn, Lehn, Sorger and van Straten from twisted cubic curves in a cubic fourfold non containing a plane, as moduli spaces of Bridgeland stable objects in the Kuznetsov component. As a consequence, we reprove the categorical version of Torelli Theorem for cubic fourfolds, we obtain the identification of the period point of LLSvS eightfold with that of the Fano variety, and we discuss derived Torelli Theorem for cubic fourfolds.

The global moduli theory of symplectic varieties

We develop the global moduli theory of symplectic varieties in the sense of Beauville. We prove a number of analogs of classical results from the smooth case, including a global Torelli theorem. In particular, this yields a new proof of Verbitsky's global Torelli theorem in the smooth case (assuming $b_2\geq 5$) which does not use the existence of a hyperkähler metric or twistor deformations.

超凯勒流形族中的丰度与SYZ猜想

The abundance and SYZ conjectures in families of hyperkahler manifolds

研究超凯勒流形上具有nef且非大第一陈类的全纯线丛的半丰性，通过引入Teichmuller空间并证明全局Torelli定理，建立半丰性的形变不变性。

Integral constraints on the monodromy group of the hyperkahler resolution of a symmetric product of a K3 surface

Let M be a 2n-dimensional Kahler manifold deformation equivalent to the Hilbert scheme of length n subschemes of a K3 surface S. Let Mon be the group of automorphisms of the cohomology ring of M, which are induced by monodromy operators. The second integral cohomology of M is endowed with the Beauville-Bogomolov bilinear form. We prove that the restriction homomorphism from Mon to the isometry group O[H^2(M)] is injective, for infinitely many n, and its kernel has order at most 2, in the remaining cases. For all n, the image of Mon in O[H^2(M)] is the subgroup generated by reflections with res…

Birational symplectic manifolds and their deformations

The known counterexamples to the global Torelli theorem for higher-dimensional hyperkahler manifolds are provided by birational manifolds. We address the question whether two birational hyperkahler manifolds (i.e. irreducible symplectic) manifolds always define non-separated points in the moduli space of marked manifolds. An affirmative answer is given for the cases of Mukai's elementary transformations and birational correspondences which are isomorphic in codimension two. The techniques are applied to show that the moduli spaces of rank two sheaves on a K3 surface are deformation equivalent …

Applications of the affine structures on the Teichmüller spaces

We prove the existence of global sections trivializing the Hodge bundles on the Hodge metric completion space of the Torelli space of Calabi--Yau manifolds, a global splitting property of these Hodge bundles. We also prove that a compact Calabi--Yau manifold can not be deformed to its complex conjugate. These results answer certain open questions in the subject. A general result about the period map to be bi-holomorphic from the Hodge metric completion space of the Torelli space of Calabi--Yau type manifolds to their period domains is proved and applied to the cases of K$3$ surfaces, cubic fou…

Hyperbolic geometry of the ample cone of a hyperkahler manifold

Let $M$ be a compact hyperkahler manifold with maximal holonomy (IHS). The group $H^2(M, R)$ is equipped with a quadratic form of signature $(3, b_2-3)$, called Bogomolov-Beauville-Fujiki (BBF) form. This form restricted to the rational Hodge lattice $H^{1,1}(M,Q)$, has signature $(1,k)$. This gives a hyperbolic Riemannian metric on the projectivisation of the positive cone in $H^{1,1}(M,Q)$, denoted by $H$. Torelli theorem implies that the Hodge monodromy group $Γ$ acts on $H$ with finite covolume, giving a hyperbolic orbifold $X=H/Γ$. We show that there are finitely many geodesic hypersurfac…