Renat Abugaliev
Let $Y$ be a smooth hypersurface in a projective irreducible holomorphic symplectic manifold X of dimension 2n. The characteristic foliation $F$ is the kernel of the symplectic form restricted to Y. In this article we prove that a generic leaf of the characteristic foliation is dense in Y if Y has positive Beauville-Bogomolov-Fujiki square.
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A criterion for associated bundles and the weak Beauville--Bogomolov decomposition
本文给出纤维丛成为伴随纤维丛的判别准则，证明满足特定条件的满射纤维丛均为伴随纤维丛，进而推出丰反典范除子代数簇的弱博韦尔-博戈莫洛夫分解。
The Beauville-Bogomolov lattice is computed for a simplest singular symplectic manifold of dimension 4, obtained as a partial desingularization of the quotient $S^{[2]}/ι$, where $S^{[2]}$ is the Hilbert square of a K3 surface $S$ and $ι$ is a symplectic involution on it. This result applies, in particular, to the singular symplectic manifolds of dimension 4, constructed by Markushevich-Tikhomirov as compactifications of families of Prym varieties of a linear system of curves on a K3 surface with an anti-symplectic involution.
We study the global geometry of the ten dimensional O'Grady irreducible symplectic variety. We determine its second Betti number, its Beauville form and its Fujiki constant.
The weak Beauville--Bogomolov decomposition in characteristic $p\geq 0$
本文在特征p≥0下研究了弱普通K-平凡变种的Beauville-Bogomolov分解，并探讨了其在正特征下的有理点和基本群行为。
We generalize Fujiki relation of Beauville-Bogomolov quadratic form on a projective symplectic variety. As an application, we study a fibre space structure of a projective symplectic variety.
Beauville--Bogomolov--Fujiki positivity and Moishezonness of complex symplectic manifolds in Fujiki class $\mathscr C$
本文研究Fujiki C类中超Fujiki流形的BBF正性与Moishezon性的关系，证明Moishezon超Fujiki流形的相关性质，给出超Fujiki 4-fold等的对应判据并应用于光滑族的Moishezon轨迹描述。
Beauville-Bogomolov decomposition for klt varieties
本文给出了具有对数终端奇点和零第一陈类的紧致Kähler簇的Beauville-Bogomolov分解定理的奇异版本的几乎自包含证明。
Fujiki relations and fibrations of irreducible symplectic varieties
本文针对推广不可约辛流形的奇异复射影簇，证明广义Beauville-Bogomolov形式满足Fujiki关系且秩与光滑情形一致，并用新定义验证其纤维化表现与不可约辛流形相似。
A strong counterexample to the log canonical Beauville--Bogomolov decomposition
本研究构造了维数不小于4的对数典范K-平凡簇作为对数典范情形下Beauville--Bogomolov分解的强反例，同时给出相关拟射影簇构造，并证明反典范类nef的对数典范配对的Albanese态射为局部稳定族。
We give a simplified proof (in characteristic zero) of the decomposition theorem for complex projective varieties with klt singularities and numerically trivial canonical bundle. The proof rests in an essential way on most of the partial results of the previous proof obtained by many authors, but avoids those in positive characteristic by S. Druel. The single to some extent new contribution is an algebraicity and bimeromorphic splitting result for generically locally trivial fibrations with fibres without holomorphic vector fields. We give first the proof in the easier smooth case, following t…
The Beauville-Bogomolov class as a characteristic class
本文在形变等价于 K3 曲面 Hilbert 概形的紧致 Kähler 流形的乘积上构造自反扭曲层，并用其限制的示性类刻画 Beauville–Bogomolov 类。
In 1995, Dan Guan constructed examples of non-Kahler, simply-connected holomorphically symplectic manifolds. An alternative construction, using the Hilbert scheme of Kodaira-Thurston surface, was given by F. Bogomolov. We investigate topology and deformation theory of Bogomolov-Guan manifolds and show that it is similar to that of hyperkahler manifolds. We prove the local Torelli theorem, showing that holomorphically symplectic deformations of BG-manifolds are unobstructed, and the corresponding period map is locally a diffeomorphism. Using the local Torelli theorem, we prove the Fujiki formul…
We investigate the existence of Lagrangian fibrations on the generalized Kummer varieties of Beauville. For a principally polarized abelian surface A of Picard number one we find the following: The Kummer variety K^n(A) is birationally equivalent to another irreducible symplectic variety admitting a Lagrangian fibration, if and only if n is a perfect square. And this is the case if and only if K^n(A) carries a divisor with vanishing Beauville-Bogomolov square.
Beauville--Bogomolov--Yau decomposition for Kähler generalized pairs
本文在Kähler框架下建立广义对的Beauville--Bogomolov--Yau分解，通过有限拟平展覆盖和局部常数纤维化，将Calabi--Yau型广义klt对分解为有理连通纤维与Calabi--Yau基，并应用于消除单值得到标准因子分解。
Let X be a projective irreducible holomorphic symplectic manifold. The second integral cohomology of X is a lattice with respect to the Beauville-Bogomolov pairing. A divisor E on X is called a prime exceptional divisor, if E is reduced and irreducible and of negative Beauville-Bogomolov degree. Let E be a prime exceptional divisor on X. We first observe that associated to E is a monodromy involution of the integral cohomology of X, which acts on the second cohomology lattice as the reflection by the cohomology class of E (Theorem 1.1). We then specialize to the case that X is deformation equi…
A generic K3 surface of degree 2t is a general complex projective K3 surface whose Picard group is generated by the class of an ample divisor whose with respect to the intersection form is 2t. We show that if X is the Hilbert square of a generic K3 surface of degree 2t, with $t > 2$, such that X admits an ample divisor with $q_X(D)=2$, where $q_X$ Beauville-Bogomolov-Fujiki form, then X is a double EPW sextic.
Let X be an irreducible symplectic manifold and L a divisor on X. Assume that L is isotropic with respect to the Beauville-Bogomolov quadratic form. We define the rational Lagrangian locus and the movable locus on the universal deformation space of the pair (X, L). We prove that the rational Lagrangian locus is empty or coincide with the movable locus.
In this paper we partly extend the Beauville-Bogomolov decomposition theorem to the singular setting. We show that any complex projective variety of dimension at most five with canonical singularities and numerically trivial canonical class admits a finite cover, étale in codimension one, that decomposes as a product of an Abelian variety, and singular analogues of irreducible Calabi-Yau and irreducible holomorphic symplectic varieties.
Irreducible symplectic manifolds are one of the three building blocks of compact Kähler manifolds with numerically trivial canonical bundle by the Beauville-Bogomolov decomposition theorem. There are several singular analogues of irreducible symplectic manifolds, in particular in the context of compact Kähler orbifolds, and in the context of normal projective varieties with canonical singularities. In this paper we will collect their definitions, analyze their mutual relations and provide a list of known examples.
We propose a general framework governing the intersection properties of extremal rays of irreducible holomorphic symplectic manifolds under the Beauville-Bogomolov form. Our main thesis is that extremal rays associated to Lagrangian projective subspaces control the behavior of the cone of curves. We explore implications of this philosophy for examples like Hilbert schemes of points on K3 surfaces and generalized Kummer varieties. We also collect evidence supporting our conjectures in specific cases.
Let X be an irreducible symplectic manifold and L a nef line bundle on X which is isotropic with respect to the Beauville-Bogomolov quadratic form. It is known that a subgroup Aut(X,L) of an automorphism group of X which fix L is almost abelian. We give a formula of the rank of Aut(X,L) in terms of MBM divisors. We also prove that the nef cone of X cut out MBM classes, which is a generalization of Kovac's structure theorem of nef cones of K3 surfaces
Thirty-three deformation classes of compact hyperkähler orbifolds
本文研究了通过商映射得到的不可约辛流形的变形类，探讨了在不同维度下通过终端化得到的orbifold例子，并计算了其奇异性质。
We construct a new 20-dimensional family of projective 6-dimensional irreducible holomorphic symplectic manifolds. The elements of this family are deformation equivalent with the Hilbert scheme of three points on a K3 surface and are constructed as natural double covers of special codimension 3 subvarieties of the Grassmanian G(3,6). These codimension 3 subvarieties are defined as Lagrangian degeneracy loci and their construction is parallel to that of EPW sextics, we call them the EPW cubes. As a consequence we prove that the moduli space of polarized IHS sixfolds of K3-type, Beauville-Bogomo…
Varieties admitting a holomorphic symplectic form: LLV algebras and derived equivalences
利用Beauville-Bogomolov分解计算具有全纯辛形式的复光滑射影簇的LLV代数，推广超凯勒簇和阿贝尔簇的已知结果，并证明Orlov猜想对许多此类簇成立，特别对所有四维情形成立。
Contents 1. Algebraicity criterion: statement 2. Proof of the algebraicity criterion. 3. Pseudoeffectivity and movable classes. 4. Harder-Narasimhan filtrations and pseudo-effectivity. 5. Pseudo-effectivity of relative canonical bundles. 6. Rational curves and non-pseudoeffectivity of the canonical/cotangent bundles. 7. Birational stability of the cotangent bundle 8. Shafarevich-Viehweg conjecture. 9. Numerically trivial foliations. 10. The Beauville-Bogomolov-Yau decomposition 11. Some questions 11.1. Pseudoeffectivity of determinants. 11.2. Variations on Abundance. 11.3. Special manifolds. R…
Polygons of Newton-Okounkov type on irreducible holomorphic symplectic manifolds
该研究在射影不可约全纯辛流形上为任意大R-除子构造了二维Newton-Okounkov型凸多边形，推导其体积与Beauville-Bogomolov-Fujiki型、除子式Zariski分解正部的关系，发现其性质类似光滑复射影曲面的对应Newton-Okounkov体。
Riemann-Roch Polynomials, MBM Classes and Poor IHS Manifolds
本文研究贫乏不可约全纯辛流形，证明其自然锥重合且无全纯叶理，并利用Riemann-Roch多项式根给出检测MBM类及有理曲线存在的统一准则，同时刻画了此类流形的形变空间。
Characteristic foliations -- a survey
本文综述了紧致 hyperkähler 流形上光滑除子特征叶状结构的几何最新进展，包含完整证明，并指出四维下其叶行为由 Beauville-Bogomolov 平方决定，高维部分结果依赖丰度猜想。
We classify the cohomology classes of Lagrangian 4-planes $¶^4$ in a smooth manifold $X$ deformation equivalent to a Hilbert scheme of 4 points on a $K3$ surface, up to the monodromy action. Classically, the cone of effective curves on a $K3$ surface $S$ is generated by nonegative classes $C$, for which $(C,C)\geq0$, and nodal classes $C$, for which $(C,C)=-2$; Hassett and Tschinkel conjecture that the cone of effective curves on a holomorphic symplectic variety $X$ is similarly controlled by "nodal" classes $C$ such that $(C,C)=-γ$, for $(\cdot,\cdot)$ now the Beauville-Bogomolov form, where …
A decomposition theorem for singular Kähler spaces with trivial first Chern class of dimension at most four
本文证明了具有klt奇点和消失第一陈类的紧Kähler四重折叠存在Beauville-Bogomolov分解，并通过提出Lipman-Zariski猜想的新版本，证明了其具有小射影形变及射影基本群。
In this article we investigate Hirzebruch-Riemann-Roch formulae for line bundles on irreducible symplectic Kähler manifolds. As Huybrechts has shown, for every irreducible complex Kähler manifold $X$ of dimension $2n$, there are numbers $a_0, a_2, ..., a_{2n}$ such that $χ(L) = \sum_{k = 0}^n a_{2k}/(2k)! q_X(c_1(L))^k$ for the Euler characteristic of a line bundle $L$, where $q_X: H^2(X, \mathbbm C) \to \mathbbm C$ is the Beauville-Bogomolov quadratic form of $X$. Using Rozansky-Witten classes similar to Hitchin and Sawon, we obtain a formula expressing the $a_{2k}$ in terms of Chern numbers …
On the Boucksom-Zariski decomposition for irreducible symplectic varieties and bounded negativity
本文研究了不可约辛流形上Boucksom-Zariski分解的性质，并证明了在光滑情况下负数有界猜想的有效版本，从而得出关于辛流形上大线包的有效应性结论。
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