Daisuke Matsushita
Let X be a projective irreducible symplectic manifold and L a non trivial nef divisor on X. Assume that the nef dimension of L is strictly less than the dimension of X. We prove that L is semiample
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Under some conditions on the deformation type, which we expect to be satisfied for arbitrary irreducible symplectic varieties, we describe which big and nef line bundles on irreducible symplectic varieties have base divisors. In particular, we show that such base divisors are always irreducible and reduced. This is applied to understand the behaviour of divisorial base components of big and nef line bundles under deformations and for K3$^{[n]}$-type and Kum$^n$-type.
An elementary description of nef cone for irreducible holomorphic symplectic manifolds
该研究描述K3型和Kummer型不可约全纯辛流形的MBM类，分析其与相关理论的联系，并将低维数值方法应用于这两类情况，为nef锥研究提供基础结果。
We show that the nefness of the canonical bundle of compact Kähler threefolds is invariant under deformed symplectic diffeomorphisms.
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
We approach non-divisorial base loci of big and nef line bundles on irreducible symplectic varieties. While for K3 surfaces, only divisorial base loci can occur, nothing was known about the behaviour of non-divisorial base loci for more general irreducible symplectic varieties. We determine the base loci of all big and nef line bundles on the Hilbert scheme of two points on very general K3 surfaces of genus two and on their birational models. Remarkably, we find an ample line bundle with a non-trivial base locus in codimension two. We deduce that, generically in the moduli spaces of polarized …
In this note, we prove that every fibre space structures of a projective irreducible symplectic manifold is a lagrangian fibration.
We give a characterization of irreducible symplectic fourfolds which are given as Hilbert scheme of points on a K3 surface.
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.
Let M_g be the moduli space of smooth curves of genus g >= 3, and \bar{M}_g the Deligne-Mumford compactification in terms of stable curves. Let \bar{M}_g^{[1]} be an open set of \bar{M}_g consisting of stable curves of genus g with one node at most. In this paper, we determine the necessary and sufficient condition to guarantee that a Q-divisor D on \bar{M}_g is nef over \bar{M}_g^{[1]}, that is, (D . C) >= 0 for all irreducible curves C on \bar{M}_g with C \cap \bar{M}_g^{[1]} \not= \emptyset.
This paper is concerned with non-symplectic involutions of irreducible symplectic manifolds of $K3^{[n]}$-type. We will give a criterion for deformation equivalence and use this to give a lattice-theoretic description of all deformation types. While moduli spaces of $K3^{[n]}$-type manifolds with non-symplectic involutions are not necessarily Hausdorff, we will construct quasi-projective moduli spaces for a certain well-behaved class of such pairs.
We show that the Andre motive of an irreducible, symplectic variety, which is deformation equivalent to the Hilbert scheme of points on a K3 surface, is an object the category generated by its motive truncated in degree two.
We prove that the integral Hodge conjecture holds for 1-cycles on irreducible holomorphic symplectic varieties of K3 type and of Generalized Kummer type. As an application, we give a new proof of the integral Hodge conjecture for cubic fourfolds.
We show the termination of any log-minimal model program for a pair $(X,Δ)$ of a symplectic manifold $X$ and an effective $\mathbb R$-divisor $Δ$.
We study the monodromy operators on the betti cohomologies associated to a good degeneration of irreducible symplectic manifold and we show that the unipotency of the monodromy operator on the middle cohomology is at least the half of the dimension.
On the kernels of Lefschetz operators
该研究刻画了丰富类完全交产生的莱夫谢茨算子的核，证明了相关混合莱夫谢茨性质、核的对角性质，建立了核解耦定理，并应用其证明了Shenfeld-van Handel与Hu-Xiao的猜想。
We generalize Nikulin's and Dolgachev's lattice-theoretical mirror symmetry for K3 surfaces to lattice polarized higher dimensional irreducible holomorphic symplectic manifolds. In the case of fourfolds of $K3^{\left[2\right]}-$type we then describe mirror families of polarized fourfolds and we give an example with mirror non-symplectic involutions.
We prove the existence of a universal family over every component of the moduli space of marked irreducible holomorphic symplectic manifolds. The analogous result follows for the Teichmuller spaces.
In the definition of irreducible holomorphic symplectic manifolds the condition of being simply connected can be replaced by vanishing irregularity. We discuss finite quotients X of complex tori where the space of reflexive 2-forms is generated by a holomorphic symplectic form, and their Lagrangian fibrations. Neither X nor the base can be smooth unless X is a 2-torus.
There is two group actions on the Fano scheme of lines such that the quotient becomes an irreducible symplectic manifold. We showed that both quotients are birational to the generalized Kummer variety or the 2-points Hilbert scheme of a K3 surface.
In this paper we analyse the birational geometry of O'Grady ten dimensional manifolds, giving a characterisation of Kaehler classes and lagrangian fibrations. Moreover, we study symplectic compactifications of intermediate jacobian fibrations of smooth cubic fourfolds.
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.
We study O'Grady examples of irreducible symplectic varieties: we establish that both of them can be deformed into lagrangian fibrations. We analyse in detail the topology of the six dimensional example: in particular we compute its Euler characteristic and determine its Beauville form.
Exploiting recent results on the ample cone of irreducible symplectic manifolds, we provide a different point of view for the computation of their monodromy groups. In particular, we give the final step in the computation of the monodromy group for generalised Kummer manifolds and we prove that the monodromy of O'Grady's ten dimensional manifold is smaller than what was expected.
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 determine the wall divisors on irreducible symplectic orbifolds which are deformation equivalent to a special type of examples, called Nikulin orbifolds. The Nikulin orbifolds are obtained as partial resolutions in codimension 2 of a quotient by a symplectic involution of a Hilbert scheme of 2 points on a K3 surface. This builds on the previous article arXiv:2009.04873 in which the theory of wall divisors was generalized to orbifold singularities.
We show that the moduli space of polarized irreducible symplectic manifolds of $K3^{[n]}$-type, of fixed polarization type, is not always connected. This can be derived as a consequence of Eyal Markman's characterization of polarized parallel-transport operators of $K3^{[n]}$-type.
We show that the graded Chow rings of two birational irreducible symplectic varieties are isomorphic. This lifts a result known for the cohomology algebras to the level of Chow rings, despite the non-injectivity the cycle class map. In the special case of general Mukai flops, we present an alternative approach based on explicit calculations.
Irreducible symplectic varieties via K3-del Pezzo double covers
对于2≤n≤10，构造第二Betti数为16≤b₂≤24的2n维不可约辛簇，作为一般K3-del Pezzo双重覆盖上Beauville-Mukai系统的有限辛商群的非平凡终端化。
We propose a generalization of Verbitsky's global Torelli theorem in the framework of compact Kähler irreducible holomorphically symplectic orbifolds by adapting Huybrechts' proof (arXiv:1106.5573). As intermediate step needed, we also provide a generalization of the twistor space and the projectivity criterion based on works of Campana (arXiv:math/0402243) and Huybrechts (arXiv:math/0106014) respectively.
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。
Irreducible symplectic varieties via relative Prym varieties
本文利用带反辛对合的 K3 曲面及商曲面线性系构造相对 Prym 簇，给出其为本原或不可约辛簇的判据，并在无穷多维度中产生不可约辛簇序列。
Irreducible symplectic varieties with a large second Betti number
本文证明了非紧Lagrangian fibration存在不可约辛紧化，并应用于证明包含固定cubic fourfold的cubic fivefolds的relative Jacobian fibration可被第二Betti数至少为24的不可约辛簇紧化，属于新形变类型。
Moduli spaces of polarised (with fixed polarisation type) irreducible holomorphic symplectic manifolds deformation equivalent to Hilbert schemes of points on $K3$ surfaces are not connected in general and A. Apostolov counted the number of their components. We extend Apostolov's results to the case of manifolds deformation equivalent to generalised Kummer varieties.
We show that the moduli spaces of sheaves on a projective K3 surface are irreducible symplectic varieties, and that the same holds for the fibers of the Albanese map of moduli spaces of sheaves on an Abelian surface.
On the Kähler cone of irreducible symplectic orbifolds
本文在不可约辛轨形上推广了全局Torelli定理及Kähler锥相关结果，并新定义了镜像对称作为模空间上的对合。
Given a projective irreducible symplectic manifold $M$ of dimension $2n$, a projective manifold $X$ and a surjective holomorphic map $f:M \to X$ with connected fibers of positive dimension, we prove that $X$ is biholomorphic to the projective space of dimension $n$. The proof is obtained by exploiting two geometric structures at general points of $X$: the affine structure arising from the action variables of the Lagrangian fibration $f$ and the structure defined by the variety of minimal rational tangents on the Fano manifold $X$.
On the finiteness of twists of irreducible symplectic varieties
本文利用不可约辛簇的锥猜想，证明了在特征0的固定有限域扩张下不可约辛簇扭曲的有限性，并讨论了非闭域锥猜想、导出等价扭曲有限性及正特征情形。
Rational curves on Hilbert schemes of points on $K3$ surfaces and generalised Kummer manifolds are constructed by using Brill-Noether theory on nodal curves on the underlying surface. It turns out that all wall divisors can be obtained, up to isometry, as dual divisors to such rational curves. The locus covered by the rational curves is then described, thus exhibiting algebraically coisotropic subvarieties that deform to general small deformations of the manifold. This provides strong evidence for a conjecture by Voisin concerning the Chow ring of irreducible holomorphic symplectic manifolds. …
We introduce the notion of induced automorphisms in order to state a criterion to determine whether a given automorphism on a manifold of $K3^{[n]}$ type is, in fact, induced by an automorphism of a $K3$ surface and the manifold is a moduli space of stable objects on the $K3$. This criterion is applied to the classification of non-symplectic prime order automorphisms on manifolds of $K3^{[2]}$ type and we prove that almost all cases are covered. Variations of this notion and the above criterion are introduced and discussed for the other known deformation types of irreducible symplectic manifol…
Supersingular Tate conjecture for irreducible symplectic varieties of known types: I
本文证明多种已知类型的超奇异不可约辛簇在特定Artin不变量条件下具有Tate或超奇异阿贝尔Chow motive，进而其乘积满足超奇异Tate猜想。
We shall give an example of irreducible symplectic manifolds X and Y which are not bimeromorphic, but have the same periods in the second cohomologies. More explicitly, X and Y are generalized Kummer varieties of dim 4 for a general complex torus T of dim 2 and its dual torus T^*. A key result is an observation of Shioda for the periods of T and T^*. As a concluding remark, we shall discuss the difference between our case and Kummer surfaces.
The Cone Conjecture for Primitive Symplectic Varieties over a Field of Characteristic Zero and an Application
本文证明了在特征零域上第二贝蒼地数大于五的有理数终端极小原始辛簇的 Kawamata-Morrison 锥猜想，并应用于证明某些条件下纤维丛的相对可动和相对 nef 锥猜想。
Let M be a compact, holomorphically symplectic Kahler manifold, and $η$ a (1,1)-current which is nef (a limit of Kahler forms). Assume that the cohomology class of $η$ is parabolic, that is, its top power vanishes. We prove that all Lelong sets of $η$ are coisotropic. When M is generic, this is used to show that all Lelong numbers of $η$ vanish. We prove that any hyperkahler manifold with Pic(M) of rank 1 has non-trivial coisotropic subvarieties, if a generator of Pic(M) is parabolic.
We study the cohomological properties of the fixed locus $X^G$ of an automorphism group $G$ of prime order $p$ acting on a variety $X$ whose integral cohomology is torsion-free. We obtain an precise relation between the mod $p$ cohomology of $X^G$ and natural invariants for the action of $G$ on the integral cohomology of $X$. We apply these results to irreducible holomorphic symplectic manifolds of deformation type of the Hilbert scheme of two points on a K3 surface: the main result of this paper is a formula relating the dimension of the mod $p$ cohomology of $X^G$ with the rank and the discr…
In the context of irreducible holomorphic symplectic manifolds, we say that (anti)holomorphic (anti)symplectic involutions are brane involutions since their fixed point locus is a brane in the physicists' language, i.e. a submanifold which is either complex or lagrangian submanifold with respect to each of the three Kähler structures of the associated hyperkähler structure. Starting from a brane involution on a K3 or abelian surface, one can construct a natural brane involution on its moduli space of sheaves. We study these natural involutions and their relation with the Fourier--Mukai transfo…
We show that there is a good notion of irreducible sympelectic varieties of $\mathrm{K3}^{[n]}$-type over an arbitrary field of characteristic zero or $p > n + 1$. Then we construct mixed characteristic moduli spaces for these varieties. Our main result is a generalization of Ogus' crystalline Torelli theorem for supersingular K3 surfaces. For applications, we answer a slight variant of a question asked by F. Charles on moduli spaces of sheaves on K3 surfaces and give a crystalline Torelli theorem for supersingular cubic fourfolds.
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。
Irreducible Holomorphic Symplectic manifolds with an action of $\mathbb Z^4_3 : \mathcal A_6$
本文刻画了同时具有最大阶辛群 $\mathbb Z_3^4 : \mathcal A_6$ 作用和非辛自同构的 K3$^{[2]}$ 型不可约全纯辛流形，并证明 Fermat 三次四维流形的直线 Fano 簇达到自同构群最大阶 174960。
We study the boundary behavior of the invariant of $K3^{[2]}$-type manifolds with antisymplectic involution, which we obtained using equivariant analytic torsion. We show the algebraicity of the singularity of the invariant by using the asymptotic of equivariant Quillen metrics and equivariant $L^2$-metrics. We prove that, in some cases, the invariant coincides with Yoshikawa's invariant for 2-elementary K3 surfaces. Hence, in these cases, our invariant is expressed as the Petersson norm of a Borcherds product and a Siegel modular form.
A symplectic fourfold
本文通过研究超Kähler流形非自然群作用的终端化，构造了具有b2=4和非商奇点的四维不可约辛流形，为已知全球Torelli定理不成立的ISV提供了具体例子。
In this note we derive from deep results due to Clozel-Ullmo the density of Noether-Lefschetz loci inside the moduli space of marked (polarized) irreducible holomorphic symplectic (IHS) varieties. In particular we obtain the density of Hilbert schemes of points on projective $K3$ surfaces and of projective generalized Kummer varieties in their moduli spaces. We present applications to the existence of rational curves on projective deformations of such varieties, to the study of relevant cones of divisors, and a refinement of Hassett's result on cubic fourfolds whose Fano variety of lines is is…
Boucksom-Zariski and Weyl chambers on irreducible holomorphic symplectic manifolds
受 Bauer 等人启发，本文为射影不可约全纯辛流形的大锥建立腔室分解，并研究其与 Weyl 腔室及体积函数的关系，最终确定伪有效锥结构。
In this note, we investigate fibre space structures of a projective irreducible symplectic manifold. We prove that an 2n-dimensional projective irreducible symplectic manifold admits only an n-dimensional fibration over a Fano variety which has only Q-factorial log-terminal singularities and whose Picard number is one. Moreover we prove that a general fibre is an abelian variety up to finite unramified cover, especially, a general fibre is an abelian surface for 4-fold.
We study symplectic varieties defined over fields of positive characteristics, especially the supersingular ones, generalizing the theory of supersingular K3 surfaces. In this work, we are mainly interested in the following two types of symplectic varieties over an algebraically closed field of positive characteristic, under natural numerical conditions: (1) smooth moduli spaces of sheaves on K3 surfaces and (2) smooth Albanese fibers of moduli spaces of sheaves on abelian surfaces. Several natural definitions of the supersingularity for symplectic varieties are discussed, which are proved to …
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。
Moduli spaces on Kuznetsov components are Irreducible Symplectic Varieties
本文研究立方四胞体Kuznetsov分支中具非本原非OG10型Mukai向量的Bridgeland半稳定对象模空间，证明其为具因子终端奇点的射影不可约辛簇，形变类由m和v₀²唯一确定，推广并拓展了此前相关模空间的研究结果。
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 …
By results of the author there exists a projective (holomorphic) symplectic desingularization of the moduli space of rank-two torsion-free sheaves on a genus-two Jacobian with $c_1=0$ and $c_2=2$. This desingularization has a natural map to the self-product of the Jacobian. We show that the fiber over $(0,0)$ is a 6-dimensional projective irreducible symplectic variety (and hence a 12-dimensional compact Hyperkahler manifold) with second Betti number equal to 8. Thus it is not deformation equivalent to any of the (few) known examples of irreducible symplectic varieties, even up to birational e…
Analytic torsion for irreducible holomorphic symplectic fourfolds with involution, III: relation with the BCOV invariant
本文比较了Camere-Garbagnati-Mongardi型Calabi-Yau四维流形的BCOV不变量与对应K3⁽²⁾型带对合流形的不变量，证明特殊情况下前者可表为某Borcherds乘积的Petersson范数。
Analytic torsion for irreducible holomorphic symplectic fourfolds with involution, I: Construction of an invariant
本文利用等变解析挠率，为具有反辛对合的 $K3^{[2]}$ 型不可约全纯辛流形构造了一个新不变量，并给出了该不变量对数的复 Hessian 公式。
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