Fedor Bogomolov, Yuri Tschinkel
We construct examples of simply connected nonalgebraic symplectic fourfolds with a prescribed number of nonintersecting symplectic curves with positive self-intersections.
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We prove that there are at most two possibilities for the base of a Lagrangian fibration from a complex projective irreducible symplectic fourfold.
We classify the polarized symplectic automorphisms of Fano varieties of smooth cubic fourfolds (equipped with the Plücker polarization) and study the fixed loci.
Cubic fourfolds with a symplectic automorphism of prime order
本文研究了具有素数阶辛自同构的立方四维流形，证明其为理性的条件，并展示了两个非等变理性的有理立方四维流形家族。
We analyze the ample and moving cones of holomorphic symplectic manifolds, in light of recent advances in the minimal model program. As an application, we establish a numerical criterion for ampleness of divisors on fourfolds deformation-equivalent to punctual Hilbert schemes of K3 surfaces.
The present paper proves that finite symplectic groups of automorphisms of hyperkähler fourfolds deformation equivalent to the Hilbert scheme of two points on a $K3$ surface are contained in the simple group $Co_1$. Then we give an example of a symplectic automorphism of order 11 on the Fano scheme of lines of a cubic fourfold.
We study the birational geometry of irreducible holomorphic symplectic varieties arising as varieties of lines of general cubic fourfolds containing a cubic scroll. We compute the ample and moving cones, and exhibit a birational automorphism of infinite order explaining the chamber decomposition of the moving cone.
Non-symplectic automorphisms of prime order of O'Grady's tenfolds and cubic fourfolds
本文分类 OG10 型不可约全纯辛流形的素数阶非辛自同构，刻画其由三次四维流形诱导的情形，并用于探讨相关有理性猜想。
Let X be a holomorphic symplectic fourfold such that b_2=23 and i a symplectic involution of X . The fixed locus F of i is a smooth symplectic submanifold of X; we show that F contains at least 12 isolated points and 1 smooth surface. We conjecture that F is made of 28 isolated fixed points and 1 K3 surface and we provide evidences for the conjecture in some examples, as the Hilbert scheme of a K3 surface, the Fano variety of a cubic in P^5 and the double cover of an EPW sextic.
Cubic fourfolds with symplectic automorphisms
本文通过分类特定群的表示，确定具有辛自同构的立方四维流形的射影方程及模空间的不可约分量数量，并探讨了其定义域。
We prove that for any polarized symplectic automorphism of the Fano variety of lines of a cubic fourfold (equipped with the Plücker polarization), the induced action on the Chow group of 0-cycles is identity, as predicted by Bloch-Beilinson conjecture.
This note concerns hyperkähler fourfolds $X$ having a non-symplectic involution $ι$. The Bloch-Beilinson conjectures predict the way $ι$ should act on certain pieces of the Chow groups of $X$. The main result is a verification of this prediction for Fano varieties of lines on certain cubic fourfolds. This has consequences for the Chow ring of the quotient $X/ι$.
Let $X$ be a hyperkähler variety, and assume that $X$ admits a non-symplectic automorphism $σ$ of order $k>{1\over 2}\dim X$. Bloch's conjecture predicts that the quotient $X/<σ>$ should have trivial Chow group of $0$-cycles. We verify this for Fano varieties of lines on certain cubic fourfolds having an order $3$ non-symplectic automorphism.
Equivariant Kuznetsov components for cubic fourfolds with a symplectic involution
研究带辛对合的立方四重簇的等变库兹涅佐夫分量，证明其等价于K3曲面的导出范畴，S为固定轨迹分量。
Let $X$ be a hyperkähler variety, and assume $X$ has a non-symplectic automorphism $σ$ of order $>{1\over 2}\dim X$. Bloch's conjecture predicts that the quotient $X/<σ>$ should have trivial Chow group of $0$-cycles. We verify this for Fano varieties of lines on certain special cubic fourfolds having an order $3$ non--symplectic automorphism.
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。
We classify non symplectic prime order automorphisms and all finite order symplectic automorphism groups of generalised Kummer fourfolds using lattice theory and recent results on ample cones and monodromy groups. We study various geometric realisations of the automorphisms obtained in the classification. In the case of higher dimensional generalised Kummers, we provide full results for finite symplectic automorphisms groups and we sketch the classification for prime order non-symplectic groups.
This article is about hyperkähler fourfolds $X$ admitting a non-symplectic involution $ι$. The Bloch-Beilinson conjectures predict the way $ι$ should act on certain pieces of the Chow groups of $X$. The main result is a verification of this prediction for Fano varieties of lines on certain cubic fourfolds. This has some interesting consequences for the Chow ring of the quotient $X/ι$.
Let F be a polarized irreducible holomorphic symplectic fourfold, deformation equivalent to the Hilbert scheme parametrizing length-two zero-dimensional subschemes of a K3 surface. The homology group H^2(F,Z) is equipped with an integral symmetric nondegenerate form, the Beauville form. We give precise conjectures for the structure of the cone of effective curves - and by duality - the cone of ample divisors. Formally they are completely analogous to the results known for K3 surfaces, and they are expressed entirely in terms of the integers represented by the Beauville form restricted to Pic(F…
A 10-dimensional symplectic moduli space of torsion sheaves on the cubic 4-fold is constructed. It parametrizes the stable rank 2 vector bundles on the hypeplane sections of the cubic 4-fold which are obtained by Serre's construction from normal elliptic quintics. The natural projection to the dual projective 5-space parametrizing the hyperplane sections is a Lagrangian fibration. The symplectic structure is closely related (and conjecturally, is equal) to the quasi-symplectic one, induced by the Yoneda pairing on the moduli space.
Rational cubic fourfolds with a symplectic group of automorphisms
本文证明了具有非2幂阶循环辛自同构群的三次四重体为有理且属于Hassett因子C_d（d=14,42），并描述了相关Lech对结构。
There are three types of involutions on a cubic fourfold; two of anti-symplectic type, and one symplectic. Here we show that cubics with involutions exhibit the full range of behaviour in relation to rationality conjectures. Namely, we show a general cubic fourfold with symplectic involution has no associated K3 surface and is conjecturely irrational. In contrast, we show a cubic fourfold with a particular anti-symplectic involution has an associated K3, and is in fact rational. We show such a cubic is contained in the intersection of all non-empty Hassett divisors; we call such a cubic Hasset…
We classify the symplectic automorphism groups for cubic fourfolds. The main inputs are the global Torelli theorem for cubic fourfolds and the classification of the fixed-point sublattices of the Leech lattice. Among the highlights of our results, we note that there are 34 possible groups of symplectic automorphisms, with 6 maximal cases. The six maximal cases correspond to 8 non-isomorphic, and isolated in moduli, cubic fourfolds; six of them previously identified by other authors. Finally, the Fermat cubic fourfold has the largest possible order (174,960) for the automorphism group (non-nece…
We prove that a very general cubic fourfold containing a plane can be embedded into a holomorphic symplectic eightfold as a Lagrangian submanifold. We construct the desired holomorphic symplectic eightfold as a moduli space of Bridgeland stable objects in the derived category of the twisted K3 surface corresponding to the cubic fourfold containing a plane.
We apply the methods of C{ă}ld{ă}raru to construct a twisted Fourier-Mukai transform between a pair of holomorphic symplectic four-folds. More precisely, we obtain an equivalence between the derived category of coherent sheaves on a certain four-fold and the derived category of twisted sheaves on its `mirror' partner. As corollaries, we show that the two spaces are connected by a one-parameter family of deformations through Lagrangian fibrations, and we extend the original Fourier-Mukai transform to degenerations of abelian surfaces.
We show that the irreducible holomorphic symplectic eightfold Z associated to a cubic fourfold Y not containing a plane is deformation-equivalent to the Hilbert scheme of four points on a K3 surface. We do this by constructing for a generic Pfaffian cubic Y a birational map Z ---> Hilb^4(X), where X is the K3 surface associated to Y by Beauville and Donagi. We interpret Z as a moduli space of complexes on X and observe that at some point of Z, hence on a Zariski open subset, the complex is just the ideal sheaf of four points.
The Fano variety of lines on singular cyclic cubic fourfolds
本文研究奇异循环三次四重簇上直线Fano簇的辛解消，特别关注沿具有$A_i$型孤立奇点的三次三重簇分支的情形，并发现覆盖自同构诱导的三阶非辛自同构。
We study lines and twisted cubics on cubic fourfolds with simple isolated singularities. We show that the Hilbert scheme compactification of the total space of Starr's fibration on the space of twisted cubics on a cubic fourfold with simple isolated singularities and not containing a plane admits a contraction to a singular projective symplectic variety of dimension eight. The latter admits a crepant resolution which is a deformation of the symplectic eightfold constructed from the space of twisted cubics on a smooth cubic fourfold. Thereby we give another proof of the fact that the latter is …
We give a classification of all non-symplectic automorphisms of prime order p acting on irreducible holomorphic symplectic fourfolds deformation equivalent to the Hilbert scheme of two points on a K3 surface, for p=2,3 and 7\leq p \leq 19. Our classification relates the isometry classes of two natural lattices associated to the action of the automorphism on the second cohomology group with integer coefficients with some invariants of the fixed locus and we provide explicit examples. As an application, we find new examples of non-natural non-symplectic automorphisms.
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.
We prove that the Fano variety of lines of a cuspidal cyclic cubic fourfold is a symplectic variety with transversal A2-singularities and we study the properties of the nonsymplectic order three automorphism induced by the covering automorphism on the irreducible holomorphic symplectic manifold obtained by blowing up the singular locus.
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.
The base surface $B$ of a Lagrangian fibration $X\twoheadrightarrow B$ of a projective, irreducible symplectic fourfold $X$ is shown to be isomorphic to ${\mathbb P}^2$.
Terminalizations of quotients of Fano varieties of lines on cubic fourfolds
本文研究了立方四维流形上线的Fano簇商的终端化，计算了相关拓扑不变量，并识别出新的四维不可约解析辛流形变形类。
In this paper we study the Lagrangian fibrations for projective irreducible symplectic fourfolds and exclude the case of non-smooth base. Our method could be extended to the higher-dimensional cases.
Non-symplectic Indices of Automorphism Groups of Smooth Cubic Fourfolds
研究具有给定辛自同构群的光滑四次三维流形的全自同构群，通过群论和格论方法限制非辛指数，并分类秩19余不变格的所有可能自同构群对。
Let $X$ be an irreducible holomorphic symplectic fourfold and $D$ a smooth hypersurface in $X$. It follows from a result by Amerik and Campana that the characteristic foliation (that is the foliation given by the kernel of the restriction of the symplectic form to $D$) is not algebraic unless $D$ is uniruled. Suppose now that the Zariski closure of its general leaf is a surface. We prove that $X$ has a lagrangian fibration and $D$ is the inverse image of a curve on its base.
Counting Fourier-Mukai partners of cubic fourfolds
本文开发算法计算三次四维簇的虚拟Fourier-Mukai伙伴数，证明温和假设下虚拟计数可恢复实际计数，应用于带辛自同构的三次簇得相关结论，证明辛自同构非Fourier-Mukai不变量。
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。
We construct a new twenty-dimensional family of projective eight-dimensional irreducible holomorphic symplectic manifolds: the compactified moduli space M_3(Y) of twisted cubics on a smooth cubic fourfold Y that does not contain a plane is shown to be smooth and to admit a contraction M_3(Y) -> Z(Y) to a projective eight-dimensional symplectic manifold Z(Y). The construction is based on results on linear determinantal representations of singular cubic surfaces.
Automorphism Groups of Smooth Cubic Fourfolds through Lattice Theory
本文利用格论和OSCAR枚举算法，完成了42族光滑四次三维簇的所有可能自同构群分类，基于周期映射已知结果。
Lagrangian tens of planes, Enriques surfaces and holomorphic symplectic fourfolds
本文研究Enriques曲面Fano模型产生的P^5中十个相互相交平面的模空间，建立其与含该十元组的三次四维流形、Beauville–Donagi全纯辛四维流形及双重EPW六次超曲面等十维模空间的联系，并由此证明带超标记Enriques曲面模空间的不可约性。
The present work is concerned with the study of four-dimensional irreducible holomorphic symplectic manifolds with second Betti number $23$. We describe their birational geometry and their relations to EPW sextics.
We give a characterization of irreducible symplectic fourfolds which are given as Hilbert scheme of points on a K3 surface.
Higher Chow cycles, cyclic cubic fourfolds and Lagrangian subvarieties
本文研究了holomorphic李代数流形上的高维Chow循环，通过构造显式不可分解的(2,1)和(4,1)循环，首次在holomorphic李代数流形上提供了此类循环的例子，并探讨了循环在拉格朗日子变种上的限制现象。
O'Grady showed that certain special sextics in $\mathbb{P}^5$ called EPW sextics admit smooth double covers with a holomorphic symplectic structure. We propose another perspective on these symplectic manifolds, by showing that they can be constructed from the Hilbert schemes of conics on Fano fourfolds of degree ten. As applications, we construct families of Lagrangian surfaces in these symplectic fourfolds, and related integrable systems whose fibers are intermediate Jacobians.
Using the Fredholm setup of [12], we study genus zero (and higher) relative Gromov-Witten invariants with maximum tangency of symplectic log Calabi-Yau fourfolds. In particular, we give a short proof of [23, Conjecture 6.2] that expresses these invariants in terms of certain integral invariants by considering generic almost complex structures to obtain a geometric count. We also revisit the localization calculation of the multiple-cover contributions in [23, Proposition 6.1] and recalculate a few terms differently to provide more details and illustrate the computation of deformation/obstructio…
Classification of Symplectic Birational Involutions of Manifolds of $OG10$ type
本文通过三种技术（Leech格、三次四维流形、改进的Kneser邻域算法）完整分类了$OG10$型流形的辛双有理对合，得到三个几何实现的对合和三个例外对合。
In this note, which is an addendum to the e-print math.AG/9810121, we prove that the variety VSP(F,10) of presentations of a general cubic form F in 6 variables as a sum of 10 cubes is a smooth symplectic 4-fold, which is deformation equivalent to the Hilbert square of a K3 surface of genus 8 but different from the family of lines on a cubic 4-fold. This provides a new geometric construction of a compact complex symplectic fourfold, different from a Hilbert square of a K3 surface, a generalized Kummer 4-fold, the variety of lines on a cubic 4-fold and the recent examples of O'Grady (see Duke M…
Generalized Nikulin surfaces and irreducible symplectic fourfolds
研究K3^{[2]}型超Kähler流形上辛对合的不动点集的二维分支，给出射影K3曲面为广义Nikulin曲面的充要条件，并证明超越格之间的Hodge等距。
A surgery of a real symplectic manifold $X_{\mathbb R}$ along a real Lagrangian sphere $S$ is a modification of the symplectic and real structure on $X_{\mathbb R}$ in a neigborhood of $S$. Genus 0 Welschinger invariants of two real symplectic $4$-manifolds differing by such a surgery have been related in a previous work in collaboration with N. Puignau. In the present paper, we explore some particular situations where these general formulas greatly simplify. As an application, we complete the computation of genus 0 Welschinger invariants of all del~Pezzo surfaces, and of all $\mathbb R$-minim…
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。
Moduli of sheaves on fourfolds as derived Lagrangian intersections
本文证明 $(-2)$-移位辛导出概形可局部表示为光滑经典概形的 $(-1)$-移位余切栈中两个 Lagrangian 的导出交，并讨论其对 CY 四维流形层模栈的意义。
The main outcome of this paper is that the variety VSP(F,10) of presentations of a general cubic form F in 6 variables as a sum of 10 cubes is a smooth symplectic 4-fold obtained a deformation of the Hilbert square of a K3 surface of genus 8. After publishing it in Trans. Am. Math. Soc. 353, No.4, 1455-1468 (2001), it was noted to us by Eyal Markman that in Theorem 3.17 we conclude without proof that VSP(F,10) should be the 4-fold of lines on another cubic 4-fold. We correct this in the e-print "Addendum to K3 surfaces of genus 8 and varieties of sums of powers of cubic fourfolds" (math.AG/061…
We explore the connection between $K3$ categories and 0-cycles on holomorphic symplectic varieties. In this paper, we focus on Kuznetsov's noncommutative $K3$ category associated to a nonsingular cubic 4-fold. By introducing a filtration on the $\mathrm{CH}_1$-group of a cubic 4-fold $Y$, we conjecture a sheaf/cycle correspondence for the associated $K3$ category $\mathcal{A}_Y$. This is a noncommutative analog of O'Grady's conjecture concerning derived categories of $K3$ surfaces. We study instances of our conjecture involving rational curves in cubic 4-folds, and verify the conjecture for sh…
In this article, we study the rational cohomology rings of Voisin's punctual Hilbert schemes $X^{[n]}$ associated to a symplectic compact fourfold $X$. We prove that these rings can be universally constructed from $H^*(X,\mathbb{Q})$ and $c_1(X)$, and that Ruan's crepant resolution conjecture holds if $c_1(X)$ is a torsion class. Next, we prove that for any almost-complex compact fourfold $X$, the complex cobordism class of $X^{[n]}$ depends only on the cobordism class of $X$.
We address the problem of classification of hyper-Kähler fourfolds with $b_2=23$. In particular we prove some special cases of the Conjecture of O'Grady about hyper-Kähler $4$-folds numerically equivalent to the Hilbert scheme of two points on a $K3$ surface.
The contents of this paper is merged into math.RT/0601155. To avoid possible confusion, I withdraw this paper.
This paper is a sequel to math.RT/0601155. Let G be a complex symplectic group. In math.RT/0601155, we constructed a certain G-variety N = N_1, which we call the (1-) exotic nilpotent cone. In this paper, we study the set of G-orbits of the variety N. It turns out that the variety N gives a variant of the Springer correspondence for the Weyl group of type C, but shares a similar flavor with that of type A case. (I.e. there appears no non-trivial local system and the correspondence is bijective.) As an application, we present one sufficient condition for the bijectivity of our exotic Deligne-La…
We introduce the $2$-nodal spherical deformation of certain singular fibers of genus $2$ fibrations, and use such deformations to construct various examples of simply connected minimal symplectic $4$-manifolds with small topology. More specifically, we construct new exotic minimal symplectic $4$-manifolds homeomorphic but not diffeomorphic to ${\mathbb{CP}}^{2}\#6({\overline{\mathbb{CP}}^{2}})$, ${\mathbb{CP}}^{2}\#7({\overline{\mathbb{CP}}^{2}})$, and $3{\mathbb{CP}}^{2}\#k({\overline{\mathbb{CP}}^{2}})$ for $k=16, 17, 18, 19$ using combinations of such deformations, symplectic blowups, and (…
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