A projective hyperkähler manifold of Kummer-type is said to be twisted modular if it is birational to the Albanese fiber of a moduli space of twisted sheaves on an abelian surface. We prove that, with the exception of certain cases of Picard rank 3, any projective Kummer-type manifold admitting a finite-order symplectic birational self-map that acts nontrivially on its second cohomology group is twisted modular. We provide a complete characterization of these exceptions in terms of their Néron-Severi lattices. We then investigate symplectic birational self-maps of modular Kummer-type manifolds, determining exactly which Mukai vectors allow the birational transformation induced by crossing the vertical wall, which acts on cohomology as a reflection, to correspond to a finite-order symplectic birational self-map. Additionally, we prove in an appendix several results concerning moduli spaces of twisted sheaves on abelian surfaces which were not readily available in the literature.
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.
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.
Nikulin proved that the isometries induced on the second cohomology group of a K3 surface $X$ by a finite abelian group $G$ of symplectic automorphisms are essentially unique. Moreover he computed the discriminant of the sublattice of $H^2(X, \Z)$ which is fixed by the isometries induced by $G$. However for certain groups these discriminants are not the same of those found for explicit examples. Here we describe Kummer surfaces for which this phenomena happens and we explain the difference.
We find formulas for the birational maps from a Kummer surface K and its dual K^* to their common minimal desingularization S. We show how the nodes of K blow up. Then we give a description of the group of linear automorphisms of S.
In the first part of this paper we give a survey of classical results on Kummer surfaces with Picard number 17 from the point of view of lattice theory. We prove ampleness properties for certain divisors on Kummer surfaces and we use them to describe projective models of Kummer surfaces of (1,d)-polarized Abelian surfaces for d=1,2,3. As a consequence we prove that in these cases the Néron--Severi group can be generated by lines. In the second part of the paper we use Kummer surfaces to obtain results on K3 surfaces with a symplectic action of the group (\Z/2\Z)^4. In particular we describe th…
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.
We prove the conjecture that two projective symplectic resolutions for a symplectic variety $W$ are related by Mukai's elementary transformations over $W$ in codimension 2 in the following cases: (i). nilpotent orbit closures in a classical simple complex Lie algebra; (ii). some quotient symplectic varieties.
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.
We give a necessary and sufficient condition for an automorphism of the Hilbert scheme of points on a K3 surface (non necessarily algebraic) to be induced by an automorphism of the surface. We prove furthermore that the group of birational transformations of a projective irreducible holomorphic symplectic manifold is finitely generated.
Unfortunately, some proofs in the first version of this paper were incorrect. In this revised version, some minor gaps are fixed, one serious mistake found. The main theorem is now claimed only under a restrictive technical assumption. This invalidates the application to quotient singularities by the Weyl group of type $G_2$. Everything else still stands (in particular, the claim that every symplectic resolution is semismall).
In this paper we shall study symplectic resolutions of a nilpotent orbit closure of a complex simple Lie algebra \g. We shall introduce an equivalence relation in the set of parabolic subgroups of $G$ in terms of marked Dynkin diagrams. We start with a nilpotent orbit closure which admits a Springer resolution with a parabolic subgroup $P_0$ of $G$. Then we prove that all symplectic resolution of the nilpotent closure are Springer resolutions with $P$ which are equivalent to $P_0$. Here all symplectic resolutions are connected by Mukai flops. We need three types of Mukai flops (types A, D and …
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.
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.
We classify the group of birational automorphisms of Hilbert schemes of points on algebraic K3 surfaces of Picard rank one. We study whether these automorphisms are symplectic or non-symplectic and if there exists a hyperkähler birational model on which they become biregular. We also present new geometrical constructions of these automorphisms.
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 …
A symplectic manifold $(M,ω)$ is called {\em (symplectically) uniruled} if there is a nonzero genus zero GW invariant involving a point constraint. We prove that symplectic uniruledness is invariant under symplectic blow-up and blow-down. This theorem follows from a general Relative/Absolute correspondence for a symplectic manifold together with a symplectic submanifold. A direct consequence is that symplectic uniruledness is a symplectic birational invariant. Here we use Guillemin and Sternberg's notion of cobordism as the symplectic analogue of the birational equivalence.
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.
This is a survey on symplectic birational geometry. In arbitrary dimension, this subject is centered around the notion of uniruledness. In low dimensions, we will also discuss Kodaira dimension and minimality.
Following Bayer and Macrì, we study the birational geometry of singular moduli spaces $M$ of sheaves on a K3 surface $X$ which admit symplectic resolutions. More precisely, we use the Bayer-Macrì map from the space of Bridgeland stability conditions $\mathrm{Stab}(X)$ to the cone of movable divisors on $M$ to relate wall-crossing in $\mathrm{Stab}(X)$ to birational transformations of $M$. We give a complete classification of walls in $\mathrm{Stab}(X)$ and show that every birational model of $M$ obtained by performing a finite sequence of flops from $M$ appears as a moduli space of Bridgeland …
Let $f\colon X\dashrightarrow X$ be a bimeromorphic transformation of a complex irreducible symplectic manifold $X$. Some important dynamical properties of $f$ are encoded by the induced linear automorphism $f^*$ of $H^2(X,\mathbb Z)$. Our main result is that a bimeromorphic transformation such that $f^*$ has at least one eigenvalue with modulus $>1$ doesn't admit any invariant fibration (in particular its generic orbit is Zariski-dense).
We study the group of symplectic birational transformations of the plane. It is proved that this group is generated by $\mathrm{SL}(2,\mathbb{Z})$, the torus and a special map of order $5$, as it was conjectured by A. Usnich. Then we consider a special subgroup $H$, of finite type, defined over any field which admits a surjective morphism to the Thompson group of piecewise linear automorphisms of $\mathbb{Z}^2$. We prove that the presentation for this group conjectured by Usnich is correct.
In view of a potential interpretation of the role of the Mathieu group M_24 in the context of strings compactified on K3 surfaces, we develop techniques to combine groups of symmetries from different K3 surfaces to larger 'overarching' symmetry groups. We construct a bijection between the full integral homology lattice of K3 and the Niemeier lattice of type (A_1)^24, which is simultaneously compatible with the finite symplectic automorphism groups of all Kummer surfaces lying on an appropriate path in moduli space connecting the square and the tetrahedral Kummer surfaces. The Niemeier lattice …
We investigate boundedness results for families of holomorphic symplectic varieties up to birational equivalence. We prove the analogue of Zarhin's trick by for $K3$ surfaces by constructing big line bundles of low degree on certain moduli spaces of stable sheaves, and proving birational versions of Matsusaka's big theorem for holomorphic symplectic varieties. As a consequence of these results, we give a new geometric proof of the Tate conjecture for $K3$ surfaces over finite fields of characteristic at least $5$, and a simple proof of the Tate conjecture for $K3$ surfaces with Picard number a…
In this paper, we shall prove that any two (projective) symplectic resolutions of a nilpotent orbit closure in a classical simple Lie algebra are connected by a finite sequence of diagrams which are locally trivial families of Mukai flops of type A or of type D. A Mukai flop of type A is the same as a stratified Mukai flop in our previous paper "Mukai flops and derived categories II". On the other hand, a Mukai flop of type D comes from a nilpotent orbit in so(2k). Let G^+ and G^- be two connected components of the orthogonal Grassmannian G_{iso}(k,2k). Then the closure of this orbit has two S…
We survey recent results on ample cones and birational contractions of holomorphic symplectic varieties of K3 type, focusing on explicit constructions and concrete examples.
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.
We investigate the stability of fibers of coisotropic fibrations on holomorphic symplectic manifolds and generalize Voisin's result on Lagrangian subvarieties to this framework. We present applications to the moduli space of holomorphic symplectic manifolds which are deformations equivalent to Hilbert schemes of points on a $K3$ surface or to generalized Kummer manifolds.
We provide a method, based on Nikulin's lattice gluing techniques, which identifies the symplectic automorphisms of Kummer surfaces as permutation groups on 24 elements preserving the Golay code. In other words, we explicitly realise these symplectic automorphism groups as subgroups of the Mathieu group M_24. The example of the tetrahedral Kummer surface is treated in detail, confirming the existence proofs of Mukai and Kondo, that its group of symplectic automorphisms is a subgroup of one of eleven subgroups of the sporadic group known as Mathieu group M_23. Kondo's lattice construction, whic…
Terminalizations of quotients of compact hyperkähler manifolds by induced symplectic automorphisms
Università di Genova（热那亚大学）；Université Paris-Saclay（巴黎萨克雷大学）；Alma Mater studiorum Università di Bologna（博洛尼亚大学）；Université Paris Cité and Sorbonne Université（巴黎西岱大学和索邦大学）
We give a construction of completely integrable 4-dimensional Hamiltonian systems with cubic Hamilton functions. Applying to the corresponding pairs of commuting quadratic Hamiltonian vector fields the so called Kahan-Hirota-Kimura discretization scheme, we arrive at pairs of birational 4-dimensional maps. We show that these maps are symplectic with respect to a symplectic structure that is a perturbation of the standard symplectic structure on $\mathbb R^4$, and possess two independent integrals of motion, which are perturbations of the original Hamilton functions. Thus, these maps are comple…
The notion of a holomorphically symplectic manifold can be generalized to the singular one. This paper studies the birational contraction maps between symplectic varieties, and then describes the deformation of a symplectic variety which has a symplectic resolution. Some examples are also included.
We construct a finite dimensional Kaehler manifold with a holomorphic, symplectic circle action whose symplectic reduced spaces may be identified with the tau-vortex moduli spaces (or tau-stable pairs). The Morse theory of the circle action induces natural birational maps between the reduced spaces for different values of tau which in the case of rank two bundles can be canonically resolved in a sequence of blow-ups and blow-downs.
We introduce the notion of induced birational transformations of irreducible holomorphic symplectic sixfolds of the sporadic deformation type discovered by O'Grady. We give a criterion to determine when a manifold of $OG_6$ type is birational to a moduli space of sheaves on an abelian surface. Then we determine when a birational transformation of the moduli space is induced by an automorphism of the abelian surface. Referring to the Mongardi--Rapagnetta--Saccá birational model of manifolds of $OG_6$ type, we give a result to determine when a birational transformation is induced at the quotient…
We show that any two birational projective Calabi-Yau manifolds have isomorphic small quantum cohomology algebras after a certain change of Novikov rings. The key tool used is a version of an algebra called symplectic cohomology, which is constructed using Hamiltonian Floer cohomology. Morally, the idea of the proof is to show that both small quantum products are identical deformations of symplectic cohomology of some common open affine subspace. Part of the proof uses the fact that subvarieties of positive codimension are stably displaceable.
The notion of d-critical loci was introduced by Joyce in order to give classical shadows of $(-1)$-shifted symplectic derived schemes. In this paper, we discuss birational geometry for d-critical loci, by introducing notions such as `d-critical flips', `d-critical flops', etc. They are not birational maps of the underlying spaces, but rather should be understood as virtual birational maps. We show that several wall-crossing phenomena of moduli spaces of stable objects on Calabi-Yau 3-folds are described in terms of d-critical birational geometry. Among them, we show that wall-crossing diagrams…
We give a construction of completely integrable ($2n$)-dimensional Hamiltonian systems with symplectic brackets of the Lie-Poisson type (linear in coordinates) and with quadratic Hamilton functions. Applying to any such system the so called Kahan-Hirota-Kimura discretization scheme, we arrive at a birational ($2n$)-dimensional map. We show that this map is symplectic with respect to a symplectic structure that is a perturbation of the original symplectic structure on $\mathbb R^{2n}$, and possesses $n$ independent integrals of motion, which are perturbations of the original Hamilton functions …
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 …
We give a construction of completely integrable $(2m)$-dimensional Hamiltonian systems with cubic Hamilton functions. The construction depends on a constant skew-Hamiltonian matrix $A$, that is, a matrix satisfying $A^{\rm T}J=JA$, where $J$ is a non-degenerate skew-symmetric matrix defining the standard symplectic structure on the phase space $\mathbb R^{2m}$. Applying to any such system the so called Kahan-Hirota-Kimura discretization scheme, we arrive at a birational $(2m)$-dimensional map. We show that this map is symplectic with respect to a symplectic structure that is a perturbation of …
