The hyper-Kummer construction, discovered by the first-named author, associates a hyper-Kähler manifold of $\mathrm{K}3^{[3]}$-type with any hyper-Kähler sixfold of generalized Kummer type. We regard this construction as a higher-dimensional analog of the classical Kummer construction of K3 surfaces from abelian surfaces. In this spirit, we prove several results which parallel those known for the classical Kummer construction: we characterize the hyper-Kummer $\mathrm{K}3^{[3]}$-manifolds up to birational equivalence in terms of their Hodge lattices, and establish a McKay correspondence for their derived categories and Chow motives. We propose a recipe to construct locally complete families of projective varieties of $\mathrm{Kum}^3$-type starting from a family of varieties of $\mathrm{K}3^{[3]}$-type equipped with 16 prime divisors in a certain Kummer lattice configuration. We also compare hyper-Kummer $\mathrm{K}3^{[3]}$-manifolds with the Mongardi-Rapagnetta-Saccà double covers of O'Grady's six-dimensional hyper-Kähler manifolds. The hyper-Kummer construction produces a rich configuration of hyper-Kähler manifolds of $\mathrm{K}3^{[2]}$-type and K3 surfaces canonically associated with a manifold of $\mathrm{Kum}^3$-type, in particular the hyper-Kummer K3 surfaces, which form countably many $4$-dimensional families of generic Picard rank 16. We prove abelianity of Chow motives for infinitely many 4-dimensional families of hyper-Kummer K3 surfaces, thereby proving Kimura-O'Sullivan finite-dimensionality conjecture for many new K3 surfaces of Picard rank 16. As other applications, we prove Beauville's weak splitting conjecture for all varieties of $\mathrm{Kum}^3$-type, and, building on previous results, we prove the Hodge and Tate conjectures for all powers of any of the varieties involved in the hyper-Kummer construction.
We prove that the complex cobordism class of any hyper-Kähler manifold of dimension $2n$ is a unique combination with rational coefficients of classes of products of punctual Hilbert schemes of $K3$ surfaces. We also prove a similar result using the generalized Kummer varieties instead of punctual Hilbert schemes. As a key step, we establish a closed formula for the top Chern character of their tangent bundles.
G. Höhn and G. Mason classified all finite groups acting faithfully and symplectically on a hyper-K{ä}hler fourfolds of type K3$^{[2]}$. There are 15 maximal among them, call them $\widetilde{G}_1,\ldots, \widetilde{G}_{15}$. Every manifold of type K3$^{[2]}$ admitting an action of $\widetilde{G}_i$ for some $i$ must necessarily have Picard rank 21 which is maximal. This fact allows us to use lattice-theoretic methods to classify all the finite groups $G$ acting faithfully on a hyper-K{ä}hler fourfold of type K3$^{[2]}$ $X$ such that $G$ contains $\widetilde{G}_i$ as a proper subgroup and $\wi…
In this work we study the Humbert-Edge's curves of type 5, defined as a complete intersection of four diagonal quadrics in $\mathbb{P}^5$. We characterize them using Kummer surfaces and using the geometry of these surfaces we construct some vanishing thetanulls on such curves. In addition, we describe an argument to give an isomorphism between the moduli space of Humbert-Edge's curves of type 5 and the moduli space of hyperelliptic curves of genus 2, and we let see how this argument can be generalized to state an isomorphism between the moduli space of hyperelliptic curves of genus $g=\frac{n-…
We show that a Hilbert scheme of conics on a Fano fourfold double cover of $\mathbb{P}^2\times\mathbb{P}^2$ ramified along a divisor of bidegree $(2,2)$ admits a $\mathbb{P}^1$-fibration with base being a hyper-Kähler fourfold. We investigate the geometry of such fourfolds relating them with degenerated EPW cubes, with elements in the Brauer groups of $K3$ surfaces of degree $2$, and with Verra threefolds studied in [Ver04]. These hyper-Kähler fourfolds admit natural involutions and complete the classification of geometric realizations of anti-symplectic involutions on hyper-Kähler $4$-folds o…
Given a smooth projective variety $M$ endowed with a faithful action of a finite group $G$, following Jarvis-Kaufmann-Kimura and Fantechi-Göttsche, we define the orbifold motive (or Chen-Ruan motive) of the quotient stack $[M/G]$ as an algebra object in the category of Chow motives. Inspired by Ruan, one can formulate a motivic version of his Cohomological HyperKähler Resolution Conjecture. We prove this motivic version, as well as its K-theoretic analogue conjectured by Jarvis-Kaufmann-Kimura, in two situations related to an abelian surface $A$ and a positive integer $n$. Case (A) concerns Hi…
In this paper, we first generalize to any hyper-Kähler manifold with nonzero third Betti number results proved by O'Grady for hyper-Kähler manifolds of generalized Kummer type. In the second part, we restrict to hyper-Kähler manifolds of generalized Kummer type and prove, using results of Markman, that their Kuga-Satake correspondence is algebraic.
Let X be a hyperkaehler manifold of Kummer type. We study the 4 dimensional intermediate Jacobian J^3(X) constructed out of the 3rd cohomology of X, motivated by the desire to understand the Kuga-Satake torus associated to X. We prove that there is an embedding of Hodge structures of H^2(X) into H^2(J^3(X)). It follows that if X is projective with polarization L, the projective Kuga-Satake torus of (X,L) is isogenous to the fourth power of J^3(X). By studying the cohomology ring of X, we find that there is a natural smooth quadric Q(X) in the projectivization of H^3(X), with a natural choice o…
We derive formulas for the construction of all inequivalent Jacobian elliptic fibrations on the Kummer surface of two non-isogeneous elliptic curves from extremal rational elliptic surfaces by rational base transformations and quadratic twists. We then show that each such decomposition yields a description of the Picard-Fuchs system satisfied by the periods of the holomorphic two-form as either a tensor product of two Gauss' hypergeometric differential equations, an Appell hypergeometric system, or a GKZ differential system. As the answer must be independent of the fibration used, identities r…
We construct Zariski K3 surfaces of Artin invariant 1, 2 and 3 in many characteristics. In particular, we prove that any supersingular Kummer surface is Zariski if the characteristic is not congruent to 1 modulo 12. Our methods combine different approaches such as quotients by the group scheme $α_p$, Kummer surfaces, and automorphisms of elliptic and hyperelliptic curves.
We prove that the natural representaion, on the total Betti cohomolgy group, of the automorphism group of a hyperkähler manifold deformation equivalent to a generalized Kummer manifold is faithful. This is a sort of generalization of an earlier work of Beauville and a recent work of Boissére, Nieper-Wisskirchan and Sarti, concerning the representation on the second cohomology group.
For an Abelian surface $A$ with a symplectic action by a finite group $G$, one can define the partition function for $G$-invariant Hilbert schemes \[Z_{A, G}(q) = \sum_{d=0}^{\infty} e(\text{Hilb}^{d}(A)^{G})q^{d}.\] We prove the reciprocal $Z_{A,G}^{-1}$ is a modular form of weight $\frac{1}{2}e(A/G)$ for the congruence subgroup $Γ_{0}(|G|)$, and give explicit expressions in terms of eta products. Refined formulas for the $χ_{y}$-genera of $\text{Hilb}(A)^{G}$ are also given. For the group generated by the standard involution $τ: A \to A$, our formulas arise from the enumerative geometry of t…
We complete the classification of order $5$ nonsymplectic automorphisms on hyper-Kähler fourfolds deformation equivalent to the Hilbert square of a K3 surface. We then compute the topological Lefschetz number of natural automorphisms of generalized Kummer fourfolds and we describe the geometry of their fix loci.
We use the arithmetic of the Kummer surface associated to the Jacobian of a hyperelliptic curve to study the primality of integers of the form $4m^2 5^n-1$. We provide an algorithm capable of proving the primality or compositeness of most of the integers in these families and discuss in detail the necessary steps to implement this algorithm in a computer. Although an indetermination is possible, in which case another choice of initial parameters should be used, we prove that the probability of reaching this situation is exceedingly low and decreases exponentially with $n$.
In the context of K3 mirror symmetry, the Greene-Plesser orbifolding method constructs a family of K3 surfaces, the mirror of quartic hypersurfaces in $\mathbb{P}^3$, starting from a special one-parameter family of K3 varieties known as the quartic Dwork pencil. We show that certain K3 double covers obtained from the three-parameter family of quartic Kummer surfaces associated with a principally polarized abelian surface generalize the relation of the Dwork pencil and the quartic mirror family. Moreover, for the three-parameter family we compute a formula for the rational point-count of its ge…
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）
Let $X_1$ and $X_2$ be deformation equivalent projective hyperkähler manifolds. We prove that the André motive of $X_1$ is abelian if and only if the André motive of $X_2$ is abelian. Applying this to manifolds of $\mbox{K3}^{[n]}$, generalized Kummer and OG6 deformation types, we deduce that their André motives are abelian. As a consequence, we prove that all Hodge classes in arbitrary degree on such manifolds are absolute. We discuss applications to the Mumford-Tate conjecture, showing in particular that it holds for even degree cohomology of such manifolds.
In this paper we first show that each Kummer quartic surface (a quartic surface $X$ with 16 singular points) is, in canonical coordinates, equal to its dual surface, and that the Gauss map induces a fixpoint free involution $γ$ on the minimal resolution $S$ of $X$. Then we study the corresponding Enriques surfaces $S/ γ$. We also describe in detail the remarkable properties of the most symmetric Kummer quartic, which we call the Cefalú quartic. We also investigate the Kummer quartic surfaces whose associated Abelian surface is isogenous to a product of elliptic curves through an isogeny with k…
Coble type hypersurfaces and hyperkähler fourfolds
Universität des Saarlandes（萨尔兰大学）；Université Paris Cité and Sorbonne Université（巴黎西岱大学和索邦大学）；Universitat de Barcelona（巴塞罗那大学）；Università degli Studi di Milano（米兰大学）
We show the existence of a complex K3 surface $X$ which is not a Kummer surface and has a one-parameter family of Levi-flat hypersurfaces in which all the leaves are dense. We construct such $X$ by patching two open complex surfaces obtained as the complements of tubular neighborhoods of elliptic curves embedded in blow-ups of the projective planes at general nine points.
We introduce a new ascending filtration, that we call the co-radical filtration in analogy with the basic theory of co-algebras, on the Chow groups of pointed smooth projective varieties. In the case of zero-cycles on projective hyper-Kähler manifolds, we conjecture it agrees with a filtration introduced by Voisin. This is established for moduli spaces of stable objects on K3 surfaces, for generalized Kummer varieties and for the Fano variety of lines on a smooth cubic fourfold. Our overall strategy is to view the birational motive of a smooth projective variety as a co-algebra object with res…
