arXiv subjects
Publications and source records attributed to Giovanni Mongardi.
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 construct several new families of Fano varieties of K3 type. We give a geometrical explanation of the K3 structure and we link some of them to projective families of irreducible holomorphic symplectic manifolds.
We provide a criterion for when Hilbert schemes of points on K3 surfaces are birational. In particular, this allows us to generate a plethora of examples of non-birational Hilbert schemes which are derived equivalent.
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 isomorphic to a Hilbert scheme of 2 points on a K3 surface. We also discuss Voisin's conjecture on the existence of coisotropic subvarieties on IHS varieties and relate it to a stronger statement on Noether-Lefschetz loci in their moduli spaces.
We study the geometry of some moduli spaces of twisted sheaves on K3 surfaces. In particular we introduce induced automorphisms from a K3 surface on moduli spaces of twisted sheaves on this K3 surface. As an application we prove the unirationality of moduli spaces of irreducible holomorphic symplectic manifolds of $K3^{[2]}$-type admitting non symplectic involutions with invariant lattices $U(2)\oplus D_4(-1)$ or $U(2)\oplus E_8(-2)$. This complements the results obtained in [Mongardi and Wandel 2015], [Bossiere et al 2016], and the results from [arXiv:1603.00403] about the geometry of IHS fourfolds constructed using the Hilbert scheme of $(1,1)$ conics on Verra fourfolds. As a byproduct we find that IHS fourfolds of $K3^{[2]}$-type with Picard lattice $U(2)\oplus E_8(-2)$ naturally contain non-nodal Enriques surfaces.
The aim of this paper is to construct Calabi-Yau 4-folds as crepant resolutions of the quotients of a hyperkähler 4-fold $X$ by a non symplectic involution $α$. We first compute the Hodge numbers of a Calabi-Yau constructed in this way in a general setting and then we apply the results to several specific examples of non symplectic involutions, producing Calabi-Yau 4-folds with different Hodge diamonds. Then we restrict ourselves to the case where $X$ is the Hilbert scheme of two points on a K3 surface $S$ and the involution $α$ is induced by a non symplectic involution on the K3 surface. In this case we compare the Calabi-Yau 4-fold $Y_S$, which is the crepant resolution of $X/α$, with the Calabi-Yau 4-fold $Z_S$, constructed from $S$ through the Borcea--Voisin construction. We give several explicit geometrical examples of both these Calabi--Yau 4-folds describing maps related to interesting linear systems as well as a rational $2:1$ map from $Z_S$ to $Y_S$.
In this note we characterize polarized parallel transport operators on irreducible holomorphic symplectic varieties which are deformations of generalized Kummer varieties. We then apply such characterization to show the existence of ample uniruled divisors on these varieties and derive some interesting consequences on their Chow group of 0-cycles.
We realize O'Grady's six dimensional example of irreducible holomorphic symplectic manifold as a quotient of an IHS manifold of K3$^{[3]}$-type by a birational involution, thereby computing its Hodge numbers.
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.
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. Some general results concerning the birational geometry of irreducible holomorphic symplectic manifolds are also proved, such as a non-projective contractibility criterion for wall divisors.
In this article, we focus on a new perspective of automorphisms of complex 2-tori, reviewing previous works from a lattice-theoretic point of view. In particular, we give a classification of families of symplectic and non-symplectic automorphisms of prime order.
We determine the subgroup of automorphisms acting trivially on the second integral cohomology for hyperkaehler manifolds which are deformation equivalent to O'Grady's sporadic examples. In particular, we prove that this subgroup is trivial in the ten dimensional case and isomorphic to $(\mathbb{Z}/2\mathbb{Z})^{\times 8}$ in the six dimensional case. Furthermore we study the fixed loci of the automorphisms in the six dimensional case.
Severi varieties and Brill-Noether theory of curves on K3 surfaces are well understood. Yet, quite little is known for curves on abelian surfaces. Given a general abelian surface $S$ with polarization $L$ of type $(1,n)$, we prove nonemptiness and regularity of the Severi variety parametrizing $δ$-nodal curves in the linear system $|L|$ for $0\leq δ\leq n-1=p-2$ (here $p$ is the arithmetic genus of any curve in $|L|$). We also show that a general genus $g$ curve having as nodal model a hyperplane section of some $(1,n)$-polarized abelian surface admits only finitely many such models up to translation; moreover, any such model lies on finitely many $(1,n)$-polarized abelian surfaces. Under certain assumptions, a conjecture of Dedieu and Sernesi is proved concerning the possibility of deforming a genus $g$ curve in $S$ equigenerically to a nodal curve. The rest of the paper deals with the Brill-Noether theory of curves in $|L|$. It turns out that a general curve in $|L|$ is Brill-Noether general. However, as soon as the Brill-Noether number is negative and some other inequalities are satisfied, the locus $|L|^r_d$ of smooth curves in $|L|$ possessing a $g^r_d$ is nonempty and has a component of the expected dimension. As an application, we obtain the existence of a component of the Brill-Noether locus $\mathcal{M}^r_{p,d}$ having the expected codimension in the moduli space of curves $\mathcal{M}_p$. For $r=1$, the results are generalized to nodal curves.
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 manifolds. Furthermore we provide a description of the Picard lattice of several irreducible symplectic manifolds having a lagrangian fibration.
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 prove that there exists a holomorphic symplectic manifold deformation equivalent to the Hilbert scheme of two points on a K3 surface that admits a non-symplectic automorphism of order 23, that is the maximal possible prime order in this deformation family. The proof uses the theory of ideal lattices in cyclomotic fields.
The present paper is devoted to the classification of symplectic automorphisms of some hyperkähler manifolds. The results contained here are an explicit classification of prime order automorphisms on manifolds of $K3^{[n]}$ type and a proof that all finite groups of symplectic automorphisms of such manifolds are contained in Conway's group $Co_1$.
In the present paper we prove that, on a hyperkähler manifold, walls of the kähler cone and extremal rays of the Mori cone are determined by all divisors satisfying certain numerical conditions.