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Antonio Rapagnetta

Publications and source records attributed to Antonio Rapagnetta.

17 recordsLinked to original sources
arXiv2026

Hyperkähler sixfolds, abelian fourfolds of Weil type and a Hodge class

There are now several proofs of the Hodge conjecture for the general abelian fourfold of Weil type with trivial discriminant. This paper provides another one. The abelian fourfolds under consideration allow a map to a hyperkähler sixfold of K3$^{[3]}$ type. The pull-back of the second Chern class of the tangent bundle of the sixfold is an algebraic class in codimension two that is not an intersection of divisor classes and the main result follows. After recalling the basic facts on abelian fourfolds of Weil type we establish the existence of the map using results on the birational geometry of these hyperkähler manifolds and deformation theory.

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arXiv2026

On the genus of a curve in a projective $3$-fold

Let $X\subset \mathbb P^r$ be a projective factorial variety of dimension $3$, degree $n$, with at worst isolated singularities. Assume that the Picard group of $X$ is generated by the hyperplane section class. Let $C\subset X$ be a projective subscheme of dimension $1$, degree $d\gg n$, and arithmetic genus $p_a(C)$. Improving a recent result by Liu, we exhibit a Castelnuovo's bound for $p_a(C)$. In the case $X$ is Calabi-Yau, our bound gives a step forward for a certain conjecture concerning the vanishing of Gopakumar-Vafa invariants of $X$.

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arXiv2023

Locally trivial monodromy of moduli spaces of sheaves on K3 surfaces

In this paper we study monodromy operators on moduli spaces $M_v(S,H)$ of sheaves on K3 surfaces with non-primitive Mukai vectors $v$. If we write $v=mw$, with $m>1$ and $w$ primitive, then our main result is that the inclusion $M_w(S,H)\to M_v(S,H)$ as the most singular locus induces an isomorphism between the monodromy groups of these symplectic varieties, allowing us to extend to the non-primitive case a result of Markman.

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arXiv2021

The Hodge numbers of O'Grady 10 via Ngô strings

We determine the Hodge numbers of the hyper-Kähler manifold known as O'Grady 10 by studying some related modular Lagrangian fibrations by means of a refinement of the Ngô Support Theorem.

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arXiv2020

The second integral cohomology of moduli spaces of sheaves on K3 and Abelian surfaces

In this paper we study the second integral cohomology of moduli spaces of semistable sheaves on projective K3 surfaces. If $S$ is a projective K3 surface, $v$ a Mukai vector and $H$ a $v-$generic polarization on $S$, we show that $H^{2}(M_{v},\mathbb{Z})$ is a free $\mathbb{Z}-$module of rank 23 carrying a pure weight-two Hodge structure and a lattice structure, with respect to which $H^{2}(M_{v},\mathbb{Z})$ is Hodge isometric to the Hodge sublattice $v^{\perp}$ of the Mukai lattice of $S$. Similar results are proved for Abelian surfaces.

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arXiv2020

Monodromy and birational geometry of O'Grady's sixfolds

We prove that the bimeromorphic class of a hyperkähler manifold deformation equivalent to O'Grady's six dimensional one is determined by the Hodge structure of its Beauville-Bogomolov lattice by showing that the monodromy group is maximal. As applications, we give the structure for the Kähler and the birational Kähler cones in this deformation class and we prove that the existence of a square zero divisor implies the existence a rational lagrangian fibration with fixed fibre types.

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arXiv2017

Fourier-Mukai and autoduality for compactified Jacobians. I

To every singular reduced projective curve X one can associate, following E. Esteves, many fine compactified Jacobians, depending on the choice of a polarization on X, each of which yields a modular compactification of a disjoint union of the generalized Jacobian of X. We prove that, for a reduced curve with locally planar singularities, the integral (or Fourier-Mukai) transform with kernel the Poincare' sheaf from the derived category of the generalized Jacobian of X to the derived category of any fine compactified Jacobian of X is fully faithful, generalizing a previous result of D. Arinkin in the case of integral curves. As a consequence, we prove that there is a canonical isomorphism (called autoduality) between the generalized Jacobian of X and the connected component of the identity of the Picard scheme of any fine compactified Jacobian of X and that algebraic equivalence and numerical equivalence coincide on any fine compactified Jacobian, generalizing previous results of Arinkin, Esteves, Gagne', Kleiman, Rocha, Sawon. The paper contains an Appendix in which we explain how our work can be interpreted in view of the Langlands duality for the Higgs bundles as proposed by Donagi-Pantev.

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arXiv2016

The Hodge diamond of O'Grady's 6-dimensional example

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.

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arXiv2015

Fine compactified Jacobians of reduced curves

To every singular reduced projective curve X one can associate many fine compactified Jacobians, depending on the choice of a polarization on X, each of which yields a modular compactification of a disjoint union of the generalized Jacobian of X. We investigate the geometric properties of fine compactified Jacobians focusing on curves having locally planar singularities. We give examples of nodal curves admitting non isomorphic (and even non homeomorphic over the field of complex numbers) fine compactified Jacobians. We study universal fine compactified Jacobians, which are relative fine compactified Jacobians over the semiuniversal deformation space of the curve X. Finally, we investigate the existence of twisted Abel maps with values in suitable fine compactified Jacobians.

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arXiv2015

On minimal models of projective Hyperkaehler manifolds

Any minimal model of a projective Hyperkaehler manifold is a projective Hyperkaehler manifold. As a consequence, moduli spaces of sheaves on a k3 that don't admit a symplectic resolution are not birational to Hyperkaehler manifolds.

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arXiv2011

Factoriality properties of moduli spaces of sheaves on abelian and K3 surfaces

In this paper we complete the determination of the index of factoriality of moduli spaces of semistable sheaves on an abelian or projective K3 surface $S$. If $v=2w$ is a Mukai vector, $w$ is primitive, $w^{2}=2$ and $H$ is a generic polarization, let $M_{v}(S,H)$ be the moduli space of $H-$semistable sheaves on $S$ with Mukai vector $v$. First, we describe in terms of $v$ the pure weight-two Hodge structure and the Beauville form on the second integral cohomology of the symplectic resolutions of $M_{v}(S,H)$ (when $S$ is K3) and of the fiber $K_{v}(S,H)$ of the Albanese map of $M_{v}(S,H)$ (when $S$ is abelian). Then, if $S$ is K3 we show that $M_{v}(S,H)$ is either locally factorial or $2-$factorial, and we give an example of both cases. If $S$ is abelian, we show that $M_{v}(S,H)$ and $K_{v}(S,H)$ are $2-$factorial.

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arXiv2010

Surfaces with nontrivial surjective endomorphisms of any given degree

We present a complete classification of complex projective surfaces $X$ with nontrivial self-maps (i.e. surjective morphisms $f:X\rightarrow X$ which are not isomorphisms) of any given degree. The starting point of our classification are results contained in Fujimoto and Nakayama that provide a list of surfaces that admit at least one nontrivial self-map. We then proceed by a case by case analysis that blends geometrical and arithmetical arguments in order to exclude that certain prime numbers appear as degrees of nontrivial self-maps of certain surfaces.

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arXiv2010

Deformation of the O'Grady moduli spaces

In this paper we study moduli spaces of sheaves on an abelian or projective K3 surface. If $S$ is a K3, $v=2w$ is a Mukai vector on $S$, where $w$ is primitive and $w^{2}=2$, and $H$ is a $v-$generic polarization on $S$, then the moduli space $M_{v}$ of $H-$semistable sheaves on $S$ whose Mukai vector is $v$ admits a symplectic resolution $\widetilde{M}_{v}$. A particular case is the $10-$dimensional O'Grady example $\widetilde{M}_{10}$ of irreducible symplectic manifold. We show that $\widetilde{M}_{v}$ is an irreducible symplectic manifold which is deformation equivalent to $\widetilde{M}_{10}$ and that $H^{2}(M_{v},\mathbb{Z})$ is Hodge isometric to the sublattice $v^{\perp}$ of the Mukai lattice of $S$. Similar results are shown when $S$ is an abelian surface.

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