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Giulia Saccà

Publications and source records attributed to Giulia Saccà.

18 recordsLinked to original sources
arXiv2026

Sections of Jacobian fibrations over lines

Let $|H|$ be a linear system on a smooth surface $S$. We study the cohomology classes of sections of the universal Jacobian over lines in $|H|$. When $S$ is a K3 surface, the universal compactified Jacobian is a hyperkähler manifold, and we do not see how to reconcile our results with some of the steps in [BKV25]. v.2: Confusing typo fixed: H^g should be H^{2g}. v3: Note added: Conjecture 7 is now corrected and solved by Pascal Autissier and Andrea Fanelli (On a conjecture by Kollár and Saccà, arXiv:2606.26983).

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arXiv2026

The geometry of antisymplectic involutions, II

We continue our study of fixed loci of antisymplectic involutions on projective hyper-Kähler manifolds of $\mathrm{K3}^{[n]}$-type induced by an ample class of square 2 in the Beauville-Bogomolov-Fujiki lattice. We prove that if the divisibility of the ample class is 2, then one connected component of the fixed locus is a Fano manifold of index 3, thus generalizing to higher dimensions the case of the LLSvS 8-fold associated to a cubic fourfold. We also show that, in the case of the LLSvS 8-fold associated to a cubic fourfold, the second component of the fixed locus is of general type, thus answering a question by Manfred Lehn.

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arXiv2025

Relative and absolute Lefschetz standard conjectures for some Lagrangian fibrations

We show that the hyper-Kähler varieties of OG10-type constructed by Laza-Saccà-Voisin (LSV) verify the Lefschetz standard conjecture. This is an application of a more general result, stating that certain Lagrangian fibrations verify this conjecture. The main technical assumption of this general result is that the Lagrangian fibration satisfies the hypotheses of Ngô's support theorem. Verifying that the LSV tenfolds do satisfy those hypotheses is of independent interest. Another point of independent interest of the paper is the definition and the study of the Lefschetz standard conjecture in the relative setting, and its relation to the classical absolute case.

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arXiv2025

Compactifying Lagrangian fibrations

We suggest a general framework for compactifing quasi-projective Lagrangian fibrations of geometric origin by holomorphic symplectic varieties. This framework includes a compactification criterion, which we then apply to various fibrations of geometric origin, and a discussion on holomorphic forms that are defined via correspondences in geometric examples. As application, we show that given a Lagrangian fibration $X \to B$ admitting local sections over an open subset $V$ with codimension $\ge 2$ complement, there exists a (possibly singular) holomorphic symplectic compactification of the Albanese fibration $A \to V$ (which we show exists as a smooth commutative algebraic group with connected fibers acting on $X_{V}$), as well as of any other torsor over $A$, or over any smooth commutative group scheme over $B$ with connected fibers that is isogenous to $A$.

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arXiv2024

Irreducible symplectic varieties via relative Prym varieties

Generalizing work of Markushevich--Tikhomirov and Arbarello--Saccà--Ferretti, we use relative Prym varieties to construct Lagrangian fibered symplectic varieties in infinitely many dimensions. We then give criteria for when the construction yields primitive symplectic varieties, respectively, irreducible symplectic varieties. The starting point of the construction is a K3 surface endowed with an anti-symplectic involution and an effective linear system on the quotient surface. We give sufficient conditions on the linear system to ensure that the relative Prym varieties satisfy the criteria above. As a consequence, we produce infinite series of irreducible symplectic varieties.

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arXiv2024

Moduli spaces on Kuznetsov components are Irreducible Symplectic Varieties

This article studies moduli spaces of Bridgeland semistable objects in the Kuznetsov component of a cubic fourfold that don't admit a symplectic resolution, i.e., moduli spaces of objects with non-primitve Mukai vector v=mv_0 that is not of OG10-type and where v_0^2 >0. For a generic stability condition, it is shown that these moduli spaces are projective irreducible symplectic varieties with factorial terminal singularities and that their deformation class is uniquely determined by the integers m and v_0^2. On the one hand, this generalizes the results of arXiv:1703.10839, arXiv:1912.06935, arXiv:2007.14108, which deal with moduli spaces of objects in the Kuznetsov component of a cubic fourfold which are smooth or of OG10-type; on the other hand, this extends to the Kuznetsov component of a cubic fourfold the results of arXiv:1802.01182, arXiv:2012.10649, on Gieseker moduli spaces of sheaves on K3 surfaces with non-primitive Mukai vector.

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arXiv2023

The geometry of antisymplectic involutions, I

We study fixed loci of antisymplectic involutions on projective hyperkähler manifolds of $\mathrm{K3}^{[n]}$-type. When the involution is induced by an ample class of square 2 in the Beauville-Bogomolov-Fujiki lattice, we show that the number of connected components of the fixed locus is equal to the divisibility of the class, which is either 1 or 2.

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arXiv2023

Singularities of Bridgeland moduli spaces for K3 categories: an update

This survey is a continuation of the study undertaken in \cite{AS18}. We examine the local structure of Bridgeland moduli spaces $M_σ(v,\D)$, where the relevant triangulated category $\D$ is either the bounded derived category $\D=\D^b(X)$ of a K3 surface $X$, or the Kuznetsov component $\D=\Ku(Y)\subset \D ^b(Y)$ of a smooth cubic fourfold $Y\subset \PP^5$. For these moduli spaces, building on \cite{Bmm19}, \cite{Bmm21} we give a direct proof of formality and, using their local isomorphism with quiver varieties, we establish their normality and their irreducibility, as long as $σ$ does not lie on a totally semistable wall. We then connect the variation of GIT quotients for quiver varieties with the changing of stability conditions on moduli spaces.

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arXiv2021

Birational geometry of the intermediate Jacobian fibration of a cubic fourfold

We show that the intermediate Jacobian fibration associated to any smooth cubic fourfold $X$ admits a hyper-Kähler compactification $J(X)$ with a regular Lagrangian fibration $J \to \mathbb P^5$. This builds upon arXiv:1602.05534, where the result is proved for general $X$, as well as on the degeneration techniques on arXiv:1704.02731 and techniques from the minimal model program. We then study some aspects of the birational geometry of $J(X)$: for very general $X$ we compute the movable and nef cones of $J(X)$, showing that $J(X)$ is not birational to the twisted version of the intermediate Jacobian fibration arXiv:1611.06679, nor to an OG$10$-type moduli space of objects in the Kuznetsov component of $X$; for any smooth $X$ we show, using normal functions, that the Mordell-Weil group $MW(π)$ of the abelian fibration $π: J \to \mathbb P^5$ is isomorphic to the integral degree $4$ primitive algebraic cohomology of $X$, i.e., $MW(π) = H^{2,2}(X, \mathbb Z)_0$.

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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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arXiv2019

Relative compactified Jacobians of linear systems on Enriques surfaces

We study certain moduli spaces of sheaves on Enriques surfaces thereby obtaining, in every odd dimension, new examples of Calabi-Yau manifolds. We describe the geometry (canonical bundle, fundamental group, second Betti number and certain Hodge numbers) of these moduli spaces showing, in partial analogy to the well-known case of sheaves on K3 or Abelian surfaces, how the geometry of the surface reflects that of the moduli space itself.

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arXiv2018

Remarks on degenerations of hyper-Kähler manifolds

Using the Minimal Model Program, any degeneration of K-trivial varieties can be arranged to be in a Kulikov type form, i.e. with trivial relative canonical divisor and mild singularities. In the hyper-Kähler setting, we can then deduce a finiteness statement for monodromy acting on $H^2$, once one knows that one component of the central fiber is not uniruled. Independently of this, using deep results from the geometry of hyper-Kähler manifolds, we prove that a finite monodromy projective degeneration of hyper-Kähler manifolds has a smooth filling (after base change and birational modifications). As a consequence of these two results, we prove a generalization of Huybrechts' theorem about birational versus deformation equivalence, allowing singular central fibers. As an application, we give simple proofs for the deformation type of certain geometric constructions of hyper-Kähler manifolds (e.g. Debarre--Voisin or Laza--Saccà--Voisin). In a slightly different direction, we establish some basic properties (dimension and rational homology type) for the dual complex of a Kulikov type degeneration of hyper-Kähler manifolds.

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arXiv2018

Singularities of moduli spaces of sheaves on K3 surfaces and Nakajima quiver varieties

The aim of this paper is to study the singularities of certain moduli spaces of sheaves on K3 surfaces by means of Nakajima quiver varieties. The singularities in question arise from the choice of a non--generic polarization, with respect to which we consider stability, and admit natural symplectic resolutions corresponding to choices of general polarizations. For sheaves that are pure of dimension one, we show that these moduli spaces are, locally around a singular point, isomorphic to a quiver variety and that, via this isomorphism, the natural symplectic resolutions correspond to variations of GIT quotients of the quiver variety.

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arXiv2017

A hyper-Kähler compactification of the Intermediate Jacobian fibration associated to a cubic fourfold

For a general cubic fourfold, it was observed by Donagi and Markman that the relative intermediate Jacobian fibration associated to the family of its hyperplane sections carries a natural holomorphic symplectic form making the fibration Lagrangian. In this paper, we obtain a smooth projective compactification of the intermediate Jacobian fibration giving a ten-dimensional compact hyper-Kähler manifold, which we then show to be deformation equivalent to the exceptional example of O'Grady. This proves a conjecture by Markushevich.

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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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arXiv2016

Explicit Brill-Noether-Petri general curves

Let $p_1,\dots, p_9$ be the points in $\mathbb A^2(\mathbb Q)\subset \mathbb P^2(\mathbb Q)$ with coordinates $$(-2,3),(-1,-4),(2,5),(4,9),(52,375), (5234, 37866),(8, -23), (43, 282), \Bigl(\frac{1}{4}, -\frac{33}{8} \Bigr)$$ respectively. We prove that, for any genus $g$, a plane curve of degree $3g$ having a $g$-tuple point at $p_1,\dots, p_8$, and a $(g-1)$-tuple point at $p_9$, and no other singularities, exists and is a Brill-Noether general curve of genus $g$, while a general curve in that $g$-dimensional linear system is a Brill-Noether-Petri general curve of genus $g$.

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arXiv2013

Relative Prym varieties associated to the double cover of an Enriques surface

Given an Enriques surface $T$, its universal K3 cover $f: S\to T$, and a genus $g$ linear system $|C|$ on $T$, we construct the relative Prym variety $P_H=\Prym_{v, H}(\D/\CC)$, where $\CC\to |C|$ and $\D\to |f^*C|$ are the universal families, $v$ is the Mukai vector $(0,[D], 2-2g)$ and $H$ is a polarization on $S$. The relative Prym variety is a $(2g-2)$-dimensional possibly singular variety, whose smooth locus is endowed with a hyperkähler structure. This variety is constructed as the closure of the fixed locus of a symplectic birational involution defined on the moduli space $M_{v,H}(S)$. There is a natural Lagrangian fibration $η: P_H \to |C|$, that makes the regular locus of $P_H$ into an integrable system whose general fiber is a $(g-1)$-dimensional (principally polarized) Prym variety, which in most cases is not the Jacobian of a curve. We prove that if $|C|$ is a hyperelliptic linear system, then $P_H$ admits a symplectic resolution which is birational to a hyperkähler manifold of K3$^{[g-1]}$-type, while if $|C|$ is not hyperelliptic, then $P_H$ admits no symplectic resolution. We also prove that any resolution of $P_H$ is simply connected and, when $g$ is odd, any resolution of $P_H$ has $h^{2,0}$-Hodge number equal to one.

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