arXiv subjects
Publications and source records attributed to Annalisa Grossi.
In this short note we correct the lattice theoretic classification in [Rev. Mat. Iberoam. 38 (2022), no. 4, 1199--1218] of the first named author, including some missing cases and explaining how to recover them.
Given a holomorphic or anti-holomorphic involution on a complex variety, the Smith inequality says that the total $\mathbb{F}_2$-Betti number of the fixed locus is no greater than the total $\mathbb{F}_2$-Betti number of the ambient variety. The involution is called maximal when the equality is achieved. In this paper, we investigate maximality of involutions of compact hyper-Kähler manifolds and of Hilbert schemes of points on surfaces. We obtain both positive and negative results. On one hand, given a smooth projective surface $S$ with $H^1(S, \mathbb{F}_2)=0$ equipped with a holomorphic (resp.~anti-holomorphic) involution $σ$, we establish the following necessary and sufficient condition for the maximality of the induced involution on the $n$th Hilbert scheme of points: the induced involution is maximal if and only if $σ$ is a maximal involution of $S$ and it acts on $H^2(S, \mathbb{Z})$ trivially (resp.~as $-\operatorname{id}$). This generalizes and completes previous partial results of Fu and Kharlamov--R\u asdeaconu. On the other hand, we show that for $n\geq 2$, a hyper-Kähler manifold of K3$^{[n]}$-deformation type admits neither maximal anti-holomorphic involutions (i.e.~real structures), nor maximal holomorphic (symplectic or anti-symplectic) involutions. In other words, such hyper-Kähler manifolds do not admit maximal (AAB), (ABA), (BAA) or (BBB) brane involutions in the sense of Kapustin--Witten.
We determine the algebraic and transcendental lattices of a general cubic fourfold with a symplectic automorphism of prime order. We prove that cubic fourfolds admitting a symplectic automorphism of order at least three are rational, and we exihibit two families of rational cubic fourfolds that are not equivariantly rational with respect to their group of automorphisms. As an application, we determine the cohomological action of symplectic birational transformations of manifolds of OG10 type that are induced by prime order sympletic automorphisms of cubic fourfolds.
Terminalizations of symplectic quotients are sources of new deformation types of irreducible symplectic varieties. We classify all terminalizations of quotients of Hilbert schemes of K3 surfaces or of generalized Kummer varieties, by finite groups of symplectic automorphisms induced from the underlying K3 or abelian surface. We determine their second Betti number and the fundamental group of their regular locus. In the Kummer case, we prove that the terminalizations have quotient singularities, and determine the singularities of their universal quasi-étale cover. In particular, we obtain at least nine new deformation types of irreducible symplectic varieties of dimension four. Finally, we compare our deformation types with those in [FM21; Men22]. The smooth terminalizations are only three and of K$3^{[n]}$-type, and surprisingly they all appeared in different places in the literature [Fuj83; Kaw09; Flo22].
We study weighted Fano fourfolds of K3 type realized as hypersurfaces in weighted projective spaces. Under the additional assumption that the singular locus has dimension at most one, we prove that only finitely many such families exist. We provide the complete list and analyze their rationality properties, as well as their singularities.
We use the LLV algebra to describe the action of a finite order automorphism on the total cohomology of a manifold of OG10 type. As an application, we prove that no Enriques manifolds arise as étale quotients of hyper-Kähler manifolds of OG10 type. This answers a question raised by Pacienza and Sarti.
We give a lattice-theoretic classification of non-symplectic automorphisms of prime order of irreducible holomorphic symplectic manifolds of OG10 type. We determine which automorphisms are induced by a non-symplectic automorphism of prime order of a cubic fourfold on the associated LSV manifolds, giving a geometric and lattice-theoretic description of the algebraic and transcendental lattices of the cubic fourfold. As an application we discuss the rationality conjecture for a general cubic fourfold with a non-symplectic automorphism of prime order.
We give a lattice-theoretic characterization for a manifold of $\mathrm{OG}10$ type to be birational to some moduli space of (twisted) sheaves on a K3 surface. We apply it to the Li-Pertusi-Zhao variety of $\mathrm{OG}10$ type associated to any smooth cubic fourfold. Moreover we determine when a birational transformation is induced by an automorphism of the K3 surface and we use this to classify all induced birational symplectic involutions.
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.
We prove that any symplectic automorphism of finite order of an irreducible holomorphic symplectic manifold of O'Grady's 10-dimensional deformation type is trivial.
We prove that any ample class on a primitive symplectic variety that is locally trivial deformation of O'Grady's singular 6 dimensional example is proportional to the first Chern class of a uniruled divisor. This result answers a question of Lehn, Mongardi and Pacienza, extending their result for primitive symplectic varieties of this deformation type.
We classify nonsymplectic automorphisms of prime order on irreducible holomorphic symplectic manifolds of O'Grady's 6-dimensional defamation type. More precisely, we give a classification of the invariant and coinvariant sublattices of the second integral cohomology group.
We prove that any symplectic automorphism of finite order on a manifold of type OG6 acts trivially on the Beauville--Bogomolov--Fujiki lattice and that any birational transformation of finite order acts trivially on its discriminant group. Moreover, we classify all possible invariant and coinvariant sublattices.
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 give an application of these criteria in the nonsymplectic case.
In this paper we classify non-symplectic automorphisms of order 8 on complex K3 surfaces in case that the fourth power of the automorphism has only rational curves in its fixed locus. We show that the fixed locus is the disjoint union of a rational curve and 10 isolated points or it consists in 4 isolated fixed points. We give examples corresponding to the case with a rational curve in the fixed locus and to the case with only isolated points in the fixed locus.