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View article: Peskine sixfolds and Debarre-Voisin fourfolds with associated cubic fourfolds
Peskine sixfolds and Debarre-Voisin fourfolds with associated cubic fourfolds Open
We develop the notion of Peskine sixfolds with associated K3 surfaces and cubic fourfolds and work out numerical conditions for when these associations occur. In discriminant 24, the first family for which there is an associated cubic four…
View article: Extremal divisors on moduli spaces of K3 surfaces
Extremal divisors on moduli spaces of K3 surfaces Open
We establish criteria for when Noether--Lefschetz divisors generate an extremal ray in the cone of pseudoeffective divisors of an orthogonal modular variety. In particular, we exhibit many extremal rays of the cone of pseudoeffective divis…
View article: Equivariant Kuznetsov components for cubic fourfolds with a symplectic involution
Equivariant Kuznetsov components for cubic fourfolds with a symplectic involution Open
We study the equivariant Kuznetsov component $\mathrm{Ku}_G(X)$ of a general cubic fourfold $X$ with a symplectic involution. We show that $\mathrm{Ku}_G(X)$ is equivalent to the derived category $D^b(S)$ of a $K3$ surface $S$, where $S$ i…
View article: Cones of Noether-Lefschetz divisors and moduli spaces of hyperkähler manifolds
Cones of Noether-Lefschetz divisors and moduli spaces of hyperkähler manifolds Open
We give a general formula for generators of the NL-cone, the cone of effective linear combinations of irreducible components of Noether-Lefschetz divisors, on an orthogonal modular variety. We then fully describe the NL-cone and its extrem…
View article: The geometry of antisymplectic involutions, II
The geometry of antisymplectic involutions, II Open
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 divisi…
View article: Kodaira dimension of moduli spaces of hyperkähler varieties
Kodaira dimension of moduli spaces of hyperkähler varieties Open
We study the Kodaira dimension of moduli spaces of polarized hyperkähler varieties deformation equivalent to the Hilbert scheme of points on a K3 surface or to O'Grady's ten dimensional variety. This question was studied by Gritsenko-Hulek…
View article: Monodromy of Kodaira fibrations of genus 3
Monodromy of Kodaira fibrations of genus 3 Open
A Kodaira fibration is a non‐isotrivial fibration from a smooth algebraic surface S to a smooth algebraic curve B such that all fibers are smooth algebraic curves of genus g . Such fibrations arise as complete curves inside the moduli spac…
View article: On product identities and the Chow rings of holomorphic symplectic varieties
On product identities and the Chow rings of holomorphic symplectic varieties Open
For a moduli space $M$ of stable sheaves over a $K3$ surface $X$, we propose a series of conjectural identities in the Chow rings $CH_\star (M \times X^\ell),\, \ell \geq 1,$ generalizing the classic Beauville-Voisin identity for a $K3$ su…
View article: Period integrals and Hodge modules
Period integrals and Hodge modules Open
We define a map $\mathcal{P}_M$ attached to any polarized Hodge module $M$ such that the restriction of $\mathcal{P}_M$ to a locus on which $M$ is a variation of Hodge structures induces the usual period integral pairing for this variation…
View article: Chow motives associated to certain algebraic Hecke characters
Chow motives associated to certain algebraic Hecke characters Open
Shimura and Taniyama proved that if is a potentially CM abelian variety over a number field with CM by a field linearly disjoint from F, then there is an algebraic Hecke character of such that . We consider a certain converse to their…
View article: Geometry of Schreieder's varieties and some elliptic and K3 moduli\n curves
Geometry of Schreieder's varieties and some elliptic and K3 moduli\n curves Open
We study the geometry of a class of $n$-dimensional smooth projective\nvarieties constructed by Schreieder for their noteworthy Hodge-theoretic\nproperties. In particular, we realize Schreieder's surfaces as elliptic modular\nsurfaces and …
View article: Unavoidable Sets of Partial Words of Uniform Length
Unavoidable Sets of Partial Words of Uniform Length Open
A set X of partial words over a finite alphabet A is called unavoidable if every two-sided infinite word over A has a factor compatible with an element of X. Unlike the case of a set of words without holes, the problem of deciding whether …
View article: Chow motives associated to certain algebraic Hecke characters
Chow motives associated to certain algebraic Hecke characters Open
Shimura and Taniyama proved that if $A$ is a potentially CM abelian variety over a number field $F$ with CM by a field $K$ linearly disjoint from F, then there is an algebraic Hecke character $λ_A$ of $K$ such that $L(A/F,s)=L(λ_A,s)$. We …
View article: Hodge Structures with Hodge Numbers (n,0,...,0,n) and their Geometric Realizations
Hodge Structures with Hodge Numbers (n,0,...,0,n) and their Geometric Realizations Open
The focus of this thesis is $\\mathbb{Q}$-Hodge structures with Hodge numbers $(n,0,\\ldots,0,n)$, which are $\\mathbb{Q}$-vector spaces $V$ equipped with a decomposition into $n$-dimensional complex subspaces$V\\otimes _\\mathbb{Q}\\mathb…
View article: Some explicit elliptic modular surfaces
Some explicit elliptic modular surfaces Open
We consider algebraic surfaces, recently constructed by Schreieder, that are smooth models of the quotient of the self-product of a complex hyperelliptic curve by a $(\mathbb{Z}/3^c\mathbb{Z})$-action. We show that these surfaces are ellip…
View article: Schreieder's Surfaces Are Elliptic Modular
Schreieder's Surfaces Are Elliptic Modular Open
We consider algebraic surfaces, recently constructed by Schreieder, which are smooth models of the quotient of the self-product of a complex hyperelliptic curve by a $\mathbb{Z}/3^c\mathbb{Z}$-action. We show that these surfaces are ellipt…