Related papers: On generalized Beauville decompositions

A $K3$ surface with an ample divisor of self-intersection 2 is a double cover of the plane branched over a sextic curve. We conjecture that a similar statement holds for the generic couple $(X,H)$ with $X$ a deformation of $(K3)^{[n]}$ and…

We investigate F-theory models with a discrete $\mathbb{Z}_2$ gauge symmetry and $SU(n)$ gauge symmetries. We utilize a class of rational elliptic surfaces lacking a global section, known as Halphen surfaces of index 2, to yield genus-one…

Most applications of the hard Lefschetz theorem related to combinatorial properties of simplicial complexes involve their $h$-vectors. In the context of positivity properties involving $h$-vectors of flag spheres, $f$-vectors with a…

We analyse the geometry of Hilbert schemes of points on abelian surfaces and Beauville's generalized Kummer varieties in positive characteristics. The main result is that, in characteristic two, the addition map from the Hilbert scheme of…

We investigate a class of Leibniz algebroids which are invariant under diffeomorphisms and symmetries involving collections of closed forms. Under appropriate assumptions we arrive at a classification which in particular gives a…

On the complement $X= {\mathbb C}^2 - \bigcup_{i=1}^n L_i$ to a central plane line arrangement $\bigcup_{i=1}^n L_i \subset {\mathbb C}^2$, a locally constant sheaf of complex vector spaces $\mathcal L_a$ is associated to any multi-index $a…

A theorem by Mumford implies that every automorphic line bundle on a pure open Shimura variety, equipped with an invariant smooth metric, can be uniquely extended as a line bundle on a toroidal compactification of the variety, in such a way…

We explore the connection between $K3$ categories and 0-cycles on holomorphic symplectic varieties. In this paper, we focus on Kuznetsov's noncommutative $K3$ category associated to a nonsingular cubic 4-fold. By introducing a filtration on…

We show that a natural isomorphism between the rational cohomology groups of the two zero-dimensional Hilbert schemes of $n$-points of two surfaces, the affine plane minus the axes and the cotangent bundle of an elliptic curve, exchanges…

Given a non-singular variety with a K3 fibration f : X --> S we construct dual fibrations Y --> S by replacing each fibre X_s of f by a two-dimensional moduli space of stable sheaves on X_s. In certain cases we prove that the resulting…

We study versions of homological mirror symmetry for hypersurface cusp singularities and the three hypersurface simple elliptic singularities. We show that the Milnor fibres of each of these carries a distinguished Lefschetz fibration; its…

We study the Tannakian properties of the category of perverse sheaves on elliptic curves endowed with the convolution product. We establish that for certain sheaves with unipotent local monodromy over seven points the corresponding Tannaka…

Let $V:f=0$ be a hypersurface of degree $d \geq 3$ in the complex projective space $\mathbb{P}^n$, $n \geq 3$, having only isolated singularities. Let $M(f)$ be the associated Jacobian algebra and $H: \ell=0$ be a hyperplane in…

Beauville introduced an integrable Hamiltonian system whose general level set is isomorphic to the complement of the theta divisor in the Jacobian of the spectral curve. This can be regarded as a generalization of the Mumford system. In…

We establish a compact analog of the P = W conjecture. For a holomorphic symplectic variety with a Lagrangian fibration, we show that the perverse numbers associated with the fibration match perfectly with the Hodge numbers of the total…

We study the Diophantine equation $a^5+b^5=c^5+d^5$ under the linear slicing constraint $(c+d)-(a+b)=h$. We first prove the necessary congruence $30\mid h$. After symmetrization, the associated discriminant equation defines, for each fixed…

We compare different constructions of mirrors of del Pezzo surfaces, focusing on degree $d \leq 3$. In particular, we extract Lefschetz fibrations, with associated exceptional collections, from the mirrors obtained via the Hori-Vafa and…

We prove that the perverse Leray filtration for the Hitchin morphism is locally constant in families, thus providing some evidence towards the validity of the $P=W$ conjecture due to de Cataldo, Hausel and Migliorini in non Abelian Hodge…

We prove that the first order deformations of two smooth projective K3 surfaces are derived equivalent under a Fourier--Mukai transform if and only if there exists a special isometry of the total cohomology groups of the surfaces which…

We study birational maps among 1) the moduli space of semistable torsion sheaves of Hilbert polynomial $4m+2$ on a smooth quadric surface, 2) the moduli space of semistable torsion sheaves of Hilbert polynomial $m^{2}+3m+2$ on…