Related papers: Polytope duality for families of $K3$ surfaces and…
The paper presents a proof of the Hodge Riemann relations for the combinatorial intersection cohomology of a polytope, as fist given by K.Karu, in terms of geometric operations on polytopes.
Using toric geometry, lattice theory, and elliptic surface techniques, we compute the Picard Lattice of certain K3 surfaces. In particular, we examine the generic member of each of M. Reid's list of 95 families of Gorenstein K3 surfaces…
The main aim of this survey paper is to gather together some results concerning the Calabi type duality discovered by Hojoo Lee between certain families of (spacelike) graphs with constant mean curvature in Riemannian and Lorentzian…
We study complex algebraic K3 surfaces with finite automorphism groups and polarized by rank-fourteen, 2-elementary lattices. Three such lattices exist, namely $H \oplus E_8(-1) \oplus A_1(-1)^{\oplus 4}$, $H \oplus E_8(-1) \oplus D_4(-1)$,…
For a family of K3 surfaces we implement a variation of a general construction of towers of algebraic curves over finite fields given in a previous paper. As a result we get a good tower over $k=\mathbb{F}_{p^2}$, that is optimal if $p=3$.
The conjectural equivalence of curve counting on Calabi-Yau 3-folds via stable maps and stable pairs is discussed. By considering Calabi-Yau 3-folds with K3 fibrations, the correspondence naturally connects curve and sheaf counting on K3…
We introduce filtered cohomologies of differential forms on symplectic manifolds. They generalize and include the cohomologies discussed in Paper I and II as a subset. The filtered cohomologies are finite-dimensional and can be associated…
The present article is concerned with mirror symmetry for generalized K3 surfaces, with particular emphasis on complex and K\"ahler rigid structures. Inspired by the works of Dolgachev, Aspinwall-Morrison and Huybrechts, we introduce a…
We exhibit automorphisms of a certain K3 surface in $\mathbb{P}^1\times \mathbb{P}^1 \times \mathbb{P}^1$ with an isolated fixed point at which the induced action on the stalk of the structure sheaf is arbitrarily close to the identity.…
The paper discusses quantum graphs with a vertex coupling which interpolates between the common one of the $\delta$ type and a coupling introduced recently by two of the authors which exhibits a preferred orientation. Describing the…
We prove that any hyper-K\"{a}hler sixfold $K$ of generalized Kummer type has a naturally associated manifold $Y_K$ of $\mathrm{K}3^{[3]}$-type. It is obtained as crepant resolution of the quotient of $K$ by a group of symplectic…
We construct non-geometric compactifications by using the F-theory dual of the heterotic string compactified on a two-torus, together with a close connection between Siegel modular forms of genus two and the equations of certain K3…
We consider a semistable degeneration of K3 surfaces, equipped with an effective divisor that defines a polarisation of degree two on a general fibre. We show that the map to the relative log canonical model of the degeneration maps every…
Intuitively, an envelope of a family of curves is a curve that is tangent to a member of the family at each point. Here we use envelopes of families of circles to study objects from matrix theory and hyperbolic geometry. First we explore…
In a Type III degeneration of K3-surfaces the dual graph of the central fibre is a triangulation of the 2-sphere. We realise the tetrahedral, octahedral and especially the icosahedral triangulation in families of K3-surfaces, preferably…
Motivated by the Beauville-Voisin conjecture about Chow rings of powers of $K3$ surfaces, we consider a similar conjecture for Chow rings of powers of EPW sextics. We prove part of this conjecture for the very special EPW sextic studied by…
We study the distribution of algebraic points on K3 surfaces.
We utilise the two principles of decoupling introduced in [arXiv:2407.16108] to prove decoupling for two types of surfaces exhibiting radial symmetry. The first type are surfaces of revolution in $\mathbb R^n$ generated by smooth surfaces…
We study K3 surfaces over non-closed fields and relate the notion of derived equivalence to arithmetic problems.
Based on a novel application of an archimedean type pairing to the geometry and deformation theory of $K3$ surfaces, we construct a regulator indecomposable $K_1$-class on a self-product of a $K3$ surface. In the Appendix, we explain how…