Related papers: Bubbling limits of non collapsing polarized K3 sur…
We prove a non-collapsing property for curvature flows of embedded hypersurfaces in the sphere and in hyperbolic space.
We prove existence of thick geodesic triangulations of hyperbolic 3-manifolds and use this to prove existence of universal bounds on the principal curvatures of surfaces embedded in hyperbolic 3-manifolds.
This is an improved version of the eprint previously entitled "Unexpected isomorphisms between hyperk\"ahler fourfolds." We study smooth projective hyperk\"ahler fourfolds that are deformations of Hilbert squares of K3 surfaces and are…
We continue our study on the hypergeometric system $E(3,6)$ which describes period integrals of the double cover family of K3 surfaces. Near certain special boundary points in the moduli space of the K3 surfaces, we construct the local…
We povide a test for numerical simulations for the collapse of regular tubes carried by a 3D incompressible flow. In particular, we obtain necessary conditions for 3D Euler to have a vortex tube collapse in finite time.
In this note we prove the Borel Conjecture for closed, irreducible and sufficiently collapsed three-dimensional Alexandrov spaces. We also pose several questions related to characterization of fundamental groups of three-dimensional…
We study the number and the length of systoles on complete finite area orientable hyperbolic surfaces. In particular, we prove upper bounds on the number of systoles that a surface can have (the so-called kissing number for hyperbolic…
In this paper we investigate when the generic member of a family of K3 surfaces admitting a non--symplectic automorphism of finite order admits also a symplectic automorphism of the same order. We give a complete answer to this question if…
We compute the hybrid limit (in the sense of Boucksom-Jonsson) of the family of K\"ahler-Einstein volume forms on a degeneration of canonically polarized manifolds. The limit measure is a weighted sum of Dirac masses at divisorial…
In this paper, we study the period mappings for the families of $K3$ surfaces derived from the $3$-dimensional $5$-verticed reflexive polytopes. We determine the lattice structures, the period differential equations and the projective…
Building on an idea of Borcherds, Katzarkov, Pantev, and Shepherd-Barron (who treated the case $e=14$), we prove that the moduli space of polarized K3 surfaces of degree $2e$ contains complete curves for all $e\geq 62$ and for some sporadic…
This is mainly a review of my results related to the title. We discuss, how many elliptic fibrations and elliptic fibrations with infinite automorphism group (or the Mordell-Weil group) an algebraic K3 surface over an algebraically closed…
We extend the Kuga-Satake construction to the case of limit mixed Hodge structures of K3 type. We use this to study the geometry and Hodge theory of degenerations of Kuga-Satake abelian varieties, associated to polarized variations of K3…
We utilize total-internal reflection to isolate the two-dimensional `surface foam' formed at the planar boundary of a three-dimensional sample. The resulting images of surface Plateau borders are consistent with Plateau's laws for a truly…
We prove the existence of singular del Pezzo surfaces that are neither K-semistable nor contain any anticanonical polar cylinder.
The more recent paper "Generic strange duality for K3 surfaces" by the authors contains stronger results.
A sufficient integral criterion for a blow-up solution of the Hopf equations (the Euler equations with zero pressure) is found. This criterion shows that a certain positive integral quantity blows up in a finite time under specific initial…
Strange duality is shown to hold over generic $K3$ surfaces in a large number of cases. The isomorphism for elliptic $K3$ surfaces is established first via Fourier-Mukai techniques. Applications to Brill-Noether theory for sheaves on $K3$s…
We establish a relationship between mirror symmetry for K3 surfaces and Arnold's strange duality for K3 surfaces. We compute various examples of mirror families. Among them the mirror moduli family for the moduli space of degree 2n…
Motivated by the Swampland Distance and the Emergent String Conjecture of Quantum Gravity, we analyse the infinite distance degenerations in the complex structure moduli space of elliptic K3 surfaces. All complex degenerations of K3…