相关论文: Hypergeometric decomposition of Delsarte K3 pencil…
We study the hypergeometric functions associated to five one-parameter deformations of Delsarte K3 quartic hypersurfaces in projective space. We compute all of their Picard--Fuchs differential equations; we count points using Gauss sums and…
We give an expression for number of points for the family of Dwork K3 surfaces $$X_{\lambda}^4: \hspace{.1in} x_1^4+x_2^4+x_3^4+x_4^4=4\lambda x_1x_2x_3x_4$$ over finite fields of order $q\equiv 1\pmod 4$ in terms of Greene's finite field…
We construct an explicit, multiplicative Chow-K\"unneth decomposition for the Hilbert scheme of points of a K3 surface. We further refine this decomposition with respect to the action of the Looijenga-Lunts-Verbitsky Lie algebra.
The aim of these notes is to acquaint the reader with important objects in complex algebraic geometry: K3 surfaces and their higher-dimensional analogs, hyperk\"ahler manifolds. These manifolds are interesting from several points of view:…
We study K3 surfaces of degree 6 containing two sets of 12 skew lines such that each line from a set intersects exactly six lines from the other set. These surfaces arise as hyperplane sections of the cubic line complex associated with the…
In this paper, we will study the connections between the mirror symmetry of K3 surfaces and the geometry of the Legendre family of elliptic curves. We will prove that the mirror map of the Dwork family is equal to the period map of the…
The elliptic genera of the K3 surfaces, both compact and non-compact cases, are studied by using the theory of mock theta functions. We decompose the elliptic genus in terms of the N=4 superconformal characters at level-1, and present an…
For all k>0 integer, we consider the regularised I-function of the family of del Pezzo surfaces of degree 8k+4 in P(2,2k+1,2k+1, 4k+1), first constructed by Johnson and Koll\'ar. We show that this function, which is of hypergeometric type,…
We study the decomposability of a Lagrangian homology class on a K3 surface into a sum of classes represented by special Lagrangian submanifolds, and develop criteria for it in terms of lattice theory. As a result, we prove the…
The purpose of this article is to give an explicit description, in terms of hypergeometric functions over finite fields, of zeta function of a certain type of smooth hypersurfaces that generalizes Dwork family. The point here is that we…
In his work studying the Zeta functions of families of hypersurfaces, Dwork came upon a one-parameter family of hypersurfaces (now known as \emph{the} Dwork family). These examples were not only useful to Dwork in his study of his…
Based on high precision computation of periods and lattice reduction techniques, we compute the Picard group of smooth surfaces. We also study the lattice reduction technique that is employed in order to quantify the possibility of…
We construct the first examples of good type III degenerations of hyperk\"ahler varieties in dimension greater than 2. These are presented as moduli of 0-dimensional subschemes on expansions of a degeneration of K3 surfaces. We prove…
We construct a new 20-dimensional family of algebraic hyper-Kaehler fourfolds and prove that they are deformation-equivalent to the second punctual Hilbert scheme of a K3 surface of degree 22.
We give a definition of generalized hypergeometric functions over finite fields using modified Gauss sums, which enables us to find clear analogy with classical hypergeometric functions over the complex numbers. We study their fundamental…
We apply the method of algebraic deformation to N-tuple of algebraic K3 surfaces. When N=3, we show that the deformed triplet of algebraic K3 surfaces exhibits a deformed hyperk\"{a}hler structure. The deformation moduli space of this…
We revisit certain one-parameter families of affine covers arising naturally from Euler's integral representation of hypergeometric functions. We introduce a partial compactification of this family. We show that the zeta function of the…
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 prove the standard conjectures for complex projective varieties that are deformations of the Hilbert scheme of points on a K3 surface. The proof involves Verbitsky's theory of hyperholomorphic sheaves and a study of the cohomology…
We seek to characterize homology classes of Lagrangian projective spaces embedded in irreducible holomorphic-symplectic manifolds, up to the action of the monodromy group. This paper addresses the case of manifolds deformation-equivalent to…