Related papers: Real and complex multiplication on K3 surfaces via…
Despite the transcendental nature of the twistor construction, the algebraic fibres of the twistor space of a K3 surface share certain arithmetic properties. We prove that for a polarized K3 surface with complex multiplication, all…
We show several examples of integrable systems related to special K3 and rational surfaces (e.g., an elliptic K3 surface, a K3 surface given by a double covering of the projective plane, a rational elliptic surface, etc.). The construction,…
We consider surfaces of geometric genus $3$ with the property that their transcendental cohomology splits into $3$ pieces, each piece coming from a $K3$ surface. For certain families of surfaces with this property, we can show there is a…
Motivated by the relation between (twisted) K3 surfaces and special cubic fourfolds, we construct moduli spaces of polarized twisted K3 surfaces of any fixed degree and order. We do this by mimicking the construction of the moduli space of…
We use a relative Fourier-Mukai transform on elliptic K3 surfaces $X$ to describe mirror symmetry. The action of this Fourier-Mukai transform on the cohomology ring of $X$ reproduces relative T-duality and provides an infinitesimal isometry…
In this paper we construct various moduli spaces of K3 surfaces $M$ equipped with a surjective holomorphic map $\pi:M\to\Pb^1$ with generic fiber a complex torus (e.g., an elliptic fibration). Examples include moduli spaces of such maps…
We give an algorithm to compute the periods of smooth projective hypersurfaces of any dimension. This is an improvement over existing algorithms which could only compute the periods of plane curves. Our algorithm reduces the evaluation of…
We study the symplectic topology of certain K3 surfaces (including the "mirror quartic" and "mirror double plane"), equipped with certain K\"ahler forms. In particular, we prove that the symplectic Torelli group may be infinitely generated,…
We study period maps for families of $K3$ surfaces those are given by anti canonical divisors of toric varieties coming from reflexive polytopes $P_2, P_4, P_5$ and $P_r$. We obtain systems of period differential equations for these…
We construct K3 surfaces over number fields that have good reduction everywhere. These do not exists over the rational numbers, by results of Abrashkin and Fontaine. Our surfaces exist for three quadratic number fields, and an infinite…
In this paper, we give Thomae type formula for \KK surfaces $\cS$ given by double covers of the projective plane branching along six lines. This formula gives relations between theta constants on the bounded symmetric domain of type…
This article examines dynamical systems on a class of K3 surfaces in $\mathbb{P}^{2} \times \mathbb{P}^{2}$ with an infinite automorphism group. In particular, this article develops an algorithm to find $\mathbb{Q}$-rational periodic points…
In this paper we show that the moduli space of nodal cubic surfaces is isomorphic to a quotient of a 4-dimensional complex ball by an arithmetic subgroup of the unitary group. This complex ball uniformization uses the periods of certain K3…
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 study families of $K3$ surfaces obtained by double covering of the projective plane branching along curves of $(2,3)$-torus type. In the first part, we study the Picard lattices of the families, and a lattice duality of them. In the…
Classical theory of Complex Multiplication (CM) shows that all abelian extensions of a complex quadratic field $K$ are generated by the values of appropriate modular functions at the points of finite order of elliptic curves whose…
We determine the Hodge endomorphism algebras of non-projective complex K3 surfaces (and more generally, hyperk\"ahler manifolds). We show that they are either totally real fields or number fields generated by Salem numbers. This is unlike…
We introduce a new algorithm for computing the periods of a smooth complex projective hypersurface. The algorithm intertwine with a new method for computing an explicit basis of the singular homology of the hypersurface. It is based on…
We give a new practical method for computing subvarieties of projective hypersurfaces. By computing the periods of a given hypersurface X, we find algebraic cohomology cycles on X. On well picked algebraic cycles, we can then recover the…
We consider real forms of relatively minimal rational surfaces F_m. Connected components of moduli of real non-singular curves in |-2K_{F_m}| had been classified recently for m=0, 1, 4 in math.AG/0312396. Applying similar methods, here we…