The transcendental lattice of the sextic Fermat surface

A nontrivial algebraic cycle in the Jacobian variety of the Fermat sextic

We compute some value of the harmonic volume for the Fermat sextic. Using this computation, we prove that some special algebraic cycle in the Jacobian variety of the Fermat sextic is not algebraically equivalent to zero.

Projective models of the supersingular K3 surface with Artin invariant 1 in characteristic 5

Let X be a supersingular K3 surface in characteristic 5 with Artin invariant 1. Then X has a polarization that realizes X as the Fermat sextic double plane. We present a list of polarizations of X with degree 2 whose intersection number with this Fermat sextic polarization is less than or equal to 5, and give the defining equations of the corresponding projective models. We also present a method to describe birational morphisms between these projective models explicitly. As a by-product, a non-projective automorphism of the Fermat sextic double plane is obtained.

Zariski-van Kampen method and transcendental lattices of certain singular K3 surfaces

We present a method of Zariski-van Kampen type for the calculation of the transcendental lattice of a complex projective surface. As an application, we calculate the transcendental lattices of complex singular K3 surfaces associated with an arithmetic Zariski pair of maximizing sextics of type $A_{10}+A_{9}$ that are defined over a real quadratic field and are conjugate to each other over the field of rational numbers.

Why should one compute periods of algebraic cycles?

In this article we show how the data of integrals of algebraic differential forms over algebraic cycles can be used in order to prove that algebraic and Hodge cycle deformations of a given algebraic cycle are equivalent. As an example, we prove that most of the Hodge and algebraic cycles of the Fermat sextic fourfold cannot be deformed in the underlying parameter space. We then take a difference of two linear cycles inside the Fermat variety, and gather evidences that the Hodge locus corresponding to this is smooth and reduced. This implies the existence of new algebraic cycles in the Fermat v…

由多项式对定义的曲面

Surfaces defined by pairs of polynomials

本文证明了任意次数的普通对角曲面及由两个双线性形式之和定义的曲面的Brauer群平凡性，并确定了Fermat曲面几何Picard群的定义域，完成了Shioda和Aoki的工作。

The modularity of K3 surfaces with non-symplectic group actions

We consider complex K3 surfaces with a non-symplectic group acting trivially on the algebraic cycles. Vorontsov and Kondo classified those K3 surfaces with transcendental lattice of minimal rank. The purpose of this note is to study the Galois representations associated to these K3 surfaces. The rank of the transcendental lattices is even and varies from 2 to 20, excluding 8 and 14. We show that these K3 surfaces are dominated by Fermat surfaces, and hence they are all of CM type. We will establish the modularity of the Galois representations associated to them. Also we discuss mirror symmetry…