Related papers: Section conjectures over $\mathbb{C}$ and Kodaira …
We define Strebel differentials for stable complex curves, prove the existence and uniqueness theorem that generalizes Strebel's theorem for smooth curves, prove that Strebel differentials form a continuous family over the moduli space of…
We address the question of existence of sections of fibrations in two settings. First, we show that a bundle with base a finite 2-complex admits a section if and only if the inclusion of the fiber is $\pi_1$-injective and the associated…
Given a family $\pi:\mc{X} \rightarrow B$ of smooth projective varieties, a closed fiber $\mc{X}_o$ and an invertible sheaf $\mc{L}$ on $\mc{X}_o$, we compare the Hodge locus in $B$ corresponding to the Hodge class $c_1(\mc{L})$ with the…
In the early 1970s, Andrew Ogg made several conjectures about the rational torsion points of elliptic curves over $\mathbb{Q}$ and the Jacobians of modular curves. These conjectures were proved shortly after by Barry Mazur as a consequence…
A famous conjecture attributed to Kodaira asks whether any compact Kaehler manifold can be approximated by projective manifolds. We confirm this conjecture on projectivized direct sums of three line bundles on three-dimensional complex tori…
We introduce in this paper the notion of Hodge similarities of transcendental lattices of hyperk\"ahler manifolds and investigate the Hodge conjecture for these Hodge morphisms. Studying K3 surfaces with a symplectic automorphism, we prove…
In this note we consider different versions of coinduction functors between categories of comodules for corings induced by a morphism of corings. In particular we introduce a new version of the coinduction functor in the case of locally…
Initiated by Gromov, the study of holomorphic curves in symplectic manifolds has been a powerfull tool in symplectic topology, however the moduli space of holomorphic curves is often very difficult to find. A common technique is to study…
Andr\'e used Hodge-theoretic methods to show that in a smooth proper family X to B of varieties over an algebraically closed field k of characteristic 0, there exists a closed fiber having the same Picard number as the geometric generic…
We prove the equidistribution of the Hodge locus for certain non-isotrivial, polarized variations of Hodge structure of weight $2$ with $h^{2,0}=1$ over complex, quasi-projective curves. Given some norm condition, we also give an asymptotic…
In [5], without giving a detailed proof, Yamauchi provided a formula to calculate the genus of a certain family of smooth complete intersection algebraic curves. That formula is used extensively in [1] to study the algebraic curves for…
In this paper we introduce and study some mathematical structures on top of transitive Lie algebroids in order to formulate gauge theories in terms of generalized connections and their curvature: metrics, Hodge star operator and integration…
The main aim of this paper is to study chains of model structures arising from cotorsion pairs in extriangulated categories. Starting with a hereditary Hovey triple, we construct further hereditary Hovey triples whose homotopy categories…
The Godement cosimplicial resolution is available for a wide range of categories of sheaves. In this paper we investigate under which conditions of the Grothendieck site and the category of coefficients it can be used to obtain fibrant…
We investigate the birational section conjecture for curves over function fields of characteristic zero and prove that the conjecture holds over finitely generated fields over Q if it holds over number fields.
We prove, following Deligne and Andr\'e, that the Hodge classes on abelian varieties of CM-type can be expressed in terms of divisor classes and split Weil classes, and we describe some consequences. In particular, we show that…
We consider the issue of when the L-polynomial of one curve over $\F_q$ divides the L-polynomial of another curve. We prove a theorem which shows that divisibility follows from a hypothesis that two curves have the same number of points…
We identify additional structure on a conservative lax monoidal functor from a closed monoidal category $\mathcal{C}$ to a Grothendieck-Verdier category $\mathcal{D}$, such that the Grothendieck-Verdier structure of $\mathcal{D}$ lifts to…
We investigate when Isomorphism Conjectures, such as the ones due to Baum-Connes, Bost and Farrell-Jones, are stable under colimits of groups over directed sets (with not necessarily injective structure maps). We show in particular that…
The paper provides a computation of the additive structure as well as a partial description of the Chern-class module structure of the cohomology of $GL_3$ over the function ring of an elliptic curve over a finite field. The computation is…