Related papers: On Grothendieck's section conjecture for orbicurve…

Period and index of a curve $X/K$ over a $p$-adic local field $K$ such that the fundamental group $\pi_1(X/K)$ admits a splitting are shown to be powers of $p$. As a consequence, examples of curves over number fields are constructed where…

This article proposes a unified analytical approach leading to a partial resolution of the Erdos-Straus, Sierpinski conjectures, and their generalization. We introduce an equivalent reformulation of these conjectures while constructing two…

Using a definition of Jordan curve similar to that of Dieudonn\'e, we prove that our notion is equivalent to that used by Berg et al. in their constructive proof of the Jordan Curve Theorem. We then establish a number of properties of…

Fuglede's conjecture in $\mathbb{Q}_p$ is proved. That is to say, a Borel set of positive and finite Haar measure in $\mathbb{Q}_p$ is a spectral set if and only if it tiles $\mathbb{Q}_p$ by translation.

We prove the main conjecture of arXiv:2102.13459 for integral coefficients in the case of tori and prove that it is t-exact. Along the way we prove that the spectral action as constructed in that manuscript is compatible with the action of…

An extension $B\subset A$ of finite dimensional algebras is bounded if the $B$-$B$-bimodule $A/B$ is $B$-tensor nilpotent, its projective dimension is finite and $\mathrm{Tor}_i^B(A/B, (A/B)^{\otimes_B j})=0$ for all $i, j\geq 1$. We show…

Drinfeld in 2010 proved the companions conjecture for smooth varieties over a finite field, generalizing L. Lafforgue's result for smooth curves. We study the obstruction to prove the conjecture for arbitrary normal varieties. To do this,…

In this article, we give a reformulation of a result from Howe, Leprevost and Poonen on a three dimensional family of abelian threefolds. We also link their result to a conjecture of Serre on a precise form of Torelli theorem for genus 3…

In this paper, we present some partial results for the geometrically m-step solvable Grothendieck conjecture in anabelian geometry. Among other things, we prove the geometrically 3-step solvable Grothendieck conjecture for genus 0 curves…

This is an expository paper based on the results in [12] and [16]. The main goal is to prove the following two conjectures for genus up to two. (1) Witten's conjecture on the relations between higher spin curves and Gelfand-Dickey…

Using meromorphic differentials with real periods, we prove Arbarello's conjecture: any compact complex cycle of dimension $g-n$ in the moduli space $\M_g$ of smooth genus $g$ algebraic curves must intersect the locus of curves having a…

The Grothendieck--Serre conjecture predicts that every generically trivial torsor under a reductive group scheme $G$ over a regular local ring $R$ is trivial. We settle it in the case when $G$ is quasi-split and $R$ is unramified. Some of…

This paper gives an algebraic conjecture which is shown to be equivalent to Thurston's Geometrization Conjecture for closed, orientable 3-manifolds. It generalizes the Stallings-Jaco theorem which established a similar result for the…

In the present note we give a new proof of a result due to Wiseman and Wilson which establishes an analogue of the Sylvester-Gallai theorem valid for curves of degree two. The main ingredients of the proof come from algebraic geometry.…

Given a morphism between complex projective varieties, we make several conjectures on the relations between the set of pseudo-effective (co)homology classes which are annihilated by pushforward and the set of classes of varieties contracted…

In this paper, we develop a new index theory for manifolds with polyhedral boundary. As an application, we prove Gromov's dihedral extremality conjecture regarding comparisons of scalar curvatures, mean curvatures and dihedral angles…

In this short note we give an elementary combinatorial argument, showing that the Conjecture of J. Fern\'andez de Bobadilla, I. Luengo, A. Melle-Hern\'andez, A. N\'emethi follows from the results of M. Borodzik and C. Livingston in the case…

We establish the Hodge conjecture for some subvarieties of a class of toric varieties. First we study quasi-smooth intersections in a projective simplicial toric variety, which is a suitable notion to generalize smooth complete intersection…

We give a proof of the openness conjecture of Demailly and Koll\'ar for positively curved singular metrics on ample line bundles over projective varieties. As a corollary it follows that the openness conjecture for plurisubharmonic…

We introduce generalized Frobenius-Schur indicators for pivotal categories. In a spherical fusion category C, an equivariant indicator of an object in C is defined as a functional on the Grothendieck algebra of the quantum double Z(C) via…