Related papers: Analytic varieties with finite volume amoebas are …

We state conjectures on the asymptotic behavior of the volumes of moduli spaces of Abelian differentials and their Siegel-Veech constants as genus tends to infinity. We provide certain numerical evidence, describe recent advances and the…

We prove that a quasi-finite endomorphism of an algebraic variety over an algebraically closed field of characteristic zero, that is injective on the complement of a closed subvariety, is an automorphism. We also prove that an endomorphism…

We investigate the real algebraic complexity of contours of amoebas associated with algebraic hypersurfaces and complete intersections in complex algebraic tori. Motivated by the foundational estimates of Lang--Shapiro--Shustin \cite{LSS},…

Let $\mathcal{A}_g$ denote the moduli stack of principally polarized abelian varieties of dimension $g$. The arithmetic height, or arithmetic volume, of $\overline{\mathcal{A}}_g$, is defined to be the arithmetic degree of the metrized…

To every finite-dimensional $\mathbb C$-algebra $\Lambda$ of finite representation type we associate an affine variety. These varieties are a large generalization of the varieties defined by "$u$ variables" satisfying "$u$-equations", first…

Given a rational monomial map, we consider the question of finding a toric variety on which it is algebraically stable. We give conditions for when such variety does or does not exist. We also obtain several precise estimates of the degree…

We extend some of the results of Agler, Knese, and McCarthy [1] to $n$-tuples of commuting isometries for $n>2$. Let $\mathbb{V}=(V_1,\dots,V_n)$ be an $n$-tuple of a commuting isometries on a Hilbert space and let Ann$(\mathbb{V})$ denote…

We use tools of mathematical logic to analyse the notion of a path on an complex algebraic variety, and are led to formulate a "rigidity" property of fundamental groups specific to algebraic varieties, as well as to define a bona fide…

We describe a class of toric varieties in the $N$-dimensional affine space which are minimally defined by no less than $N-2$ binomial equations.

Answering a question by Nisse and Sottile, we derive a formula for the dimension of the amoeba of an irreducible algebraic variety.

We give several new criteria for a quasi-projective variety to be affine. In particular, we prove that an algebraic manifold $Y$ with dimension $n$ is affine if and only if $H^i(Y, \Omega^j_Y)=0$ for all $j\geq 0$, $i>0$ and $\kappa(D,…

Any compact body in ${\mathbb R}^N$ with smooth boundary defines a two-valued function on the space of affine hyperplanes: the volumes of two parts into which these hyperplanes cut the body. This function is never algebraic if $N$ is even…

Let $X$ be a variety over a complete nontrivially valued field $K$. We construct an algebraizable formal model for the analytification of $X$ in the case $X$ admits a closed embedding into a toric variety. By algebraizable we mean that the…

We show that volumes and related limits in terms of the Kodaira-Iitaka dimension exist for line bundles and graded linear series on compact reduced complex analytic spaces.

In this paper, we consider the cohomology rings of some multiple weight varieties of type A, that is, symplectic torus quotients for a direct product of several coadjoint orbits of the special unitary group. Under some specific assumptions,…

Let $k$ be a field that is finitely generated over the field of rational numbers and $Br(k)$ the Brauer group of $k$. Let $X$ be an absolutely irreducible smooth projective variety over $k$, let $Br(X)$ be the cohomological…

This is a continuation of arXiv: 2408.03012. We answer affirmatively Question 5.10 posed in the previous article. More precisely, let $(X, \omega)$ be a conical symplectic variety of dimension $2n$ with $wt(\omega) = 2$, which has a…

Let $X$ be an algebraic variety such that the group $\text{Aut}(X)$ acts on $X$ transitively. We define the transitivity degree of $X$ as a maximal number $m$ such that the action of $\text{Aut}(X)$ on $X$ is $m$-transitive. If the action…

We study toric varieties over a field k that split in a Galois extension K/k using Galois cohomology with coefficients in the toric automorphism group. Part of this Galois cohomology fits into an exact sequence induced by the presentation…

Given an affine algebraic variety V and a quantization A of its coordinate ring, it is conjectured that the primitive ideal space of A can be expressed as a topological quotient of V. Evidence in favor of this conjecture is discussed, and…