Related papers: Higher syzygies on general polarized abelian varie…

Let $A = \prod_{1\leq i\leq n} A_i$ be the product of principally polarized abelian varieties $A_1, \ldots, A_n$ of dimensions $g_1, \ldots, g_n$, respectively, each defined over a number field $K$, and pairwise nonisogenous over…

We prove a connexity theorem for abelian varieties in characteristic $0$: if $X$ is an abelian variety and $V\rightarrow X$ and $W\rightarrow X$ two morphisms, then, under certain hypotheses, the fiber product of $V$ and $W$ over $X$ is…

Building on the theory of infinitesimal Newton--Okounkov bodies and previous work of Lazarsfeld--Pareschi--Popa, we present a Reider-type theorem for higher syzygies of ample line bundles on abelian surfaces. As an application of our…

We prove the Ax-Lindemann theorem for the coarse moduli space $\mathcal{A}_{g}$ of principally polarized abelian varieties of dimension $g\ge 1$, and affirm the Andr\'e-Oort conjecture unconditionally for $\mathcal{A}_{g}$ for $g\le 6$.

We apply Angehrn-Siu-Helmke's method to estimate basepoint freeness thresholds of higher dimensional polarized abelian varieties. We showed that a conjecture of Caucci holds for very general polarized abelian varieties in the moduli spaces…

Let L be a normally generated line bundle on X; we say L satisfies property N_p (notation after Mark Green) if the matrices in the free resolution of R (the homogeneous coordinate ring of X) over S (the homogeneous coordinate ring of the…

Let X be a proper smooth variety over the complex numbers. We consider the generalized Albanese variety Alb(X,Y) of X of modulus Y, which is a higher dimensional analogue of the generalized Jacobian variety with modulus of Rosenlicht-Serre.…

It is shown that a complex normal projective variety has non-positive Kodaira dimension if it admits a non-isomorphic quasi-polarized endomorphism. The geometric structure of the variety is described by methods of equivariant lifting and…

We make further progress towards a Kneser-type generalization of Pollard's Theorem to general abelian groups. For two sets $A$ and $B$ in an abelian group $G$, the \emph{$t$-popular sumset} of $A$ and $B$, denoted by $A+_t B$, is the set of…

Consider a very general abelian variety $A$ of dimension at least $3$ and an integer $0<d\leq \dim A$. We show that if the map $A^k\to CH_0(A)$ has a $d$-dimensional fiber then $k\geq d+(\dim A+1)/2$. This extends results of the…

In this article we we continue the study of property $N_p$ of irrational ruled surfaces begun in \cite{ES}. Let $X$ be a ruled surface over a curve of genus $g \geq 1$ with a minimal section $C_0$ and the numerical invariant $e$. When $X$…

A differential analogue of the conjecture of Reichstein, Rogalski, and Zhang in algebraic dynamics is here established: if $X$ is a projective variety over an algebraically closed field of characteristic zero which admits a global algebraic…

Let A be an abelian variety defined over a number field and of dimension g. When g<3, by the recent work of Sawin, we know the exact (nonzero) value of the density of the set of primes which are ordinary for A. In higher dimension very…

There are many equivalent ways to describe the p-torsion of a principally polarized abelian variety in characteristic p. We briefly explain these methods and then illustrate them for abelian varieties A of arbitrary dimension g in several…

In this paper, we classify the possible group structures on the set of $R$-valued points of an abelian variety, where $R$ is any real closed field. We make use of a family of abelian varieties that, in effect, allows one to quantify over…

Inspired by the work of Cherry, we introduce and study a new notion of Brody hyperbolicity for rigid analytic varieties over a non-archimedean field $K$ of characteristic zero. We use this notion of hyperbolicity to show the following…

We prove that the Albanese morphism of any normal proper variety $X$ in positive characteristic satisfying $S^0(X, \omega_X) \neq 0$ and $P_2(X) = 1$ is surjective with connected fibers, adn that $\mathrm{Alb}(X)$ is ordinary. We obtain…

In this paper, we consider the following $(A, B)$-polynomial $f$ over finite field: $$f(x_0,x_1,\cdots,x_n)=x_0^Ah(x_1,\cdots,x_n)+g(x_1,\cdots,x_n)+P_B(1/x_0),$$ where $h$ is a Deligne polynomial of degree $d$, $g$ is an arbitrary…

Let g be a semisimple Lie algebra of finite dimension. The nullcone N of g is the set of (x,y) in g x g such that x and y are nilpotents and are in the same Borel sualgebra. The main result of this paper is that N is a closed and…

We show how a natural constant introduced by Jiang and Pareschi for a polarized abelian variety encodes information about the syzygies of the section ring of the polarization. As a particular case this gives a quick and characteristic-free…