Related papers: An inequality on polarized endomorphisms

In this paper, we discuss a generalization of log canonical singularities in the non-$\mathbb{Q}$-Gorenstein setting. We prove that if a normal complex projective variety has a non-invertible polarized endomorphism, then it has log…

The following conjecture arose out of discussions between B. Harbourne, J. Ro\'e, C. Cilberto and R. Miranda: for a smooth projective surface $X$ there exists a positive constant $c_X$ such that $h^1(\mathcal O_X(C))\le c_X h^0(\mathcal…

If $V$ is a smooth projective variety defined over a local field $K$ with finite residue field, so that its \'etale cohomology over the algebraic closure $\bar{K}$ is supported in codimension 1, then the mod $p$ reduction of a projective…

Let k be an algebraically closed field and X a smooth projective k-variety. A famous theorem of A. A. Roitman states that the canonical map from the degree zero part of the Chow group of zero cycles on X to the group of k-points of its…

If $X$ is a projective, geometrically irreducible variety defined over a finite field $\F_q$, such that it is smooth and its Chow group of 0-cycles fulfills base change, i.e. $CH_0(X\times_{\F_q}\bar{\F_q(X)})=\Q$, then the second author's…

We show that if a rational map is constant on each isomorphism class of unpolarized abelian varieties of a given dimension, then it is a constant map. Our results are motivated by and shed light on a proposed construction of a cryptographic…

In this paper, we show that a general polarized abelian variety $(X,L)$ of type $(1,\dots,1,d)$ and dimension $g$ satisfies property $(N_p)$ if $ d \geq \sum_{i=0}^{g} (p+2)^i$. In particular, the case $p=0$ affirmatively solves a…

Let $X/K$ be a variety over a field, and $A/K$ an abelian variety. A regular homomorphism to $A$ (in codimension $i$) induces, for every smooth geometrically connected pointed $K$-scheme $(T,t_0)$ and every cycle class $Z \in CH^i(T\times…

We define arithmetical and dynamical degrees for dynamical systems with several rational maps on projective varieties, study their properties and relations, and prove the existence of a canonical height function associated with divisorial…

Let X be a subvariety of $P^n$ defined by equations of degrees $ d =(d_1,...,d_s)$, over an algebraically closed field k of any characteristic. We study properties of the Fano scheme $F_r(X)$ that parametrizes linear subspaces of dimension…

v2: We improved a little bit according to the referee's wishes. v1: On $X$ projective smooth over a field $k$, Pink and Roessler conjecture that the dimension of the Hodge cohomology of an invertible $n$-torsion sheaf $L$ is the same as the…

The Isomorphism Conjecture is a conceptional approach towards a calculation of the algebraic K-theory of a group ring RG, where G is an infinite group. In this paper we prove the conjecture in dimensions n<2 for fundamental groups of closed…

An action of a complex reductive group $\mathrm G$ on a smooth projective variety $X$ is regular when all regular unipotent elements in $\mathrm G$ act with finitely many fixed points. Then the complex $\mathrm G$-equivariant cohomology…

In the previous article (\cite{S}), we proved that slope stability of a holomorphic vector bundle $E$ over a polarized manifold $(X,L)$ implies Chow stability of $(\mathbb{P}E^*,\mathcal{O}_{\mathbb{P}E^*}(1)\otimes \pi^* L^k)$ for $k \gg…

Let $X$ be a smooth polarized algebraic surface over the compex number field. We discuss the invariants obtained from the moduli stacks of semistable sheaves of arbitrary ranks on $X$. For that purpose, we construct the virtual fundamental…

Let $R$ be a commutative unital ring. Given a finitely presented affine $R$-group scheme $G$ acting on a separated scheme $X$ of finite type over $R$, we show that there is a prime $p_0$ such that for any $R$-algebra $k$ which is an…

Let $f \colon X \to X$ be a surjective endomorphism of a normal projective surface. When $\operatorname{deg} f \geq 2$, applying an (iteration of) $f$-equivariant minimal model program (EMMP), we determine the geometric structure of $X$.…

Let $X$ denote an equivariant embedding of a connected reductive group $G$ over an algebraically closed field $k$. Let $B$ denote a Borel subgroup of $G$ and let $Z$ denote a $B \times B$-orbit closure in $X$. When the characteristic of $k$…

Assume that $M$ is a c.t.m. of $ZFC+CH$ containing a simplified $(\omega_1,2)$-morass, $P\in M$ is the poset adding $\aleph_3$ generic reals and $G$ is $P$-generic over $M$. In $M$ we construct a function between sets of terms in the…

A complex manifold $X$ of dimension $n$ together with an ample vector bundle $E$ on it will be called a {\sf generalized polarized variety}. The adjoint bundle of the pair $(X,E)$ is the line bundle $K_X + det(E)$. We study the positivity…