Related papers: A necessary condition for Chow semistability of po…

Let $\Delta\subset \mathbb{R}^n$ be an $n$-dimensional integral Delzant polytope. It is well-known that there exist the $n$-dimensional compact toric manifold $X_\Delta$ and the very ample $(\mathbb{C}^\times)^n$-equivariant line bundle…

For a given polarized toric variety, we define the notion of $\lambda$-stability which is a natural generalization of uniform K-stability. At the neighbourhoods of the vertices of the corresponding moment polytope $\Delta$, we consider…

The logarithmic Chow semistability is a notion of Geometric Invariant Theory for the pair consists of varieties and its divisors. In this paper we introduce a obstruction of semistability for polarized toric manifolds and its toric…

In this paper, we prove that a Gorenstein toric Fano variety $(X, -K_{X})$ is asymptotically Chow semistable then it is Ding polystable with respect to toric test configurations (Theorem 1.3). This extends the known result obtained by…

Let $X \to \mathbb P^N$ be a smooth linearly normal projective variety. It was proved by Paul that the $K$-energy of $(X, {\omega_{FS}}|_{X})$ restricted to the Bergman metrics is bounded from below if and only if the pair of (rescaled)…

Chow stability is one notion of Mumford's Geometric Invariant Theory for studying the moduli space of polarized varieties. Kapranov, Sturmfels and Zelevinsky detected that Chow stability of polarized toric varieties is determined by its…

Let $P_{\lambda\Sigma_n}$ be the Ehrhart polynomial associated to an intergal multiple $\lambda$ of the standard symplex $\Sigma_n \subset \mathbb{R}^n$. In this paper we prove that if $(M, L)$ is an $n$-dimensional polarized toric manifold…

Given a polarized manifold there are obstructions for asymptotic Chow semistability described as integral invariants. One of them is an obstruction to the existence for the first Chern class of the polarization to admit a constant scalar…

For a polarized algebraic manifold $(X,L)$, let $T$ be an algebraic torus in the group of all holomorphic automorphisms of $X$. Then strong relative K-stability will be shown to imply asymptotic relative Chow-stability. In particular, by…

Let $X$ be a Fano type variety and $(X,\Delta)$ be a log Calabi-Yau pair with $\Delta$ a Weil divisor. If $(X,\Delta)$ admits a polarized endomorphism, then we show that $(X,\Delta)$ is a finite quotient of a toric pair. Along the way, we…

We consider all complex projective manifolds X that satisfy at least one of the following three conditions: 1. There exists a pair $(C ,\varphi)$, where $C$ is a compact connected Riemann surface and $\varphi : C\to X$ a holomorphic map,…

Given a complex-projective klt pair $(X, \Delta)$ with standard coefficients and such that $K_X + \Delta$ is ample, we determine necessary and sufficient conditions for the pair $(X, \Delta)$ to be uniformized by a bounded symmetric domain.…

Let $X$ be a smooth irreducible complex algebraic variety of dimension $n$ and $L$ a very ample line bundle on $X$. Given a toric degeneration of $(X,L)$ satisfying some natural technical hypotheses, we construct a deformation $\{J_s\}$ of…

We prove that polarised manifolds that admit a constant scalar curvature K\"ahler (cscK) metric satisfy a condition we call slope semistability. That is, we define the slope $\mu$ for a projective manifold and for each of its subschemes,…

For a polarized toric variety (X,L) of dimension n less than 5, we give a lower bound of the Delta-genus by using the vanishing number of adjoint bundle of a multiple of L. We show that for a polarized toric variety of dimension n with…

Let $X$ be a smooth projective toric variety, and let $\widetilde{X}$ denote the blow-up of $X$ at finitely many distinct tours-invariant points. This paper provides an explicit combinatorial formula for the Chow weight of $\widetilde{X}$…

The holomorphic invariants introduced by Futaki as obstruction to the asymptotic Chow semistability are studied by an algebraic-geometric point of view and are shown to be the Mumford weights of suitable line bundles on the Hilbert scheme.…

In this paper, we consider an obstruction to asymptotic Chow-semistability of a polarized Kaehler algebraic manifold. Even when a linear algebraic group of positive dimension acts nontrivially and holomorphically on a polarized Kaehler…

We formulate a sufficient condition for non-displaceability (by Hamiltonian symplectomorphisms which are identity outside of a compact) of a pair of subsets in a cotangent bundle. This condition is based on micro-local analysis of sheaves…

We prove that a general rational smooth Fano threefold admits a toric model. More precisely, for a general rational smooth Fano threefold $X$, we show the existence of a boundary divisor $D$ for which $(X,D)\simeq_{\rm cbir}…