Related papers: Test configurations and Okounkov bodies

Let $\pi:\mathcal{X}\to M$ be a holomorphic fibration with compact fibers and $L$ a relatively ample line bundle over $\mathcal{X}$. We obtain the asymptotic of the curvature of $L^2$-metric and Qullien metric on the direct image bundle…

We consider an ergodic invariant measure $\mu$ for a smooth action of $Z^k$, $k \ge 2$, on a $(k+1)$-dimensional manifold or for a locally free smooth action of $R^k$, $k \ge 2$ on a $(2k+1)$-dimensional manifold. We prove that if $\mu$ is…

This is a sequel of our paper [arXiv:1809.08425] on the Quot-scheme limit and variational properties of Donaldson's functional, which established its coercivity for slope stable holomorphic vector bundles over smooth projective varieties.…

Let $\Mg$ denote the moduli space of compact Riemann surfaces of genus $g$. Mumford had proved that, for each fixed genus $g$, there are isomorphisms asserting that certain higher $DET$ bundles over $\Mg$ are certain fixed…

We explore the existence of a continuous marginal law with respect to the Lebesgue measure for each component $(X,Y,Z)$ of the solution to coupled quadratic forward-backward stochastic differential equations (QFBSDEs) {for which the drift…

We prove that an arbitrary convex body $C \subseteq \mathbf{R}^{n+1} $, whose $ k $-th anisotropic curvature measure (for $ k =0, \ldots , n-1 $) is a multiple constant of the anisotropic perimeter of C, must be a rescaled and translated…

In this paper, we prove a result on the bisection of mass assignments by parallel hyperplanes on Euclidean vector bundles. Our methods consist of the development of a novel lifting method to define the configuration space--test map scheme,…

In this paper, we rigorously prove the bulk-edge correspondence for finite two-dimensional ergodic disordered systems. Specifically, we focus on the short-range Hamiltonians with ergodic disordered on-site potentials. We first introduce the…

In this paper, we prove a result on the bisection of mass assignments by parallel hyperplanes on Euclidean vector bundles. Our methods consist of the development of a novel lifting method to define the configuration space--test map scheme,…

In this paper we will provide a new proof of the fact that for any convex body $K\subseteq\R^n$ $$ \frac{{{2n}\choose{n}}}{n^n}n\int_0^\infty r^{n-1}\vol_n(K\cap(re_n+K))dr\leq\frac{(\vol_n(K))^{n+1}}{(\vol_{n-1}(P_{e_n^\perp}(K)))^n}, $$…

Let K be the closure of a bounded open set with smooth boundary in C^n. A Fekete configuration of order p for K is a finite subset of K maximizing the Vandermonde determinant associated with polynomials of degree at most p. A recent theorem…

We show the existence of Lebesgue-equivalent conservative and ergodic $\sigma$-finite invariant measures for a wide class of one-dimensional random maps consisting of piecewise convex maps. We also estimate the size of invariant measures…

We develop a Hodge theoretic invariant for families of projective manifolds that measures the potential failure of an Arakelov-type inequality in higher dimensions, one that naturally generalizes the classical Arakelov inequality over…

In this article, we focus on the sampling of the configurational Gibbs-Boltzmann distribution, that is, the calculation of averages of functions of the position coordinates of a molecular $N$-body system modelled at constant temperature. We…

One of the basic aims of this paper is to study the relationship between the geometry of ``hypersurface like'' subsets of Euclidean space and the properties of the measures they support. In this context we show that certain doubling…

We show that, for any prime power p^k and any convex body K (i.e., a compact convex set with interior) in Rd, there exists a partition of K into p^k convex sets with equal volume and equal surface area. We derive this result from a more…

We study the sigma-finite measures in the space of vector-valued distributions on the manifold $X$ with Laplace transform $$\Psi(f)=\exp\{-\theta\int_X\ln||f(x)||dx\}, \theta>0.$$ We also consider the weak limit of Haar measures on the…

Let $(M,\omega)$ be a closed $2n$-dimensional semifree Hamiltonian $S^1$-manifold with only isolated fixed points. We prove that a density function of the Duistermaat-Heckman measure is log-concave. Moreover, we prove that $(M,\omega)$ and…

The main result of this work is a homogenization theorem via variational convergence for elastic materials with stiff checkerboard-type heterogeneities under the assumptions of physical growth and non-self-interpenetration. While the…

The Weyl principle is extended from the Riemannian to the pseudo-Riemannian setting, and subsequently to manifolds equipped with generic symmetric $(0,2)$-tensors. More precisely, we construct a family of generalized curvature measures…