Related papers: Positive degree and arithmetic bigness

We consider heights of horizontal irreducible divisors on an arithmetic surface with respect to some hermitian line bundle. We obtain both lower and upper bounds for these heights. The results are different and sometimes stronger that those…

We establish in this article convergence results of normalized Harder-Narasimhan polygons both in geometric and in arithmetic frameworks by introducing the Harder-Narasimhan filtration indexed by $\mathbb R$ and the associated Borel…

In this master thesis, we give a new proof on the pointwise asymptotic expansion for Bergman kernel of a hermitian holomorphic line bundle on the points where the curvature of the line bundle is positive and satisfy local spectral gap…

Motivated by the effective bounds of ordinary differential equations, we prove an effective version of uniform bounding for partial differential fields with commuting derivations. More precisely, we provide an upper bound for the size of…

This paper is the sequel of the paper "Continuity of volumes on arithmetic varieties", in which we established the arithmetic volume function of smooth hermitian Q-invertible sheaves and proved its continuity. The continuity of the volume…

We build on the recent techniques of Codogni and Patakfalvi, from \cite{Codogni:Patakfalvi:2021}, which were used to establish theorems about semi-positivity of the Chow Mumford line bundles for families of $\K$-semistable Fano varieties.…

In this paper, we study the dimension of cohomology of semipositive line bundles over Hermitian manifolds, and obtain an asymptotic estimate for the dimension of the space of harmonic $(0,q)$-forms with values in high tensor powers of a…

We compute explicitly the Riemannian volume, with respect to the Fubini-Study metric, of a domain bounded by a Hermitian quadric in complex projective space. The volume is a rational function of the eigenvalues of the defining quadratic…

We provide upper bounds on the size of the homology of a closed aspherical Riemannian manifold that only depend on the systole and the volume of balls. Further, we show that linear growth of mod p Betti numbers or exponential growth of…

In this note, we estimate the upper bound of volume of closed positively or nonnegatively curved Alexandrov space $X$ with strictly convex boundary. We also discuss the equality case. In particular, the Boundary Conjecture holds when the…

The space of positively curved hermitian metrics on a positive holomorphic line bundle over a compact complex manifold is an infinite-dimensional symmetric space. It is shown by Phong and Sturm that geodesics in this space can be uniformly…

We develop a probabilistic framework for large-scale dimension bounds in metric geometry, based on padded decompositions, randomized ball carving on net graphs, and the Lov\'asz Local Lemma. For metric measure spaces with volume doubling…

We study the convergence of the Hermite series of measurable functions on the real line. We characterize the norm convergence of truncated partial Hermite sums in rearrangement invariant spaces provided that the truncations vanish…

By using the $\mathbb R$-filtration approach of Arakelov geometry, one establishes explicit upper bounds for geometric and arithmetic Hilbert-Samuel function for line bundles on projective varieties and hermitian line bundles on arithmetic…

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.

We study the degeneration of semipositive smooth hermitian line bundles on open complex manifolds, assuming that the metric extends well away from a codimension two analytic subset of the boundary. Using terminology introduced by R. Hain,…

The Harder-Narasimhan theory provides a canonical filtration of a vector bundle on a projective curve whose successive quotients are semistable with strictly decreasing slopes. In this article, we present the formalization of…

In this paper, we give complex geometric descriptions of the notions of algebraic geometric positivity of vector bundles and torsion-free coherent sheaves, such as nef, big, pseudo-effective and weakly positive, by using singular Hermitian…

Given two arbitrary vector bundles on the Fargues-Fontaine curve, we give an explicit criterion in terms of Harder-Narasimhan polygons on whether they realize a semistable vector bundle as their extensions. Our argument is largely…

By using asymptotic Morse inequalities we give a lower bound for the space of holomorphic sections of high tensor powers in a positive line bundle over a q-concave domain. The curvature of the positive bundle induces a hermitian metric on…