Related papers: Differentiability of the $\chi$-volume function ov…

Given a smooth projective curve C defined over a number field and given two elliptic surfaces E_1/C and E_2/C along with sections P_i and Q_i of E_i (for i = 1,2), we prove that if there exist infinitely many algebraic points t on C such…

This is the second paper of a series. It extends the results of the first paper from number fields to finitely generated fields, based on the recent theory of adelic line bundles of the same authors. We prove an arithmetic Hodge index…

Affine varieties among all algebraic varieties have simple structures. For example, an affine variety does not contain any complete algebraic curve. In this paper we study affine related properties of strata of $k$-differentials on smooth…

We study the long-time behaviour of nonnegative solutions of the Porous Medium Equation posed on Cartan-Hadamard manifolds having very large negative curvature, more precisely when the sectional or Ricci curvatures diverge at infinity more…

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…

In this work, we show that along a particular choice of Hermitian curvature flow, the non-positivity of Chern-Ricci curvature will be preserved if the initial metric has non-positive bisectional curvature. As an application, we show that…

In this paper, we obtain the existence criteria for a geometic flow on noncompact affine Riemannian manifolds. Our results can be regarded as a real version of Lee-Tam [19]. As an application, we prove that a complete noncompact Hessian…

In this paper we detail a number of properties of the affine line of a derivator, including a number of morphisms between $\mathbb{D}$ and $\mathbb{A}^1_{\mathbb{D}}$, a monoidal structure on $\mathbb{A}^1_{\mathbb{D}}$ if $\mathbb{D}$ is…

We prove an equivariant Riemann-Roch formula for divisors on algebraic curves over perfect fields. By reduction to the known case of curves over algebraically closed fields, we first show a preliminary formula with coefficients in Q. We…

In this paper we prove local existence of a Ricci de Turck flow starting at a space with incomplete edge singularities and flowing for a short time within a class of incomplete edge manifolds. We derive regularity properties for the…

We propose a criterion of equidistribution by the differentiability of certain arithmetic invariants. Combined with the slope method and the asymptotic measures, this criterion gives a new "conceptual" proof to equidistribution results…

We study Bott-Chern and Aeppli cohomologies of a vector space endowed with two anti-commuting endomorphisms whose square is zero. In particular, we prove an inequality \`a la Fr\"olicher relating the dimensions of the Bott-Chern and Aeppli…

In the theory of algebraic function fields and their applications to the information theory, the Riemann-Roch theorem plays a fundamental role. But its use, delicate in general, is efficient and practical for applications especially in the…

For $\phi$ a metric on the anticanonical bundle, $-K_X$, of a Fano manifold $X$ we consider the volume of $X$ $$ \int_X e^{-\phi}. $$ We prove that the logarithm of the volume is concave along continuous geodesics in the space of positively…

We generalize Demailly's construction of projective jet bundles and strictly negatively curved pseudometrics on them to the logarithmic case. We establish this logarithmic generalization explicitly via coordinates, just as Noguchi's…

We prove a so called $\kappa$ non-inflating property for Ricci flow, which provides an upper bound for volume ratio of geodesic balls over Euclidean ones, under an upper bound for scalar curvature. This result can be regarded as the…

Generalizing work done by Miyaoka and others in the case of divisors and of curves, we compute the cones of effective cycles of arbitrary dimension on a projective bundle over a complex projective curve in terms of the numerical data in an…

We study the volume distribution of nodal domains of random band-limited functions on generic manifolds, and find that in the high energy limit a typical instance obeys a deterministic universal law, independent of the manifold. Some of the…

We establish an inequality comparing the height and the $\chi$-arithmetic volume of toric metrized divisors on $\mathbb{P}^1_{\mathbb{Q}}$. This gives a partial answer to a question of Burgos, Moriwaki, Philippon and Sombra ([5, remark…

Let $C$ be a curve over a non-archimedean local field of characteristic zero. We formulate algebro-geometric statements that imply boundedness of functions on the moduli space of stable bundles of rank $2$ and fixed odd degree determinant…