Related papers: Inequalities for semistable families of arithmetic…

We prove Bogomolov's inequality for semistable sheaves on product type varieties in arbitrary characteristic. This gives the first examples of varieties with positive Kodaira dimension in positive characteristic on which Bogomolov's…

We introduce the notion of a Hodge-Riemann pair of cohomology classes that generalizes the classical Hodge-Riemann bilinear relations, and the notion of a Bogomolov pair of cohomology classes that generalizes the Bogomolov inequality for…

We give a new simple proof of boundedness of the family of semistable sheaves with fixed numerical invariants on a fixed smooth projective variety. In characteristic zero our method gives a quick proof of Bogomolov's inequality for…

We prove Bogomolov's inequality on a normal projective variety in positive characteristic and we use it to show some new restriction theorems and a new boundedness result. Then we redefine Higgs sheaves on normal varieties and we prove…

We prove Bogomolov type inequalities for high Chern characters of semistable sheaves satisfying certain Frobenius semipositivity. The key ingredients in the proof are a high rank generalization of the asymptotic Riemann-Roch theorem and…

We develop a sheaf cohomology theory of algebraic varieties over an algebraically closed non-trivially valued non-archimedean field $K$ based on Hrushovski-Loeser's stable completion. In parallel, we develop a sheaf cohomology of definable…

A. Moriwaki proved the following arithmetic analogue of the Bogomolov unstability theorem. If a torsion-free hermitian coherent sheaf on an arithmetic surface has negative discriminant then it admits an arithmetically destabilising…

We prove some positivity results on the coefficients in the complexified Hilbert polynomial of a semi-stable object. After applying these results on the classical slope stability conditions, we get sequences of quadratic inequalities for…

Let f : X --> Spec(Z) be an arithmetic variety of dimension d >= 2 and (H, k) an arithmetically ample Hermitian line bundle on X. Let (E, h) be a rank r vector bundle on X. In this paper, we will prove that if E is semistable with respect…

We deduce an analogue of the Bogomolov conjecture for non-degenerate subvarieties in fibered products of families of elliptic curves from the author's recent theorem on equidistribution in families of abelian varieties. This generalizes…

We introduce the volume function for hermitian invertible sheaves on an arithmetic variety as an analogue of the geometric volume function. The main result of this paper is the continuity of the arithmetic volume function. As a consequence,…

Let K be an algebraic number field, O_K the ring of integers of K, and f : X --> Spec(O_K) an arithmetic surface. Let (E, h) be a rank r Hermitian vector bundle on X such that $E$ is semistable on the geometric generic fiber of f. In this…

We present a variation of quasi-isometry to approach the problem of defining a geometric notion equivalent to commensurability. In short, this variation can be summarized as "quasi-isometry with uniform parameters for a large enough family…

We provide a new proof of the following result: Let $X$ be a variety of finite type over an algebraically closed field $k$ of characteristic 0, let $Z\subset X$ be a proper closed subset. There exists a modification $f:X_1 \rar X$, such…

In his work on singularities, expanders and topology of maps, Gromov showed, using isoperimetric inequalities in graded algebras, that every real valued map on the $n$-torus admits a fibre whose homological size is bounded below by some…

In this article, we prove the orbifold version of the Bogomolov-Gieseker inequality for stable $\mathbb Q$-sheaves on K\"ahler varieties, generalizing our earlier work \cite{GP25} in dimension three. We also provide a characterization of…

We generalize the Bogomolov-Gieseker inequality for semistable coherent sheaves on smooth projective surfaces to smooth Deligne-Mumford surfaces. We work over positive characteristic $p>0$ and generalize Langer's method to smooth…

To a pair $(A,s)$ consisting of a smooth, cyclic $A_\infty$-algebra $A$ and a splitting $s$ of the Hodge filtration on its Hochschild homology Costello (2005) associates an invariant which conjecturally generalizes the total descendant…

We generalize the framework of tilt-stability to singular schemes and formulate the generalized Bogomolov-Gieseker inequality conjecture of Bayer-Macr\`i-Toda for singular threefolds. We also develop relative versions of these…

In this paper we show an Arakelov inequality for semi-stable families of algebraic curves of genus $g\geq 1$ over characteristic $p$ with nontrivial Kodaira-Spencer maps. We apply this inequality to obtain an upper bound of the number of…