Related papers: Elementary counterexamples to Kodaira vanishing in…
The goal of this paper is to give a new proof of a special case of the Kodaira-Saito vanishing theorem for a variation of Hodge structure on the complement of a divisor with normal crossings. The proof does not use the theory of mixed Hodge…
I consider the class of surfaces $X$ over algebraically closed fields with numerical invariants given in the title. In characteristic zero, this class contains fake projective planes which were introduced by David Mumford. I prove that in…
In this paper, we prove the weak positivity theorem in positive characteristic when the canonical ring of the geometric generic fiber $F$ is finitely generated and the Frobenius stable canonical ring of $F$ is large enough. As its…
The Kodaira principle asserts that suitable cohomological contraction maps annihilate obstructions to deforming complex structures. In this paper, we revisit these phenomena from a purely analytic point of view, developing a refined power…
We generalize a vanishing theorem for the cohomology of symmetric powers of the cotangent bundle of subvarieties of projective space due to Schneider. From this we deduce new vanishing results for Green-Griffiths jet differential bundles,…
It is well known that the Eisenbud-Goto regularity conjecture is true for arithmetically Cohen-Macaulay varieties, projective curves, smooth surfaces, smooth threefolds in $\mathbb{P}^5$, and toric varieties of codimension two. After J.…
We use homological methods to establish a formal criterion for Generic Vanishing, in the sense originated by Green and Lazarsfeld and pursued further by Hacon and the first author, but in the context of an arbitrary Fourier-Mukai…
Let X be a smooth projective curve of genus g>1 over an algebraically closed field of characteristic 2. Pull-back by the (absolute) Frobenius on X only defines a rational morphism on the moduli scheme of rank-2 vector bundles on X, because…
We consider the following conjecture: if X is a smooth projective variety over a field of characteristic zero, then there is a dense set of reductions X_s to positive characteristic such that the action of the Frobenius morphism on the top…
Using inversion of adjunction, we deduce from Nadel's theorem a vanishing property for ideals sheaves on projective varieties, a special case of which recovers a result due to Bertram--Ein--Lazarsfeld. This enables us to generalize to a…
I propose a few increasingly stronger "superadditivity" conjectures regarding the behavior of Kodaira dimension under morphisms of smooth quasi-projective complex varieties.
We introduce higher-order variants of the Frobenius-Seshadri constant due to Musta\c{t}\u{a} and Schwede, which are defined for ample line bundles in positive characteristic. These constants are used to show that Demailly's criterion for…
We prove a Kawamata-Viehweg vanishing theorem on a normal compact Kahler space X: if L is a nef line bundle with numerical dimension at least equal to 2, then the q-th cohomology group of K_X+L vanishes for q at least equal to the dimension…
We show that every holomorphic one-form on a smooth complex projective variety of general type must vanish at some point. The proof uses generic vanishing theory for Hodge D-modules on abelian varieties.
In this paper we study the derivatives of Frobenius and the derivatives of Hodge weights for families of Galois representations with triangulations. We generalize the Fontaine-Mazur L-invariant and use it to build a formula which is a…
Two conjectures relating the Kodaira dimension of a smooth projective variety and existence of number of nowhere vanishing 1-forms on the variety are proposed and verified in dimension 3.
The Kodaira-Nakano Vanishing Theorem has been generalized to the relative setting by A. Sommese. We prove a version of this theorem for non-compact manifolds. As an apllication, we prove that the cohomology of a fiber of a symplectic…
This article is concerned with the convexity properties of universal covers of projective varieties. We study the relation between the convexity properties of the universal cover of X and the properties of the pullback map sending vector…
We investigate when the filtration induced by Beilinson's spectral sequence splits non-canonically into a direct sum decomposition. We conclude that for any vector bundle $\mathcal{E}$ on a projective space over an algebraically closed…
We investigate prime avoidance for an arbitrary set of prime ideals in a commutative ring. Various necessary and/or sufficient conditions for prime avoidance are given, which yield natural classes of infinite sets of primes that satisfy…