Related papers: Vanishing and injectivity theorems for Hodge modul…
We note that the vanishing and injectivity theorems of Koll\'ar and Esnault-Viehweg can be used to give a quick algebraic proof of a strengthening of the Ein-Lazarsfeld Skoda-type division theorem for global sections of adjoint line bundles…
We study the vanishing of (co)homology along ring homomorphisms for modules that admit certain filtrations, and generalize a theorem of O. Celikbas-Takahashi. Our work produces new classes of rigid and test modules, in particular over local…
We introduce the intersection cohomology module of a matroid and prove that it satisfies Poincar\'e duality, the hard Lefschetz theorem, and the Hodge-Riemann relations. As applications, we obtain proofs of Dowling and Wilson's Top-Heavy…
We investigate an analogue of the Grothendieck $p$-curvature conjecture, where the vanishing of the $p$-curvature is replaced by the stronger condition, that the module with connection mod $p$ underlies a $\mathcal{D}_X$-module structure.…
A toric polyhedron is a reduced closed subscheme of a toric variety that are partial unions of the orbits of the torus action. We prove vanishing theorems for toric polyhedra. We also give a proof of the $E_1$-degeneration of Hodge to de…
In this paper we will first show some Kollar-Enoki type injectivity theorems on compact Kahler manifolds, by using the Hodge theory, the Bochner- Kodaira-Nakano identity and the analytic method provided by O. Fujino and S. Matsumura in [15,…
We bound the generation level of the Hodge filtration on the localization along a hypersurface in terms of its minimal exponent. As a consequence, we obtain a local vanishing theorem for sheaves of forms with log poles. These results are…
Weak coalgebra-Galois extensions are studied. A notion of an invertible weak entwining structure is introduced. It is proven that, within an invertible weak entwining structure, the surjectivity of the canonical map implies bijectivity…
We prove the Kawamata-Viehweg vanishing theorem for a large class of divisors on surfaces in positive characteristic. By using this vanishing theorem, Reider-type theorems and extension theorems of morphisms for normal surfaces are…
We prove a conjecture of Schmid and the second named author that the unitarity of a representation of a real reductive Lie group with real infinitesimal character can be read off from a canonical filtration, the Hodge filtration. Our proof…
In this paper we discuss an obstruction to the integral Hodge conjecture, which arises from certain behavior of vanishing cycles. This allows us to construct new counter-examples to the integral Hodge conjecture. One typical such…
We formulate and establish a generalization of Koll\'ar's injectivity theorem for adjoint bundles twisted by suitable multiplier ideal sheaves. As applications, we generalize Koll\'ar's torsion-freeness, Koll\'ar's vanishing theorem, and a…
This is a sequel to "Kodaira-Saito vanishing via Higgs bundles in positive characteristic" (arXiv:1611.09880). However, unlike the previous paper, all the arguments here are in characteristic zero. The main result is a Kodaira vanishing…
The purpose of this paper is to give two supplements for vanishing theorems: One is a relative version of the Kawamata-Viehweg-Nadel type vanishing theorem, which is obtained from an observation for the variation of the numerical dimension…
Building on the nonabelian Hodge theory in positive characteristic developed by Ogus, Vologodsky, and Schepler, we propose a generalization of the decomposition theorem of Deligne and Illusie from the perspective of mixed Hodge modules.…
We prove a vanishing theorem for the Hodge number h^21 of projective toric varieties provided by a certain class of polytopes. We explain how this Hodge number also gives information about the deformation theory of the toric Gorenstein…
We use the liftability of the relative Frobenius morphism of toric varieties and the strong liftability of toric varieties to prove the Bott vanishing theorem, the degeneration of the Hodge to de Rham spectral sequence and the…
We give a new proof of vanishing result of Esnault for the cohomology of constructible sheaves in the tower of ``mock'' Frobenius covers of projective space. The key idea is to use (a global form of) the perversity of nearby cycles.
We extend SL(2)-orbit theorems for degeneration of mixed Hodge structures to a situation in which we do not assume the polarizability of graded quotients. We also obtain analogous results on Deligne systems.
Given a very ample line bundle on a smooth projective variety, the variation of Hodge structure associated to the universal family of hyperplane sections can be thought of as a $D$-module with action generated by the Gauss-Manin connection.…