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Register a new userOn the decomposition theorem for intersection de Rham complexes
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We establish a positive characteristic analogue of intersection cohomology for polarized variations of Hodge structure. This includes: a) the decomposition theorem for the intersection de Rham complex; b) the $E_1$-degeneration theorem for the intersection de Rham complex of a periodic de Rham bundle: c) the Kodaira vanishing theorem for the intersection cohomology groups of a periodic Higgs bundle.
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We provide a new formalism of de Rham--Witt complexes in the logarithmic setting. This construction generalizes a result of Bhatt--Lurie--Mathew, and agrees with those of Hyodo--Kato and Matsuue for log-smooth schemes of log-Cartier type. We then apply our formalism to obtain a more direct proof of the log crystalline comparison of A_inf-cohomology in the case of semistable reduction, which is established by Cesnavicius--Koshiwara.
We will prove a Kodaira-Nakano type of vanishing theorem for the logarithmic de Rham complex of unitary local system. We will then study the weight filtration on the logarithmic de Rham complex, and prove a stronger statement for the associated graded complex.
Let X be a complex analytic manifold and D subset X a free divisor. Integrable logarithmic connections along D can be seen as locally free {cal O}_X-modules endowed with a (left) module structure over the ring of logarithmic differential operators {cal D}_X(log D). In this paper we study two related results: the relationship between the duals of any integrable logarithmic connection over the base rings {cal D}_X and {cal D}_X(log D), and a differential criterion for the logarithmic comparison theorem. We also generalize a formula of Esnault-Viehweg in the normal crossing case for the Verdier dual of a logarithmic de Rham complex.
In this article, we prove that there is a canonical Verdier self-dual intersection space sheaf complex for the middle perversity on Witt spaces that admit compatible trivializations for their link bundles, for example toric varieties. If the space is an algebraic variety our construction takes place in the category of mixed Hodge modules. We obtain an intersection space cohomology theory, satisfying Poincare duality, valid for a class of pseudomanifolds with arbitrary depth stratifications. The main new ingredient is the category of Kunneth complexes; these are cohomologically constructible complexes with respect to a fixed stratification, together with additional data, which codifies triviality structures along the strata. In analogy to what Goreski and McPherson showed for intersection homology complexes, we prove that there are unique Kunneth complexes that satisfy the axioms for intersection space complexes introduced by the first and third author. This uniqueness implies the duality statements in the same scheme as in Goreski and McPherson theory.
Over any smooth algebraic variety over a $p$-adic local field $k$, we construct the de Rham comparison isomorphisms for the etale cohomology with partial compact support of de Rham $mathbb Z_p$-local systems, and show that they are compatible with Poincare duality and with the canonical morphisms among such cohomology. We deduce these results from their analogues for rigid analytic varieties that are Zariski open in some proper smooth rigid analytic varieties over $k$. In particular, we prove finiteness of etale cohomology with partial compact support of any $mathbb Z_p$-local systems, and establish the Poincare duality for such cohomology after inverting $p$.
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