Search results for "Hodge theory"

Hodge Numbers for the Cohomology of Calabi-Yau Type Local Systems

2014

We determine the Hodge numbers of the cohomology group \(H_{L^{2}}^{1}(S, \mathbb{V}) = H^{1}(\bar{S},j_{{\ast}}\mathbb{V})\) using Higgs cohomology, where the local system \(\mathbb{V}\) is induced by a family of Calabi-Yau threefolds over a smooth, quasi-projective curve S. This generalizes previous work to the case of quasi-unipotent, but not necessarily unipotent, local monodromies at infinity. We give applications to Rohde’s families of Calabi-Yau 3-folds.

Deuring’s mass formula of a Mumford family

2015

We study the Newton polygon jumping locus of a Mumford family in char p p . Our main result says that, under a mild assumption on p p , the jumping locus consists of only supersingular points and its cardinality is equal to ( p r − 1 ) ( g − 1 ) (p^r-1)(g-1) , where r r is the degree of the defining field of the base curve of a Mumford family in char p p and g g is the genus of the curve. The underlying technique is the p p -adic Hodge theory.

The cohomology of a variation of polarized Hodge structures over a quasi-compact Kähler manifold

2007

In this article, we consider the cohomologies with coefficients in a variation of polarized Hodge structures on a quasi-compact Kaehler manifold. We show that the L 2 L^2 -Dolbeault cohomology can be identified with the L 2 L^2 cohomology; we also give several direct applications of the result above.

On Hodge theory for the generalized geometry (I)

2013

Abstract We first investigate the linear Dirac structure from the viewpoint of a mixed Hodge structure. Then we discuss a Hodge-decomposition-type theorem for the generalized Kahler manifold and study the moduli space of a generalized weak Calabi–Yau manifold. We present a holomorphic anomaly equation and a one-loop partition function in a topological B-model under the generalized geometric context.

Projective Crystalline Representations of \'Etale Fundamental Groups and Twisted Periodic Higgs-de Rham Flow

2017

This paper contains three new results. {\bf 1}.We introduce new notions of projective crystalline representations and twisted periodic Higgs-de Rham flows. These new notions generalize crystalline representations of \'etale fundamental groups introduced in [7,10] and periodic Higgs-de Rham flows introduced in [19]. We establish an equivalence between the categories of projective crystalline representations and twisted periodic Higgs-de Rham flows via the category of twisted Fontaine-Faltings module which is also introduced in this paper. {\bf 2.}We study the base change of these objects over very ramified valuation rings and show that a stable periodic Higgs bundle gives rise to a geometric…

Nonabelian Hodge theory in positive characteristic via exponential twisting

2015

An Introduction to Hodge Structures

2015

We begin by introducing the concept of a Hodge structure and give some of its basic properties, including the Hodge and Lefschetz decompositions. We then define the period map, which relates families of Kahler manifolds to the families of Hodge structures defined on their cohomology, and discuss its properties. This will lead us to the more general definition of a variation of Hodge structure and the Gauss-Manin connection. We then review the basics about mixed Hodge structures with a view towards degenerations of Hodge structures; including the canonical extension of a vector bundle with connection, Schmid’s limiting mixed Hodge structure and Steenbrink’s work in the geometric setting. Fin…

Hodge Theory and Algebraic Cycles

2006

Algebraic cycles and Hodge theory, in particular Chow groups, Deligne cohomology and the study of cycle class maps were some of the themes of the Schwerpunkt ”Globale Methoden in der Komplexen Geometrie”. In this survey we report about several projects around the structure of (higher) Chow groups CH(X,n) [3] which the author has studied with his coauthors during this time by using different methods. In my opinion there are two interesting view points: first the internal structure of higher Chow groups, i.e., the existence of interesting elements and nontriviality of parts of their Bloch-Beilinson filtrations. This case has arithmetic and geometric features, and the groups in question show d…

Holomorphic de Rham Cohomology

2017

We are going to define a natural comparison isomorphism between algebraic de Rham cohomology and singular cohomology of varieties over the complex numbers with coefficients in \(\mathbb {C}\). The link is provided by holomorphic de Rham cohomology, which we study in this chapter.

A note on the characteristic $p$ nonabelian Hodge theory in the geometric case

2012

We provide a construction of associating a de Rham subbundle to a Higgs subbundle in characteristic $p$ in the geometric case. As applications, we obtain a Higgs semistability result and a $W_2$-unliftable result.