Search results for "Hodge conjecture"
showing 6 items of 6 documents
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.
Formal Periods and the Period Conjecture
2017
Following Kontsevich (see Kontsevich in Operads and motives in deformation quantization. Lett. Math. Phys. 48(1):35–72, 1999), we now introduce another algebra \(\tilde{\mathbb {P}}(k)\) of formal periods from the same data we have used in order to define the actual period algebra of a field in Chap. 11. The main aim of this chapter is to give conceptual interpretation of this algebra of formal periods. We then use it to formulate and discuss the period conjecture.
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.
Algebraic de Rham Cohomology
2017
Let k be a field of characteristic zero. We are going to define relative algebraic de Rham cohomology for general varieties over k, not necessarily smooth.
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…