Hodge Theory and Complex Algebraic Geometry I

Abstract

The first of two volumes offering a modern introduction to Kaehlerian geometry and Hodge structure. The book starts with basic material on complex variables, complex manifolds, holomorphic vector bundles, sheaves and cohomology theory, the latter being treated in a more theoretical way than is usual in geometry. The author then proves the Kaehler identities, which leads to the hard Lefschetz theorem and the Hodge index theorem. The book culminates with the Hodge decomposition theorem. The meanings of these results are investigated in several directions. Completely self-contained, the book is ideal for students, while its content gives an account of Hodge theory and complex algebraic geometry as has been developed by P. Griffiths and his school, by P. Deligne, and by S. Bloch. The text is complemented by exercises which provide useful results in complex algebraic geometry.

Geometry and arithmetic of Maschke’s Calabi–Yau three-fold

“…We briefly recall the basics of rational Hodge structures (see for example [30], Chapter 7), representations of finite groups and their applications to algebraic geometry.…”

“…We refer to [30], chapter 12, for this section. We denote by L the total space of the family of lines in X parametrized by C + , it is a surface in the product C + × X.…”

Towards open string mirror symmetry for one-parameter Calabi-Yau hypersurfaces

“…In fact, the normal function should be viewed as a holomorphic section of the Griffiths intermediate Jacobian fibration over the moduli space of complex structures of the family Y . For more details about the mathematical properties of ν C , see [31][32][33]. The main property of interest to us is that the normal function satisfies an inhomogeneous version of the Picard-Fuchs equations [27]:…”

“…Similar to T B,α , Δ ij,α is a quantity from Hodge theory, the Griffiths infinitesimal invariant [34] of the normal function ν Cα (see also [31][32][33]). In the holomorphic limit, they are related by 17) where D i is the covariant derivative of special geometry.…”

Comparison of dynamical degrees for semi-conjugate meromorphic maps

“…In this section, we prove some useful results which can be applied to currents of integration on varieties and may have independent interest. The reader will find in Demailly [4] and Voisin [19] the basic facts on currents and on Kähler geometry. i.…”