Geometry of Kodaira moduli spaces

A general theorem on the existence of natural torsion-free affine connections on a complete family of compact complex submanifolds in a complex manifold is proved.

     

Kodaira dimension of irregular varieties

Let f:X→Y be an algebraic fiber space with general fiber F. If Y is of maximal Albanese dimension, we show that κ(X)≥κ(Y)+κ(F).     

Kodaira dimension of symmetric powers

We compute the plurigenera and the Kodaira dimension of the th symmetric power of a smooth projective variety . As an application we obtain genus estimates for the curves lying on .

     

Higher-dimensional Analogues of Inoue and Kodaira Surfaces

In this paper, by considering the Cauchy Problem for the KdV hierarchy in certain weighted Sobolev spaces, the existence of rapidly decreasing solutions is established.     

On the Kodaira dimension of minimal threefolds

Dedicated to Professor M. Nagata on his sixtieth birthday     

Kodaira dimension of holomorphic singular foliations

We introduce numerical invariants of holomorphic singular foliations under bimeromorphic transformations of surfaces. The basic invariant is a foliated version of the Kodaira dimension of compact complex manifolds.The author was supported by CNPq-Brazil in 1998 and Conseil Régional de Bourgogne in 1999.     

Kodaira vanishing theorems on non-complete algebraic manifolds

Dedicated to Hans Grauert on the occasion of his 60th birthday     

Wild automorphisms of varieties with Kodaira dimension 0

An automorphism σ of a projective variety X is said to be wild if σ(Y) ≠ Y for every non-empty subvariety Y \subsetneq X{Y \subsetneq X} . In [1] Z. Reichstein, D. Rogalski, and J.J. Zhang conjectured that if X is an irreducible projective variety admitting a wild automorphism then X is an abelian variety, and proved this conjecture for dim(X) ≤ 2. As a step toward answering this conjecture in higher dimensions we prove a structure theorem for projective varieties of Kodaira dimension 0 admitting wild automorphisms. This essentially reduces the Kodaira dimension 0 case to a study of Calabi-Yau varieties, which we also investigate. In support of this conjecture, we show that there are no wild automorphisms of certain Calabi-Yau varieties.     

Elementary counterexamples to Kodaira vanishing in prime characteristic

Using methods from the modular representation theory of algebraic groups one can construct [1] a projective homogeneous space forSL ₄, in prime characteristic, which violates Kodaira vanishing. In this note we show how elementary algebraic geometry can be used to simplify and generalize this example.     

Kodaira Type Vanishing Theorem for the Hirokado Variety

The Hirokado variety is a Calabi–Yau threefold in characteristic 3 that is not liftable either to characteristic 0 or the ring W ₂ of the second Witt vectors. Although Deligne–Illusie–Raynaud type Kodaira vanishing cannot be applied, we show that H ¹(X, L ^?1) = 0, for an ample line bundle such that L ³ has a non-trivial global section, holds for this variety.     

Kodaira fibrations and beyond: methods for moduli theory

Kodaira fibred surfaces are remarkable examples of projective classifying spaces, and there are still many intriguing open questions concerning them, especially the slope question. The topological characterization of Kodaira fibrations is emblematic of the use of topological methods in the study of moduli spaces of surfaces and higher dimensional complex algebraic varieties, and their compactifications. Our tour through algebraic surfaces and their moduli (with results valid also for higher dimensional varieties) deals with fibrations, questions on monodromy and factorizations in the mapping class group, old and new results on Variation of Hodge Structures, especially a recent answer given (in joint work with Dettweiler) to a long standing question posed by Fujita. In the landscape of our tour, Galois coverings, deformations and rigid manifolds (there are by the way rigid Kodaira fibrations), projective classifying spaces, the action of the absolute Galois group on moduli spaces, stand also in the forefront. These questions lead to interesting algebraic surfaces, for instance remarkable surfaces constructed from VHS, surfaces isogenous to a product with automorphisms acting trivially on cohomology, hypersurfaces in Bagnera-de Franchis varieties, Inoue-type surfaces.     

On the pluricanonical maps of varieties of intermediate Kodaira dimension

In this paper we will prove a uniformity result for the Iitaka fibration $f:X\rightarrow Y$ , provided that the generic fiber has a good minimal model and the Prokhorov–Shokurov conjecture holds. In particular, the result holds if the variation of $f$ is zero or $\kappa (X)=\text{ dim}X-1$ .