Variations of Mixed Hodge Structure and Semipositivity Theorems

“…Kawamata also extended his result to the case where the dimension of the base variety B is > 1 in [Kaw81], giving later a simpler proof of semipositivity in [Kaw02]. There has been a lot of literature on the subject ever since, (see the references we cited in [CatDet13], see [E-V90] for the ampleness of W m when the fibration is not birationally isotrivial, and see [FF14] and [FFS14]).…”

Section: 22mentioning

confidence: 99%

Kodaira fibrations and beyond: methods for moduli theory

Catanese¹

2017

Jpn. J. Math.

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Abstract. Kodaira fibred surfaces are a remarkable example 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.

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“…proof of [50,Lemma 14.61], see also [27] for the classical approach). In particular, it follows from Remark 3.4 that the refined limit Hodge-Deligne polynomial is a motivic invariant over K. That is, we may consider the refined Hodge-Deligne map…”

Section: And M(r)mentioning

confidence: 99%

Tropical geometry, the motivic nearby fiber, and limit mixed Hodge numbers of hypersurfaces

Katz

Stapledon

2016

Res Math Sci

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A complex variety, degenerating over a punctured disk, carries a limit mixed Hodge structure on its cohomology, encoding the action of monodromy. The associated limit mixed Hodge-Deligne polynomial can be expressed in terms of the motivic nearby fiber. Using techniques from tropical geometry, we present a new formula for the motivic nearby fiber and concentrate on the case of degenerating families of complex hypersurfaces, generalizing work of Danilov and Khovanskiȋ and Batyrev and Borisov. If these families satisfy a natural smoothness condition, called schönness, their limit mixed Hodge-Deligne numbers can be expressed in terms of new, combinatorial invariants of a polyhedral subdivision of the associated Newton polytope. These invariants are multivariable, Ehrhart-theoretic extensions of Stanley's invariants of subdivisions and are situated in his theory in a companion combinatorial paper.

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Variations of Mixed Hodge Structure and Semipositivity Theorems

Cited by 50 publications

References 59 publications

Weak positivity for Hodge modules

Weak positivity for Hodge modules

Kodaira fibrations and beyond: methods for moduli theory

Tropical geometry, the motivic nearby fiber, and limit mixed Hodge numbers of hypersurfaces

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