Compact moduli for certain Kodaira fibrations

Rollenske, Soenke

doi:10.2422/2036-2145.2010.4.08

Cited by 8 publications

(5 citation statements)

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“…Using these explicit descriptions Rollenske went further in [Rol10] and showed that, in the case where the branched cover has a cyclic Galois group, then the closure of this irreducible component inside the Kollár-Shepherd Barron-Alexeev [K-SB88] compactification is again a connected component. One should observe that there are extremely few examples where the connected components of the KSBA compact moduli space have been investigated, apart from the obvious case of rigid surfaces (see [LW12] for the case of surfaces isogenous to a product).…”

Section: And Siu [M-s80]mentioning

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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Section: And Siu [M-s80]mentioning

confidence: 99%

Kodaira fibrations and beyond: methods for moduli theory

Catanese¹

2017

Jpn. J. Math.

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“…An explicit study of compactifications of the moduli spaces of surfaces of general type was pursued in [VanOp06], [Al-Par09], [LW10], [Rol10].…”

Section: Smoothings and Surgeriesmentioning

confidence: 99%

A Superficial Working Guide to Deformations and Moduli

Catanese¹

2011

Preprint

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ContentsIntroduction 1. Analytic moduli spaces and local moduli spaces: Teichmüller and Kuranishi space 1.1. Teichmüller space 1.2. Kuranishi space 1.3. Deformation theory and how it is used 1.4. Kuranishi and Teichmüller 2. The role of singularities 2.1. Deformation of singularities and singular spaces 2.2. Atiyah's example and three of its implications 3. Moduli spaces for surfaces of general type 3.1. Canonical models of surfaces of general type. 3.2. The Gieseker moduli space 3.3. Minimal models versus canonical models 3.4. Number of moduli done right 3.5. The moduli space for minimal models of surfaces of general type 3.6. Singularities of moduli spaces 4. Automorphisms and moduli 4.1. Automorphisms and canonical models 4.2. Kuranishi subspaces for automorphisms of a fixed type 4.3. Deformations of automorphisms differ for canonical and for minimal models 4.4. Teichmüller space for surfaces of general type 5. Connected components of moduli spaces and arithmetic of moduli spaces for surfaces 5.1. Gieseker's moduli space and the analytic moduli spaces 5.2. Arithmetic of moduli spaces 5.3. Topology sometimes determines connected components 1 I owe to David Buchsbaum the joke that an expert on algebraic surfaces is a 'superficial' mathematician.The present work took place in the realm of the DFG Forschergruppe 790 "Classification of algebraic surfaces and compact complex manifolds". A major part of the article was written, and the article was completed, when the author was a visiting research scholar at KIAS. .

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“…Although M h shares many nice properties of M g , e.g., it is a DM-stack of finite type over the base field, it may possibly have many connected components that behave very differently [Cat86,Vak06]. Hence, almost all available results on the global geometry of M h pertain to specific components of the moduli of surfaces (e.g., [vO05,vO06,Liu12,Rol10,AP09,Lee00]) or special components of the moduli of log-stable varieties (e.g., [HKT06,HKT09,Ale02,Has99,Hac04]).…”

Section: Introductionmentioning

confidence: 99%

Moduli of products of stable varieties

Bhatt

Ho²,

Patakfalvi³

et al. 2013

Compositio Math.

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ABSTRACT. We study the moduli space of a product of stable varieties over the field of complex numbers, as defined via the minimal model program. Our main results are: (a) taking products gives a well-defined morphism from the product of moduli spaces of stable varieties to the moduli space of a product of stable varieties, (b) this map is always finiteétale, and (c) this map very often is an isomorphism. Our results generalize and complete the work of Van Opstall in dimension 1. The local results rely on a study of the cotangent complex using some derived algebro-geometric methods, while the global ones use some differential-geometric input.

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Compact moduli for certain Kodaira fibrations

Abstract: We explicitly describe the possible degenerations of a class of double Kodaira fibrations in the moduli space of stable surfaces. Using deformation theory we also show that under some assumptions we get a connected component of the moduli space of stable surfaces.

Cited by 8 publications

References 11 publications

Kodaira fibrations and beyond: methods for moduli theory

Kodaira fibrations and beyond: methods for moduli theory

A Superficial Working Guide to Deformations and Moduli

Moduli of products of stable varieties

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