Quasi-projective Moduli for Polarized Manifolds

Viehweg, Eckart

doi:10.1007/978-3-642-79745-3

Cited by 260 publications

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“…See [43]. Yau conjectured that Mumford-Chow stability of the canonically embedded projective manifolds is equivalent to the existence of Kähler Eimstein metric.…”

Section: Introductionmentioning

confidence: 99%

Shafarevich’s Conjecture for CY Manifolds I

Liu¹,

Todorov²,

Yau³

et al. 2005

Pure and Applied Mathematics Quarterly

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In this paper we first study the moduli spaces related to Calabi-Yau manifolds. We then apply the results to the following problem. Let C be a fixed Riemann surface with fixed finite number of points on it. Given a CY manifold with fixed topological type, we consider the set of all families of CY manifolds of the fixed topological type over C with degenerate fibres over the fixed points up to isomorphism. This set is called Shafarevich set. The analogue of Shafarevich conjecture for CY manifolds is for which topological types of CY the Shafarevich set is finite. It is well-known * This work is partially supported by the Institute of Mathematical Sciences, CUHK 1 that the analogue of Shafarevich conjecture is closely related to the study of the moduli space of polarized CY manifolds and the moduli space of the maps of fixed Riemann surface to the coarse moduli space of the CY manifolds. We prove the existence of the Teichmüller space of CY manifolds together with a universal family of marked CY manifolds. From this result we derive the existence of a finite cover of the coarse moduli space which is a non-singular quasi-projective manifold. Over this cover we construct a family of polarized CY manifolds. We study the moduli space of maps of the fixed Riemann with fixed points on it to the moduli space of CY manifolds constructed in the paper such that the maps map the fixed points on the Riemann surface to the discriminant locus. If this moduli space of maps is finite then Shafarevich conjecture holds. We relate the analogue of Shafarevich problem to the non-vanishing of the Yukawa coupling. We give also a counter example of the Shafarevich problem for a class of CY manifolds.

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“…See [43]. Yau conjectured that Mumford-Chow stability of the canonically embedded projective manifolds is equivalent to the existence of Kähler Eimstein metric.…”

Section: Introductionmentioning

confidence: 99%

Shafarevich’s Conjecture for CY Manifolds I

Liu¹,

Todorov²,

Yau³

et al. 2005

Pure and Applied Mathematics Quarterly

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“…Actually, [Vie95] considers thea priori weaker-condition: R [g] is an isomorphism for some g > 0 divisible by index(ω S ). One can see that in this case ( * ) g holds for every g divisible by index(ω S ), at least in characteristic 0; see [Kol13a].…”

Section: V-deformationsmentioning

confidence: 99%

The dualizing sheaf on first-order deformations of toric surface singularities

Altmann

Kollár

2016

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Abstract. We explicitly describe infinitesimal deformations of cyclic quotient singularities that satisfy one of the deformation conditions introduced by Wahl, Kollár-Shepherd-Barron and Viehweg. The conclusion is that in many cases these three notions are different from each other. In particular, we see that while the KSB and the Viehweg versions of the moduli space of surfaces of general type have the same underlying reduced subscheme, their infinitesimal structures are different.

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“…We refer the reader to Chapter 2 in [Vie2] for the definition and a detailed discussion of the notion of weak positivity, which will not be directly used in this paper.…”

Section: Positivity Of Direct Imagesmentioning

confidence: 99%

Rational connectedness modulo the non-nef locus

Broustet¹,

Pacienza²

2011

Comment. Math. Helv.

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Abstract. It is well known that a smooth projective Fano variety is rationally connected. Recently Zhang [Z2] (and later Hacon and McKernan [HM] as a special case of their work on the Shokurov RC-conjecture) proved that the same conclusion holds for a klt pair .X; / such that .K X C / is big and nef. We prove here a natural generalization of the above result by dropping the nefness assumption. Namely we show that a klt pair .X; / such that .K X C/ is big is rationally connected modulo the non-nef locus of .K X C /. This result is a consequence of a more general structure theorem for arbitrary pairs .X; / with .K X C / pseff. Mathematics Subject Classification (2010). 14J40.

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Quasi-projective Moduli for Polarized Manifolds

Cited by 260 publications

References 0 publications

Shafarevich’s Conjecture for CY Manifolds I

Shafarevich’s Conjecture for CY Manifolds I

The dualizing sheaf on first-order deformations of toric surface singularities

Rational connectedness modulo the non-nef locus

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