Base Spaces of Non-Isotrivial Families of Smooth Minimal Models

Viehweg, Eckart; Zuo, Kang

doi:10.1007/978-3-642-56202-0_16

Cited by 68 publications

(142 citation statements)

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“…We would also like to compare Condition 1 with some of the results obtained in [45]. In [47] some important applications of the Condition 1 are obtained.…”

Section: ) With a Proof Of The Mordell Conjecture As A Consequence"mentioning

confidence: 91%

“…Theorem 35 presumably also follows from the methods used in [45], and it seems to extend to non-isotrivial morphisms π : X C → C whose generic fibre has a semi-ample canonical sheaf. In fact, by [45], 6.2. b., it is sufficient to show that such a family is rigid. To prove this, assume that there exists a deformation π : X C×T → C × T with T a projective curve, which is smooth over C 0 ×T 0 .…”

Section: Lemma 36 Suppose That the Family (38) Satisfies The Conditiomentioning

confidence: 93%

“…Recently very important results of Viehweg and the last named author appeared in [44], [45] and [46]. Brody hyperbolicity was proved for the moduli space of canonically polarized complex manifolds.…”

Section: ) With a Proof Of The Mordell Conjecture As A Consequence"mentioning

confidence: 99%

“…Applying Cor. 6.5 in [45] to non-rigid families of CY 3-folds, the three times iterated Kodaira-Spencer map…”

Section: Theorem 48 Suppose Thatmentioning

confidence: 99%

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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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“…We would also like to compare Condition 1 with some of the results obtained in [45]. In [47] some important applications of the Condition 1 are obtained.…”

Section: ) With a Proof Of The Mordell Conjecture As A Consequence"mentioning

confidence: 91%

Section: Lemma 36 Suppose That the Family (38) Satisfies The Conditiomentioning

confidence: 93%

Section: ) With a Proof Of The Mordell Conjecture As A Consequence"mentioning

confidence: 99%

“…Applying Cor. 6.5 in [45] to non-rigid families of CY 3-folds, the three times iterated Kodaira-Spencer map…”

Section: Theorem 48 Suppose Thatmentioning

confidence: 99%

See 2 more Smart Citations

Shafarevich’s Conjecture for CY Manifolds I

Liu¹,

Todorov²,

Yau³

et al. 2005

Pure and Applied Mathematics Quarterly

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“…§13. Smooth Fibers that are Minimal of General Type and Singular Fibers Viehweg and Zuo [69], respectively Kovács [46] present different approaches to this case. Here we discuss the latter.…”

Section: §12 Logarithmic Vanishing Theoremsmentioning

confidence: 99%

Families of Varieties of General Type: the Shafarevich Conjecture and Related Problems

Kovács

2003

Bolyai Society Mathematical Studies

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Consider the curves on the (x, y)-plane parametrized by λ and given by the equation, y 2 = x 5 − 5λx + 4λ.An easy computation shows that these curves are non-singular for λ ∈ C \ {0, 1}. On the other hand the projective closure of each of these curves is singular at the point [0 : 1 : 0]. Blowing up this point and another one infinitely near this one yields a resolution, C λ . The collection of C λ forms a family of smooth projective curves over the base C \ {0, 1}.Exercise 0.1. Prove that the projection of C λ onto the y = 0 axis is the only degree 2 morphism ontoA family of curves is called isotrivial if any two general members are isomorphic. For example the blow up of the projective plane at a single point, considered as a P 1 -bundle over P 1 , is an isotrivial family since all of its members are isomorphic to the projective line. Since the genus of a curve in a family is constant, and there is only one curve of genus zero, one cannot expect much more in this case. However, for higher genus curves one can have non-isotrivial families as the above example shows. 134Sándor J. Kovács projective curves of a given genus over a given base curve. Furthermore, if there is such a family, then the base curve satisfies a certain hyperbolic condition. (For definitions and a more precise formulation, see Section §1). The conjecture was confirmed by Parshin [57] for the case of a compact base and by Arakelov [3] in general. It was recently generalized to families of higher dimensional varieties. This generalization is our main topic.It will be advantageous to work with a compactification of the family. Considering families over a compact base curve B naturally leads to a slightly different view on the problem. Instead of smooth families over a non-compact base, we are looking at arbitrary families over a compact base, and consider the locus over which the family is smooth. It is fairly obvious that these are equivalent situations. An arbitrary family over a compact base gives a smooth family over some open subset and a smooth family over an open curve can be extended to a (not necessarily smooth) family over the projective closure of the curve cf.[28], III.9.8.One could ask what can be said about the singular members of the family. On the simplest level, how many are there? In fact, Szpiro has asked this: What is the lower bound on the number of singular members if B P 1 ? This is the same as asking how close the base of the smooth family can be to being compact. Beauville [6] gave the following answer to Szpiro's question: there are always at least three singular members and there are families with exactly three. In fact, Beauville's proof also shows that there is at least one singular member if the base curve is elliptic. In short, 2g(B) − 2 + δ > 0, where g(B) is the genus of the (compact) base curve and δ is the number of singular members of the family. In other words, the base of a smooth family must be hyperbolic.Notice that the existence of Kodaira surfaces shows that there are families over high genus curves wi...

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Existence of Rational Curves on Algebraic Varieties, Minimal Rational Tangents, and Applications

Kebekus

Conde

Global Aspects of Complex Geometry

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Base Spaces of Non-Isotrivial Families of Smooth Minimal Models

Cited by 68 publications

References 15 publications

Shafarevich’s Conjecture for CY Manifolds I

Shafarevich’s Conjecture for CY Manifolds I

Families of Varieties of General Type: the Shafarevich Conjecture and Related Problems

Existence of Rational Curves on Algebraic Varieties, Minimal Rational Tangents, and Applications

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