Boundedness of pluricanonical maps of varieties of general type

Hacon, Christopher D.; McKernan, James

doi:10.1007/s00222-006-0504-1

Cited by 146 publications

(203 citation statements)

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“…[Zh06], [HM07]). The extension theorem ( [HM06], 3.17) lies at the heart of the proof. We first recall the following terminology.…”

Section: Degenerations Of Fano Varietiesmentioning

confidence: 93%

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Degenerations of rationally connected varieties

Hogadi

2009

Trans. Amer. Math. Soc.

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“…[Zh06], [HM07]). The extension theorem ( [HM06], 3.17) lies at the heart of the proof. We first recall the following terminology.…”

Section: Degenerations Of Fano Varietiesmentioning

confidence: 93%

“…The proof of Theorem 3.1 relies on the following extension theorem proved in [HM06]. The statement given below is less general than the one in [HM06].…”

Section: Degenerations Of Fano Varietiesmentioning

confidence: 99%

Degenerations of rationally connected varieties

Hogadi

2009

Trans. Amer. Math. Soc.

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“…Results in the style of Bombieri have been recently obtained by Hacon and Mckernan [205], and from these one obtains (non explicit) boundedness results for the varieties of general type with fixed invariants K n X , χ(X ) .…”

Section: Varieties With Singularitiesmentioning

confidence: 94%

Topological methods in moduli theory

Catanese

2015

Bull. Math. Sci.

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One of the main themes of this long article is the study of projective varieties which are K(H,1)'s, i.e. classifying spaces BH for some discrete group H. After recalling the basic properties of such classifying spaces, an important class of such varieties is introduced, the one of Bagnera-de Franchis varieties, the quotients of an Abelian variety by the free action of a cyclic group. Moduli spaces of Abelian varieties and of algebraic curves enter into the picture as examples of rational K(H,1)'s, through Teichmüller theory. The main trhust of the paper is to show how in the case of K(H,1)'s the study of moduli spaces and deformation classes can be achieved through by now classical results concerning regularity of classifying maps. The Inoue type varieties of Bauer and Catanese are introduced and studied as a key example, and new results are shown. Motivated from this study, the moduli spaces of algebraic varieties, and especially of algebraic curves with a group of automorphisms of a given topological type are studied in detail, following new results by the author, Michael Lönne and Fabio Perroni. Finally, the action of the absolute Galois group on the moduli spaces of such K(H,1) varieties is studied. In the case of surfaces isogenous to a product, it is shown how this yields a faifhtul action on the set of connected components of the moduli space: for each Galois automorphism of order different from 2 there is an Communicated by Efim Zelmanov.The present work took place in the realm of the DFG Forschergruppe 790 "Classification of algebraic surfaces and compact complex manifolds". Part of the article was written when the author was visiting KIAS, Seoul, as KIAS research scholar.

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“…On a smooth minimal projective surface S of general type, Bombieri [2] proved that the m-canonical map is a birational morphism for all m ≥ 5. The work of Tsuji [31], Hacon-M c Kernan [13] and Takayama [28] says that there exists a universal constant r 3 such that the r 3 -canonical map Φ r3 is birational for all smooth projective 3-folds of general type. We note that Tsuji [31] announced a very large r 3 .…”

Section: Introductionmentioning

confidence: 99%