Diffeomorphism of simply connected algebraic surfaces

Catanese, Fabrizio; Wajnryb, Bronislaw

doi:10.4310/jdg/1180135677

Cited by 19 publications

(53 citation statements)

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“…I conjectured (in [232]) that the answer should be negative, on the basis of some families of simply connected surfaces of general type constructed in [82] and later investigated in [84], [104] and [111]: these were shown to be homeomorphic by the results of Freedman (see [166,167]), and it was then relatively easy to show then [85] that there were many connected components of the moduli space corresponding to homeomorphic but non diffeomorphic surfaces. It looked like the situation should be similar even if one would fix the diffeomorphism type.…”

Section: Connected Components Of Gieseker's Moduli Spacementioning

confidence: 99%

“…In my joint work with Wajnryb [104] the DEF = DIFF question was shown to have a negative answer also for simply connected surfaces (indeed for some of the families of surfaces constructed in [82]). I refer to [106] for a rather comprehensive treatment of the above questions (and to [3,17,100,107,111] for the symplectic point of view, [37,137] for the special case of geometric genus p g = 0).…”

Section: Connected Components Of Gieseker's Moduli Spacementioning

confidence: 99%

“…The question was finally answered in the negative, and in every possible way ( [26,98,104,239,278]). The Friedman-Morgan speculation does not hold true and the DEF = DIFF question has a negative answer.…”

Section: Connected Components Of Gieseker's Moduli Spacementioning

confidence: 99%

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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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Section: Connected Components Of Gieseker's Moduli Spacementioning

confidence: 99%

Section: Connected Components Of Gieseker's Moduli Spacementioning

confidence: 99%

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Topological methods in moduli theory

Catanese

2015

Bull. Math. Sci.

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“…[Man01]) appeared, there were further counterexamples given by Catanese, Kharlamov-Kulikov, Catanese-Wajnryb, Bauer-Catanese-Grunewald (cf. [Cat03], [K-K02], [BCG05a], [CW04]). The counterexamples are of quite different nature: Manetti used (Z/2) rcovers of P 1 C × P 1 C , his surfaces have b 1 = 0, but are not 1-connected.…”

Section: Remark 74 1) It Is Nowadays Wellknown That Smooth Compact mentioning

confidence: 99%

Chebycheff and Belyi Polynomials, Dessins d’Enfants, Beauville Surfaces and Group Theory

Bauer

Catanese

Grunewald

2006

MedJM

Self Cite

161

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Abstract. We start discussing the group of automorphisms of the field of complex numbers, and describe, in the special case of polynomials with only two critical values, Grothendieck's program of 'Dessins d' enfants', aiming at giving representations of the absolute Galois group. We describe Chebycheff and Belyi polynomials, and other explicit examples. As an illustration, we briefly treat difference and Schur polynomials. Then we concentrate on a higher dimensional analogue of the triangle curves, namely, Beauville surfaces and varieties isogenous to a product. We describe their moduli spaces, and show how the study of these varieties leads to new interesting questions in the theory of finite (simple) groups.

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“…That is, if two surfaces have different fundamental groups, then they are not deformation equivalent. For surfaces X, Y denote ⋍ Y (and again -the inverse directions are not correct; see [5], [11]). …”

Section: Introductionmentioning

confidence: 99%

On fundamental groups related to the Hirzebruch surface F 1

Friedman

Teicher

2008

Sci. China Ser. A-Math.

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Abstract. Given a projective surface and a generic projection to the plane, the braid monodromy factorization (and thus, the braid monodromy type) of the complement of its branch curve is one of the most important topological invariants ([10]), stable on deformations. From this factorization, one can compute the fundamental group of the complement of the branch curve, either in C 2 or in CP 2 . In this article, we show that these groups, for the Hirzebruch surface F 1,(a,b) , are almost-solvable. That is -they are an extension of a solvable group, which strengthen the conjecture on degeneratable surfaces (see [13]).

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Diffeomorphism of simply connected algebraic surfaces

Cited by 19 publications

References 1 publication

Topological methods in moduli theory

Topological methods in moduli theory

Chebycheff and Belyi Polynomials, Dessins d’Enfants, Beauville Surfaces and Group Theory

On fundamental groups related to the Hirzebruch surface F 1

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