Gauge Theory and Four-Manifold Topology

Donaldson, Simon

doi:10.1007/978-3-0348-9328-2_4

Cited by 1 publication

(2 citation statements)

References 29 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Another proof is based on the construction of a flat bundle E corresponding to a homomorphism Z * K → {±1} with some cohomology group H i (X, E) of odd dimension, contradicting the extension of Hodge theory to flat rank 1 bundles (see [181] Here, the ends of a non compact topological space X are just the limit, as the compact subset K gets larger, of the connected components of the complement set X \K . It is interesting to observe that the previous result was extended by Donaldson [138] also to the case where X 1 , X 2 are simply connected smooth projective surfaces. One could believe that the combination of the two results implies tout court that the connected sum X 1 X 2 of two projective manifolds of dimension at least two cannot be homeomorphic to a projective manifold.…”

Section: (T W(m)) ∼ = π 1 (M)mentioning

confidence: 79%

See 1 more Smart Citation

Topological methods in moduli theory

Catanese

2015

Bull. Math. Sci.

View full text Add to dashboard Cite

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.

show abstract

Section: (T W(m)) ∼ = π 1 (M)mentioning

confidence: 79%

“…Friedman and Morgan instead made the 'speculation' that the answer to the DEF = DIFF question should be positive (1987) (see [168]), motivated by the new examples of homeomorphic but not diffeomorphic surfaces discovered by Donaldson (see [138] and [139] for a survey on this topic).…”

Section: Connected Components Of Gieseker's Moduli Spacementioning

confidence: 99%

Topological methods in moduli theory

Catanese

2015

Bull. Math. Sci.

View full text Add to dashboard Cite

show abstract

Gauge Theory and Four-Manifold Topology

Cited by 1 publication

References 29 publications

Topological methods in moduli theory

Topological methods in moduli theory

Contact Info

Product

Resources

About