## Differential Topology of Complex Surfaces: Elliptic Surfaces with Pg = 1: Smooth ClassificationThis book is about the smooth classification of a certain class of algebraicsurfaces, namely regular elliptic surfaces of geometric genus one, i.e. elliptic surfaces with b1 = 0 and b2+ = 3. The authors give a complete classification of these surfaces up to diffeomorphism. They achieve this result by partially computing one of Donalson's polynomial invariants. The computation is carried out using techniques from algebraic geometry. In these computations both thebasic facts about the Donaldson invariants and the relationship of the moduli space of ASD connections with the moduli space of stable bundles are assumed known. Some familiarity with the basic facts of the theory of moduliof sheaves and bundles on a surface is also assumed. This work gives a good and fairly comprehensive indication of how the methods of algebraic geometry can be used to compute Donaldson invariants. |

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### Contents

Certain moduli spaces for bundles on elliptic surfaces with P 1 57 | 1 |

2 | 13 |

Identification of 83rS H with 73S | 33 |

Copyright | |

6 other sections not shown

### Other editions - View all

Differential Topology of Complex Surfaces John W. Morgan,Kieran G. O'Grady No preview available - 2014 |

### Common terms and phrases

apply associated assume At(V canonical Claim Clearly completes computation conclude connected consider construction contained Corollary corresponding define definition denote determined diffeomorphism dimension divisor easily easy elementary modification elliptic surfaces equal Equation equivalent establish exact sequence exists extension fact fiber fits follows formula Furthermore Gieseker given gives Hence Hilb3(S holds identified immediate implies induced integer intersection invariant irreducible component isomorphism Item Lemma line bundle locally free Mc(S moduli space morphism multiple multiple fibers natural notation Notice open dense subset open subset pair parametrized polarization polynomials positive projective Proof properties Proposition prove rank-two reduced relative represented respectively restriction result satisfies scheme semistable sheaf sheaves slope stable smooth stratification stratum subscheme subset subsubsection Suppose surjective term Theorem torsion-free trivial universal vector bundle zero