Download Algebraic surfaces and holomorphic vector bundles by Robert Friedman PDF

By Robert Friedman

A unique function of the e-book is its built-in method of algebraic floor thought and the research of vector package conception on either curves and surfaces. whereas the 2 topics stay separate throughout the first few chapters, they develop into even more tightly interconnected because the ebook progresses. therefore vector bundles over curves are studied to appreciate governed surfaces, after which reappear within the evidence of Bogomolov's inequality for strong bundles, that is itself utilized to check canonical embeddings of surfaces through Reider's process. equally, governed and elliptic surfaces are mentioned intimately, prior to the geometry of vector bundles over such surfaces is analysed. a number of the effects on vector bundles look for the 1st time in booklet shape, subsidized by way of many examples, either one of surfaces and vector bundles, and over a hundred routines forming a vital part of the textual content. aimed toward graduates with a radical first-year direction in algebraic geometry, in addition to extra complex scholars and researchers within the components of algebraic geometry, gauge idea, or 4-manifold topology, some of the effects on vector bundles may also be of curiosity to physicists learning string thought.

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Complex Surfaces of General Type: Some Recent Progress 47 Fig. 5. The curve C with a dihedral symmetry We recall that the Coxeter elements associated to a chain are products of the type ∆ = (Tα1 )(Tα2 Tα1 ) . . (Tαn Tαn−1 . . Tα1 ) of Dehn twists associated to the curves of the chain. In order to finally prove that such product (let us call it Ψ ) of Coxeter elements and Ψ are isotopic, one observes that if one removes the above cited chains of circles from the curve C, one obtains 4 connected components which are diffeomorphic to circles.

The main question on this moduli space remains the following: Question 1. Is M6,4,0 connected? Here is a partial picture: 39 37 IVa1 Ia {{ xx { x { x {{ xx |xx  }{{ IVa2 IVb1 36  IVb2 38 IIIa { { {{ {{ }{{  V1 IIIb o || || | |  }|| V2 II  Ib This picture is partial because up to now it is not known whether all possible arrows are drawn. More precisely, M6,4,0 is connected if and only if one of the two following degenerations is possible: Ia → V1 or Ia → V2 . 26 Ingrid C. Bauer, Fabrizio Catanese, and Roberto Pignatelli This picture was done by Horikawa in [Hor78] with the exception of the horizontal line IIIb ← II, recently obtained in [BCP04].

4) If Γ does not act freely, taking the tangent representation at a fixed point, we see by 3) that A contains a cyclotomic extension whose degree divides 4. 5) Using Hasse’s theorem, one chooses A such that the set of primes where it ramifies contains, one for each possible intermediate field K between the quadratic field k of A and one of the finitely many possible cyclotomic extensions above, a prime P such that K ⊗ kP is not an integral domain: this however contradicts 4) hence shows the desired assertion.

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