By Herbert Clemens, Janos Kollár

ISBN-10: 0521562449

ISBN-13: 9780521562447

This quantity collects a chain of survey articles on advanced algebraic geometry, which within the early Nineties was once present process an important switch. Algebraic geometry has spread out to principles and connections from different fields that experience routinely been distant. This publication offers a good suggestion of the highbrow content material of the swap of path and branching out witnessed through algebraic geometry long ago few years.

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**Extra info for Current Topics in Complex Algebraic Geometry**

**Sample text**

This gives a bijection of the set of triples (E, ρ, σ) (up to isomorphism) onto SLr C((z)) . To get rid of the the trivializations, we have to mod out by the automorphism group of the trivial bundle over D and X p. We get the following diagram: {E, ρ, σ} ←→ SLr C((z)) {E, ρ} ←→ Q := SLr C((z)) / SLr (C[[z]]) π {E} ←→ SLr (AX )\ SLr C((z)) / SLr (C[[z]]) . Of course I have only constructed a bijection between the set of isomorphism classes of vector bundles on X with trivial determinant and the set of double classes SLr (AX )\ SLr C((z)) / SLr (C[[z]]); with some technical work one shows that the construction actually gives an isomorphism of stacks.

In preparing the manuscript, I received considerable help from J. Koll´ar, J. McKernan and S. Mori. The responsibility for mistakes is of course entirely mine. 2. Notation, Minimal Models, etc. The aim of this section is to introduce the basic notation and terminology to be used extensively in the rest of the paper, and to give a quick reminder of minimal model theory, including the logarithmic theory. Unless otherwise explicitly declared, we shall work with complex projective normal varieties.

If T has positive or mixed characteristic, assume moreover that X is Cohen–Macaulay (this is automatic in characteristic 0). Let x ∈ S be a point, r the index of KX at x. 2) O = O[[x, y, z]]Zr /(xy − τ ), where Zr acts with weights (a, −a, 1) for some (a, r) = 1. 3) r > 1 and O = O[[x, y, z]]Zr /(xy − F (z r )), where Zr acts with weights (a, −a, 1) for some (a, r) = 1, or ˆX,x ∼ r = 1 and O = O[[x, y, z]]/(G(x, y, z)). Let F ∈ k[z] (resp. G ∈ k[[x, y, z]]) be the reduction of F (resp. G) mod τ .

### Current Topics in Complex Algebraic Geometry by Herbert Clemens, Janos Kollár

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