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

By Herbert Clemens, Janos Kollár

This quantity collects a chain of survey articles on advanced algebraic geometry, which within the early Nineteen Nineties was once present process a big switch. Algebraic geometry has unfolded to principles and connections from different fields that experience normally been far-off. This ebook provides a good suggestion of the highbrow content material of the switch of path and branching out witnessed by way of algebraic geometry some time past few years.

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Extra resources for Current Topics in Complex Algebraic Geometry

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L3 ≥ 1000), the same conclusion holds with 2 L · S ≥ 5.  The proof of the above result is quite complicated. The Kawamata technique is still in a stage where it is rather difficult to use it to prove things, even when one has a good idea of what should be proved. In the case at hand, the general form of the statement is suggested by earlier results of Reider on algebraic surfaces (proved with Bogomolov’s stability of vector bundles). 2) Theorem. Let L be a nef and big divisor on a smooth complex projective surface X, and let x ∈ X be a given point.

0 Conformal Field Theory predicts that the connection should be (projectively) unitary. This is one of the remaining challenges for mathematicians: Question 8. , one such that the image of ρr,k is contained in P U (HX )). Here again the rank-one case is well-known, and Theorem 2 gives an answer in the case k = 1—though not a very explicit one. VECTOR BUNDLES ON CURVES AND GENERALIZED THETA FUNCTIONS 31 10. Are there generalized theta functions? I believe that the above results give some evidence that the spaces H 0 (SU X (r), Lk ) are non-abelian analogues of the spaces H 0 (JX, O(kΘ)).

Theorem 6. a) There is a canonical isomorphism H 0 (SU X (r), Lk ) −→ BX b) The dimension of both spaces is given by the Verlinde formula (given in Theorem 5). There are by now several available proofs of these results. The fact that the k dimension of BX (r) is given by the Verlinde formula follows from the work of k Tsuchiya, Ueno and Yamada [T-U-Y]. They show that the dimension of BX (r) is independent of the curve X, even if X is allowed to have double points. Then k it is not too difficult to express BX (r) in terms of analogous spaces for the normalization of X (this is called the factorization rules by the physicists).

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