AG-Soyuz (Europe-NIS algebraic geometry network)
INTAS reference number 93-2805 and 93-2805 Ext

Five highlighted research topics

 (a) Higher dimensional birational geometry (work of Corti, Iskovskikh,
Prokhorov, Pukhlikov, Reid). One of the main achievements of algebraic
geometry over the last 20 years is the work of Mori and others extending
minimal models and the Enriques-Kodaira classification to 3-folds. A lot
of progress has been made since around 1995 in understanding how to apply
Mori theory to birational geometry. The book
 [CR] Explicit birational geometry of 3-folds, A. Corti and M. Reid
 (eds.), CUP 2000, v+349 pp., ISBN: 0 521 63641 8
contains papers by Corti, Pukhlikov and Reid working out in detail the
theory of birational rigidity of Fano 3-folds, and presenting a manifesto
for a new view of birational geometry.

 (b) Derived categories of coherent sheaves and Fourier-Mukai transform
(Bondal, Bridgeland, King, Markarian, Orlov, Thomas, Reid). The work of
Bondal, Orlov and other Moscow geometers in the mid 1990s pioneered the
idea of the derived category of coherent sheaves on a variety (up to
equivalence) as a geometric invariant of the variety, analogous to K
theory or cohomology theories. Stimulated by contact during the first
period of AG-Soyuz, a group of UK mathematicians has been deeply involved
since 1996 in this Russian research speciality. The paper
 [BKR] Tom Bridgeland, Alastair King and Miles Reid, Mukai implies McKay:
 the McKay correspondence as an equivalence of derived categories,
 Dedicated to Andrei Tyurin's 60th birthday, math.AG/9908027, 27 pp.,
 submitted to J. Amer. Math. Soc.
takes up the challenge, applying the ideas of the derived category to the
geometry of a crepant resolution of a SL(3) orbifold singularity.

 (c) Special Lagrangian geometry, the Strominger-Yau-Zaslow approach to
mirror symmetry, geometric quantisation and Bohr-Sommerfeld orbits (Gross,
Hitchin, Tyurin).  The geometry of special Lagrangian subvarieties is
widely believed to be behind mirror symmetry in string theory. Work of
Tyurin since 1998 introduces a spectrum of geometries linking algebraic
geometry to special Lagrangian geometry, and relates these ideas to the
Bohr-Sommerfeld conditions of geometric quantisation.
 [Tyu] Andrei Tyurin, Special Langrangian geometry as a slight deformation
 of algebraic geometry; Izv. Ross. Akad. Nauk Ser. Mat. 64 (2000)
 141--224; in English Warwick preprint 22/1998, math.AG/9806006, 45 pp.

 (d) Classifying modules over singular curves in representation theoretic
type, enumeration of simple cases (Drozd and Greuel). The program of Drozd
and Greuel relates "simple" singularities in the sense of deformation
theory to "simple" in the algebraic sense of only a discrete set of
modules up to isomorphism. They study vector bundles
over singular projective curves, obtaining a division into finite, tame
and wild cases. The finite and tame type generalise Grothendieck and
Atiyah's famous results classifying vector bundles over the projective
line and elliptic curves.
 [DG] Y.A. Drozd and G.M. Greuel: On the classification of vector bundles
 on projective curves, preprint of Max Planck Inst. fur Math, Bonn,
 MPI/1999--130, 37 pp.

 (e) Higher dimensional local fields, adelic group and applications
(Abrashkin, Fesenko, Parshin, Vostokov). Higher dimensional local fields
have been available since the mid 1970s as a tool to attack problem in
arithmetic, and class field theory for higher dimensional arithmetic
schemes was developed during the 1980s in terms of higher local fields and
adelic groups. As applications of these ideas, Parshin and his school have
obtained spectacular recent results relating to integrable systems and to
problems of conformal field theory.
 [Pa9] A.N. Parshin, On the Krichever correspondence for algebraic
 surfaces, Funktsional. Anal. i Prilozhen., to appear 2000