By Previato E. (ed.)

Our wisdom of gadgets of algebraic geometry comparable to moduli of curves, (real) Schubert periods, basic teams of enhances of hyperplane preparations, toric types, and version of Hodge constructions, has been stronger lately via rules and structures of quantum box concept, resembling reflect symmetry, Gromov-Witten invariants, quantum cohomology, and gravitational descendants.

These are the various issues of this refereed number of papers, which grew out of the unique consultation, "Enumerative Geometry in Physics," held on the AMS assembly in Lowell, MA, April 2000. This consultation introduced jointly mathematicians and physicists who pronounced at the most up-to-date effects and open questions; all of the abstracts are integrated as an Appendix, and likewise integrated are papers by means of a few who couldn't attend.

The assortment presents an summary of cutting-edge instruments, hyperlinks that attach classical and glossy difficulties, and the most recent wisdom available.

Readership: Graduate scholars and study mathematicians attracted to algebraic geometry and comparable disciplines.

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38 §5. Toroidal compactification In [SC], Mumford and his coworkers constructed, for each bounded symmetric domain X and an arithmetic subgroup r of Aut(X), class of explicit toroidal compactifications of r\X. a In fact, the compactification is determined by a certain combinatorial data, called r-admissible rational polyhedral cone decompositions. If the cone decomposition is chosen to be a "nice" one, which is always possible by performing subdivisions, then we actually get a smooth compactification.

By r of a topological space H* denote it by r\ 9 . 1) "rational boundary components". H* = {(y,Q)ly 9 (Yl ,Q) where . ff 1 r = (Y2,Q') R ,-1 Set-theoretically, E for Q £ Hr , Q' N r,g' and £ Hr , , Yl'Y2 E Q' = Pr,g(y~lYl)·Q It is easily checked that R is an equivalence relation, SP29(Q) representative. 3) H;, For any point x in Satake topology. t. 1) then yU = U. In the above definition we can replace SP29(Z) by any subgroup r of finite index. 2). acts on H; continuously. t. exist neighborhoods U of x and U' r, then there of x' such that r·u n u' = ~.

Gl g. is not the Og. 2) < < 9 , is isomorphic no matter whether it is rational or not. 4) name ly, F r In = H*g , we have the standard rational boundary components, (1, H ) c H*. 1). For each (rational) boundary component F, let G 19F = F} N(F) {g W(F) unipotent radical of N(F) U(F) center of W(F) E 40 V(F) = W(F)/U(F). 2) Via exponential, we get a section in W(F), and can write W(F) set theoretically as V(F)·U(F). It has a structure as a complex vector space. 3) N(F) can be written as a semi-direct product direct product (no compact factor in this case) where (a) Gl · V .