xxii Notation J (c ·I ) multiplier ideal sheaf (11) (X,D) log pair, D = i aiDi, ai [0,1] Q (11) Singularities of Toric Varieties mult(σ) multiplicity of a simplicial cone, equals [Nσ : Zu1 + ··· + Zud] (6,11) parallelotope of a simplicial cone, equals { i λiui | 0 λi 1} (11) Πσ polytope related to canonical and terminal singularities of (11) Θσ the polyhedron Conv(σ N \{0}) (10,11) Σcan fan over bounded faces of Θσ, reduces to canonical singularities (11) Topology of a Toric Variety sublattice of N generated by |Σ|∩ N (12) π1(XΣ) fundamental group of XΣ, isomorphic to N/NΣ (12) SN real torus N ⊗Z S1 = HomZ(M,S1) (S1)n (12) (XΣ)≥0 nonnegative real points of a toric variety (12) f , μ algebraic and symplectic moment maps XP MR (12) μΣ symplectic moment map CΣ(1) Cl(XΣ)R (12) Singular Homology and Cohomology Hi(X,R) ith singular cohomology of X with coefficients in a ring R (9) Hi(X,R) ith reduced cohomology of X (9) Hc(X,R) i ith cohomology of X with compact supports (12) Hi(X,R) ith singular homology of X (12) H BM i (X,R) ith Borel-Moore homology of X (13) bi(X) ith Betti number of X, equals dim Hi(X,Q) (12) e(X) Euler characteristic of X, equals i (−1)i bi(X) (9,10,12) , cap and cup products (12) H•(X,R) cohomology ring p H p (X,R) under cup product (12) [W] cohomology class of a subvariety W in H2n−2k(X,Q) (12,13) [W]r refined cohomology class of W in H2n−2k(X,X \W,Q) (12,13) f! generalized Gysin map (13) X integral X : H•(X,Q) Q, equals Gysin map of X {pt} (12,13) Equivariant Cohomology for a Group Action EG a contractible space on which G acts freely (12) BG the quotient EG/G (12) EG ×G X quotient of EG × X modulo relation (e · g,x) (e,g · x) (12) H•(X,R) G equivariant cohomology ring, equals H•(EG ×G X,R) (12) ΛG, (ΛG)Q integral and rational equivariant cohomology ring of a point (12) X G fixed point set for action of G on X (12)
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