K-theoretic computations in enumerative geometry

2.1 Basic notations and constructions First recall the following notations and constructions • KD(X) the Grothendieck group of the category of D-equivariant locally free sheaves on X. • K(X) the Grothendieck group of the category of D-equivariant coherent sheaves on X. • KD(X) the Grothendieck group of D-equivariant locally free sheaves on the fixed locus X. • K(X) the Grothendieck group of category of D-equivariant locally free sheaves on the fixed locus X. • ∆, X(D): character group of D • R(D) = Z[∆]. Z-group algebra associated to the character group ∆ = X(T ), or simply the representation ring of D. It’s an integral domain. For χ ∈ ∆, we usually use e to denote the corresponding element in R(D). And let S ⊂ R(D) be the multiplicative subset generated by (1 − e) for all non-trivial χ, then we know 0 / ∈ S. • cl(F ), [F ] The class represented by F in KD(X).