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...onal (technical) assumption that it admits a global resolution, we can define a virtual fundamental class of the expected dimension. An application of the results of this work is contained in a paper =-=[3]-=- by the first author. There Gromov-Witten invariants are constructed for any genus, any target variety and the axioms listed in [4] are verified. We now give a more detailed outline of the contents of...

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...-equivariant perfect obstruction theory. The virtual fundamental class [X] vir in the expected equivariant Chow group A C∗ ∗ (X) may be constructed by the methods of Li-Tian [LT] and Behrend-Fantechi =-=[B]-=-, [BF]. The connected components Xi of the fixed point scheme carry an associated C∗-fixed perfect obstruction theory. Virtual fundamental classes in A∗(Xi) are thus determined. The virtual normal bun...

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... projective variety over C. Let M = Mg,n(X,β) be the moduli stack of stable maps to X representing the class β ∈ H2(X, Z). Let [M] vir ∈ A∗(M) denote the virtual class in the expected dimension [BF], =-=[B]-=-, [LiT]. A direct analogue of Mumford’s result holds for the universal family over M. Virtual divisors in M are of two types. First, stable splittings (11) ξ = (g1 + g2 = g,A1 ∪ A2 = [n],β1 + β2 = β) ...

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...ndations of the theory of quantum cohomology or Gromov-Witten invariants for semipositive symplectic manifolds. Recently, semipositivity condition has been removed by the work of many authors: [LT2], =-=[B]-=-, [FO],[LT3], [R5], [S]. The focus now is on calculation and applications. As quantum cohomology was developed, many examples were computed. They are all Fano manifolds. One of most important classes ...

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...curve LCI curve C ⊂ X, the space of first-order deformations and the obstruction group are given by H0(C,NC/X) and H1(C,NC/X) respectively. Suppose given an unmarked stable map f : C → X. By [BF] and =-=[B]-=-, the space of first-order deformations of the stable map and the obstruction group are given by the hypercohomology groups Def(f) = H1(C,RHomOC (Ω·f ,OC)) Obs(f) = H2(C,RHomOC (Ω·f ,OC)) where Ω·f is...

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...] and corrected in [O3], and since Illusie explicitly works in the general setting of ringed topoi, all the necessary generalizations have already been established. Specifically, in the discussion of =-=[B]-=- page 604, immediately after Proposition 4, one relies on the claim that φ : Rπ∗(f ∗ T X) ∨ → L Kg,n(X,β)/M tw g,n is a perfect relative obstruction theory. This relative case, discussed in section 7 ...