Donaldson-Thomas invariants of Calabi-Yau threefolds

Gromov-Witten, Gopakumar-Vafa, and Donaldson-Thomas invariants of Calabi-Yau threefolds are compared. In certain situations, the Donaldson-Thomas invariants are very easy to handle, sometimes easier than the other invariants. This point is illustrated in several ways, especially by revisiting computations of Gopakumar-Vafa invariants by Katz, Klemm, and Vafa in a rigorous mathematical framework. This note is based on my talk at the 2004 Snowbird Conference on String Geometry. 1 DT Invariants and GW Invariants 1.1 Generalities Let X be a nonsingular complex projective threefold, β ∈ H2(X,Z), and let n ∈ Z. We let In(X, β) denote the part of the Hilbert scheme of X parametrizing subschemes Z ⊂ X with • [Z] = β • χ(OZ) = n. The class [Z] ∈ H(X,Z) can be equivalently defined as either the dimension one component of the support cycle of Z, or as ch2(OZ). We let IZ be the ideal sheaf of Z. In [1], a perfect obstruction theory is defined on In(X, β) arising naturally from the deformation theory of the ideal sheaves IZ . The virtual dimension is given by D = dimExt0(IZ , IZ)− dimExt 2 0(IZ , IZ) = c1(X) · β. (1)