Zero dimensional Donaldson--Thomas invariants of threefolds
The recent results and conjectures of \textit{D. Maulik, N. Nekrasov, A. Okounkov} and \textit{R. Pandharipande} [Compos. Math. 142, No. 5, 1263--1285 (2006; Zbl 1108.14046), ibid. 142, No. 5, 1286--1304 (2006; Zbl 1108.14047)], that relate the invariants of the moduli of ideal sheaves of curves on a Calabi--Yau manifold to its Gromov--Witten invariants constitute major progress. Following these papers, one picks a curve class \(\beta\) and an integer \(n\), and one denotes \(I_X(\beta, n)\) the Hilbert scheme of one dimensional subschemes \(Z \subset X\) (\(X\) a Calabi--Yau threefold) satisfying \([Z] = \beta\) and \(\chi(\mathcal{O}_Z) = n\). One shows easy that \(I_X(\beta, n)\) is the moduli space \(\mathfrak{M}_X(0, I, \beta, c_3)\) with \(c_3\) the third Chern class of any ideal sheaf of \(Z \subset X\) in \(I_X(\beta, n)\). Now, one forms the generating function \(\mathcal{D} \mathcal{T}_{X, \beta}(q) = \sum_n \text{deg}[I_X(\beta, n)]^{\text{vir}} q^n\). Of the several conjectures on \(\mathcal{D}\mathcal{T}_{X, \beta}(q)\), one is about the dimension zero Donaldson - Thomas invariant \(\mathcal{D}\mathcal{T}_{X, 0} (q)\). Let \(M(q) =\prod_n \dfrac{1}{(1 - q^n)^n}\) be the three dimensional partition function and \(c_3(T_X \otimes K_X)\) be the third Chern class. The main result of this paper is the proof of this conjecture for all compact smooth complex threefolds by using a homotopy approach: Theorem 0.2. The zero-dimensional Donaldson-Thomas series \(\mathcal{D}\mathcal{T}_{X, 0}(q)\) for any compact smooth complex threefold \(X\) are of the form \(\mathcal{D}\mathcal{T}_{X, 0}(q) = M(-q)^{c_3(T_x \otimes K_X)}\). This conjecture was independently proved for the class of Calabi--Yau threefolds by \textit{K. Behrend} and \textit{B. Fantechi} [Symmetric obstruction theories and Hilbert schemes of points on threefolds, \url{arXiv:math. AG/0512556}] and for projective threefolds by \textit{M. Levine} and \textit{R. Pandharipande} [Algebraic Cobordism revisited, \url{arXiv: math. AG/0605196}].
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