Generalized Donaldson-Thomas invariants (Q2914213)

\textit{S. K. Donaldson} and \textit{R. P. Thomas} [in: The geometric universe: science, geometry, and the work of Roger Penrose. Proceedings of the symposium on geometric issues in the foundations of science, Oxford, UK, June 1996 in honour of Roger Penrose in his 65th year. Oxford: Oxford University Press. 31--47 (1998; Zbl 0926.58003)] started a program to apply the arguments in real four dimensional Donaldson theory to their complex siblings, majorly to the complex four dimensional hyper-Kähler manifolds. In [J. Differ. Geom. 54, No. 2, 367--438 (2000; Zbl 1034.14015)], \textit{R. P. Thomas} first applied their arguments to Calabi-Yau threefolds.NEWLINENEWLINECalabi-Yau manifolds are those compact complex manifolds with zero Ricci curvatures. When it is simply connected, it is a product of Calabi-Yau manifolds of complex dimension \(n\) such that \(h^{i.0} =0\) if \(i\neq 0, n\) and \(h^{n,0} =h^{0,0} =1\) with hyper-Kähler manifolds. Hyper-Kähler manifolds are those Calabi-Yau manifolds with holomorphic symplectic structures. Two Calabi-Yau manifolds \(M, N\) are mirror symmetric to each other if \(H^* (M, T_M )=H^* (N, T^*_{N})\). Because of the existence of the holomorphic symplectic structure, any hyper-Kähler manifold is mirror symmetric to itself. If a Calabi-Yau manifold is mirror symmetric to itself, it is called a hyper-Calabi-Yau manifold.NEWLINENEWLINEThere is a conjecture that for any Calabi-Yau threefold, the height \(h=h^{1,1} +h^{1,2} \leq 502\) and the half Euler number \(|k|\leq 480\), where \(k=h^{1,1} -h^{1,2}\). The reviewer observed that for all the known examples, we have \((h-246)^2 -(|k|-240)^2 \leq 4\times 1986\) or \(h\leq 246\) in 2009.NEWLINENEWLINEFor the complex four dimensional hyper-Kähler case, we have either \(b_2 =23, b_3 =0\) or \(3\leq b_2 \leq 8\) in [\textit{D. Guan}, Math. Res. Lett. 8, No. 5--6, 663--669 (2001; Zbl 1011.53039)]. At the beginning of 2000's we began a method of foliation. Let \(H\) be an ample line bundle, then the hypersurface \(H\) has a complex one dimensional foliation coming from the kernel of the holomorphic symplectic form.NEWLINENEWLINEIn [Zbl 1034.14015], let \(\tau\) be an ample line bundle on a Calabi-Yau threefold \(X\), \(E\) a coherent sheaf on \(X\). \(\alpha =[E]\) be a class in \(K(X)\). Let \(A = {\mathcal M} ^{\alpha} _{ss} (\tau )\) be the moduli space of semistable sheaf in the class \(\alpha\), \(B = {\mathcal M} ^{\alpha} _{st} (\tau )\) be the moduli space of stable sheaf in the class \(\alpha\). When \(A = B\), i.e., if all the semistable sheaves are stable, Thomas defined the Donaldson-Thomas invariant \(DT^{\alpha} (\tau )\), which is an integer. However, when \(A\neq B\), Donaldson-Thomas' method did not work.NEWLINENEWLINEIn this survey, the authors defined a rational generalized Donaldson-Thomas invariant \(\bar{DT}^{\alpha} (\tau )\) for any Calabi- Yau threefolds \(X\) and any class \(\alpha\). That is, even if \(A\neq B\). When \(A = B\), it is the same as the original Donaldson-Thomas invariant and therefore is an integer. The generalized Donaldson-Thomas invariant is invariant under the deformation of \(X\).NEWLINENEWLINEFor the entire collection see [Zbl 1230.53007].