Stability conditions, wall-crossing and weighted Gromov-Witten invariants. (English) Zbl 1216.14051

Bayer, Arend; Manin, Yu I.

Mosc. Math. J. 9, No. 1, 3-32 (2009).

The authors extend B. Hassett’s theory of weighted stable pointed curves [Adv. Math. 173, No. 2, 316–352 (2003; Zbl 1072.14014)] to weighted stable maps. The space of stability conditions is described explicitly, and the wall-crossing phenomenon is studied. This can be seen as a non-linear analog of the theory of stability conditions and wall-crossing phenomena in abelian and triangulated categories [T. Bridgeland, Ann. Math. (2) 166, No. 2, 317–345 (2007; Zbl 1137.18008)].
More precisely, let \(g\geq 0\) be a non-negative integer, \(\beta\) an effective one-dimensional class in the Chow ring of a projective variety \(V\) over a field \(k\), and \(\mathcal{A}:S\to \mathbb{Q}\cap [0,1]\) a system of weights on a finite set \(S\). The data \((g,\mathcal{A},\beta)\) are called admissible if \(\beta\neq 0\) or \(2g-2+\sum_{i\in S}\mathcal{A}(i)>0\), and if \(\beta\) is bounded by the characteristic of the field \(k\), i.e., if \(\mathrm{char}(k)=0\) or \(\beta\cdot L<\mathrm{char}(k)\) for some very ample line bundle \(L\) on \(V\). Given admissible data \((g,\mathcal{A},\beta)\), the authors introduce a natural notion of stable maps of type \((g,\mathcal{A},\beta)\) and show that the category \(\overline{\mathcal{M}}_{g,\mathcal{A}}(V,\beta)\) of stable maps of type \((g,\mathcal{A},\beta)\) with their isomorphisms is a proper Deligne-Mumford stack of finite type. When \(\mathcal{A}=\mathbf{1}_S\) is the constant weight 1 over \(S\), one obtains Kontsevich’s stack \(\overline{\mathcal{M}}_{g,S}(V,\beta)\) of \(S\)-marked stable curves, and when \(V\) is a point, one obtains Hassett’s moduli stacks \(\overline{\mathcal{M}}_{g,\mathcal{A}}\). Moreover, the behavior of \(\overline{\mathcal{M}}_{g,\mathcal{A}}(V,\beta)\) as \(\mathcal{A}\) ranges in \([0,1]^S\) is investigated, and a chamber decomposition corresponding to wall-crossing phenomena is described.
Also, one writes \(\mathcal{A}\geq \mathcal{B}\) if \(\mathcal{A}\) and \(\mathcal{B}\) are defined on the same set \(S\), and \(\mathcal{A}(i)\geq \mathcal{B}(i)\) for any \(i\) in \(S\). If \(\mathcal{A}\geq \mathcal{B}\) then there is a birational contraction \(\rho_{\mathcal{B},\mathcal{A}}:\overline{\mathcal{M}}_{g,\mathcal{A}}(V,\beta)\to \overline{\mathcal{M}}_{g,\mathcal{B}}(V,\beta)\); these satisfy \(\rho_{\mathcal{C},\mathcal{B}}\circ \rho_{\mathcal{B},\mathcal{A}}=\rho_{\mathcal{C},\mathcal{A}}\) whenever \(\mathcal{A}\geq\mathcal{B}\geq\mathcal{C}\). In particular, every moduli stack \(\overline{\mathcal{M}}_{g,\mathcal{A}}(V,\beta)\) is a birational contraction of Kontsevich’s moduli stack \(\overline{\mathcal{M}}_{g,S}(V,\beta)\). The authors define a system of virtual fundamental classes for the moduli stacks \(\overline{\mathcal{M}}_{g,\mathcal{A}}(V,\beta)\) compatible with the system of contractions \(\rho_{\mathcal{B},\mathcal{A}}\), by setting \([\overline{\mathcal{M}}_{g,\mathcal{A}}(V,\beta)]^{\mathrm{virt}}=\rho_{\mathcal{A},\mathbf{1}_S\,*}[\overline{\mathcal{M}}_{g,S}(V,\beta)]^{\mathrm{virt}}\), and use them to define weighted Gromov-Witten invariants. The compatibility of the virtual fundamental classes with the birational contractions \(\rho_{\mathcal{B},\mathcal{A}}\) implies that the weighted Gromov-Witten invariants without gravitational descendants are actually independent of the weight \(\mathcal{A}\). Finally, it is shown how, by including gravitational descendants, one obtains an \(\mathcal{L}\)-algebra in the sense of [A. Losev and Yu. Manin, Aspects of Mathematics E 36, 181–211 (2004; Zbl 1080.14066)].

Reviewer: Domenico Fiorenza (Roma)

MSC:

14N35	Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
14D22	Fine and coarse moduli spaces
53D45	Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
14H10	Families, moduli of curves (algebraic)
14E99	Birational geometry

Keywords:

weighted stable maps; Gromov-Witten invariants; gravitational descendants

Citations:

Zbl 1072.14014; Zbl 1137.18008; Zbl 1080.14066