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The virtual class of the moduli stack of stable \(r\)-spin curves. (English) Zbl 1136.14015
Let \(n\) be a non-negative integer, \(r\) be a natural number and \({\mathfrak m}=(m_1,\dots,m_n)\) an \(n\)-tuple of non-negative integers. A smooth \(n\)-pointed \(r\)-spin curve of genus \(g\) is a triple \((C,{\mathfrak p},{\mathcal F})\) consisting of a smooth genus \(g\) curve \(C\), with \(n\) marked points \({\mathfrak p}=(p_1,\dots,p_n)\) and a line bundle \({\mathcal F}\) with an isomorphism \({\mathcal F}^{\otimes r}\simeq \omega_C(-\sum m_ip_i)\). The moduli space \({\mathcal M}_{g,n}^{1/r,{\mathfrak m}}\) for such objects has been naïvely introduced by E. Witten [in: Topological methods in modern mathematics. Proc. symp. John Milnor’s sixtieth birthday, State University of New York, Stony Brook, USA, June 14-June 21, 1991. Houston, TX: Publish or Perish, Inc. 235–269 (1993; Zbl 0812.14017)], together with the description of a virtual fundamental class on a ‘reasonable’ compactification of \({\mathcal M}_{g,n}^{1/r,{\mathfrak m}}\). Later, T. J. Jarvis [Compos. Math. 110, No. 3, 291–333 (1998; Zbl 0912.14010); Int. J. Math. 11, No. 5, 637–663 (2000; Zbl 1094.14504)] was able to introduce a suitable notion of stable \(r\)-spin curve and to prove that the moduli functor for stable \(n\)-pointed \(r\)-spin curves of genus \(g\) and type \({\mathfrak m}\) is represented by a smooth Deligne-Mumford stack \(\overline{\mathcal M}_{g,n}^{1/r,{\mathfrak m}}\). Also, together with T. Kimura and A. Vaintrob, T. J. Jarvis [Contemp. Math. 276, 167–177 (2001; Zbl 0986.81105); Compos. Math. 126, No.2, 157-212 (2001; Zbl 1015.14028)] formulated a set of axioms to be satisfied by the virtual class of \(\overline{\mathcal M}_{g,n}^{1/r,{\mathfrak m}}\); later an algebro-geometric construction of these virtual classes was given by A. Polishchuk and A. Vaintrob [Contemp. Math. 276, 229–249 (2001; Zbl 1051.14007)] and A. Polishchuk [in: Aspects of Mathematics E 36, 253–264 (2004; Zbl 1105.14010)].
The author uses results by R. Seeley and I. M. Singer [J. Geom. Phys. 5, No. 1, 121–136 (1988; Zbl 0692.30038)] on the \(\overline\partial\) operator on singular Riemann surfaces to give a rigorous construction of the virtual fundamental class described by Witten, and to prove that this class satisfies the Jarvis-Kimura-Vaintrob axioms.
More precisely, let \({\mathcal M}\) (resp. \(\overline{\mathcal M}\)) denote a moduli space of smooth (resp. stable) \(r\)-spin curves. From the universal curve \({\mathcal C}\to {\mathcal M}\) one obtains the Hilbert space bundles \({\mathcal E}^0\) and \({\mathcal E}^1\) whose fibers at the point \([(C,{\mathfrak p},{\mathcal F})]\) are \(L^2(C;{\mathcal F})\) and \(L^2(C;{\mathcal F}\otimes T^{0,1})\), respectively. By the Seeley-Singer results, the operator \(\overline{\partial}\colon {\mathcal E}^0\to {\mathcal E}^1\), extends to an operator \(\overline{\partial}\colon \overline{\mathcal E}^0\to \overline{\mathcal E}^1\), over \(\overline{\mathcal M}\). A finite reduction of \(\overline{\partial}\colon \overline{\mathcal E}^0\to \overline{\mathcal E}^1\) is a pair of finite rank subbundles \(E^i\subseteq \overline{\mathcal E}^i\) such that \(\overline{\partial}(E^0)\subseteq E^1\), \(\text{ index}\{ \overline{\partial}\colon E^0\to E^1\}=\text{ index}\{ \overline{\partial}\colon \overline{\mathcal E}^0\to \overline{\mathcal E}^1\}\), and \(\overline{\partial}(\overline{\mathcal E}^0)^{\perp}\subseteq E^1\). If \(\pi\) denotes the projection to the base \(\pi\colon E^0\to \overline {\mathcal M}\), one has a natural non-trivial section \(\phi\colon E^0\to\pi^*E^1\) (the Witten map) such that \(\phi^{-1}(0)=\overline {\mathcal M}\). Hence the top Chern class of \(\pi^*E^1\to E^0\) is supported on \(\overline {\mathcal M}\); pushing forward this class on \(\overline {\mathcal M}\) one obtains an element of the cohomology group \(H^*(\overline {\mathcal M})\). Let \(d(r,{\mathfrak m})=2r^{-1}(-2g+2+\sum m_i)+2\); it is shown that the cohomology class \((-1)^{d(r,{\mathfrak m})}\pi_*c_{top}(\pi^*E^1,\phi)\in H^*(\overline{\mathcal M}_{g,n}^{1/r,{\mathfrak m}})\) is independent of the finite reduction and satisfies the Jarvis-Kimura-Vaintrob axioms for the virtual fundamental class.

14H10 Families, moduli of curves (algebraic)
Full Text: DOI
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