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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.

MSC:
14H10 Families, moduli of curves (algebraic)
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[1] Chen, W.: A Homotopy theory of orbispaces, http://arxiv.org/list/math.AT/0102020, 2001
[2] Fujita, H., Kuroda, S.T.: Functional Analysis I (in Japanese). Tokyo: Iwanami Shoten, 1983
[3] Jarvis, T.: Geometry of the moduli of higher spin curves. Internat. J. Math. 11, 637–663 (2000) · Zbl 1094.14504
[4] Jarvis, T.: Torsion-free sheaves and moduli of generalized spin curves. Composito Math. 110: 291–333 (1998) · Zbl 0912.14010
[5] Jarvis, T.: The Picard group of the moduli of higher spin curves. New York J. Math. 7, 23–47 (2001) · Zbl 0977.14010
[6] Jarvis, T., Kimura, T., Vaintrob, A.: Moduli spaces of higher spin curves and integrable hierarchies. Composito Math. 126, 157–212 (2001) · Zbl 1015.14028
[7] Jarvis, T., Kimura, T., Vaintrob, A.: Gravitational descendents and the moduli space of higher spin curves. In: Advances in algebraic geometry motivated by physics (Lowell, MA, 2000). Contemp. Math. 276, pp. 167–177, 2001 · Zbl 0986.81105
[8] Jarvis, T., Kimura, T., Vaintrob, A.: Spin Gromov-Witten Invariants. Commun. Math. Phys. 259, no. 3, 511–543 (2005) · Zbl 1094.14042
[9] Polishchuk, A.: Witten’s top Chern class on the moduli space of higher spin curves. In : Frobenius manifolds, Aspects Math. Wiesbaden: Vieweg, E36, pp. 253–264, 2004 · Zbl 1105.14010
[10] Polishchuk, A., Vaintrob, A.: Algebraic construction of Witten’s top Chern class. In: Advances in algebraic geometry motivated by physics (Lowell, MA, 2000). Contemp. Math. 276, pp. 229–249, 2001 · Zbl 1051.14007
[11] Seely, R., Singer, I.M.: Extending to Singular Riemann Surfaces. J. Geom. Phys. 5, 121–136 (1989)
[12] Witten, E., Two dimensional gravity and intersection theory on the moduli space. Surveys in Diff. Geom. 1, 243–310 (1991) · Zbl 0757.53049
[13] Witten, E.: Algebraic geometry associated with matrix models of two dimensional gravity. In: Topological methods in modern mathematics (Stony Brook, NY, 1991), Houston, TX: Publish or Perish, pp. 235–269, 1993 · Zbl 0812.14017
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