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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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References:
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