Geometric proofs of some results of Morita (Q2710514)

In the three papers: Ann. Inst. Fourier 39, 777-810 (1989; Zbl 0672.57015); Math. Proc. Camb. Philos. Soc. 105, 79-101 (1989; Zbl 0775.57001); Topology and Teichmuller spaces, Proceedings of the 37th Taniguchi symposium, Katinkulta 1995, World Scientific. 159-186 (1996; Zbl 0939.32011), \textit{S. Morita} gave relations between certain two dimensional cohomology classes of various moduli spaces of curves. The authors of the present paper reformulate and prove Morita's results in a more geometrical setting, thus providing a new enlightening point of view of the topological results of Morita.NEWLINENEWLINENEWLINELet us give an idea of these results. Let \({\mathcal M}_g\) be the coarse moduli space of non-singular genus \(g\) curves \(C\) and denote by \({\mathcal C}_g\) the universal curve over it. It is known that \(H^2({\mathcal C}_g,\mathbb{Q})\) is generated by the first Chern class \(\omega\) of the relative dualizing bundle together with the pullback to \({\mathcal C}_g\) of the first Chern class \(\lambda\) of the Hodge bundle on \({\mathcal M}_g\). Let \({\mathcal A}_g\) be the moduli spaces of principally polarized abelian varieties, then \(\lambda\) in \(H^2({\mathcal C}_g, \mathbb{Q})\) is the pullback on \({\mathcal A}_g\) via the period map. If \({\mathcal I}\to {\mathcal A}_g\) is the universal abelian variety, it turns out that there exists a closed 2-form on \({\mathcal I}\) whose restriction to each fiber is the invariant form corresponding to the polarization. Denote the cohomology class of this form by \(\varphi\in H^2({\mathcal I},\mathbb{Q}) \cong H^2(Sp_g (\mathbb{Z})\ltimes H_\mathbb{Z}, \mathbb{Q})\). The class \(\varphi\) is not rational but \(2\varphi\) is. Denote the canonical divisor on \(C\) by \(K_C\) and define \(\kappa: {\mathcal C}_g \to {\mathcal I}\) by \([C,x] \mapsto(2g-2) x-K_C\in \text{Jac} C\). The first of Morita's results proved is:NEWLINENEWLINENEWLINEFor all \(g\geq 1\): \(k^* \varphi= 2g(g-1)\omega-6 \lambda\in H^2({\mathcal C}_g, \mathbb{Q})\).NEWLINENEWLINENEWLINEPassing to the moduli space of curves with a level two structure \({\mathcal M}_g[2]\) we can choose a theta characteristic \(\alpha\) for each curve and we can define a map \(\dot I\alpha: {\mathcal C}_g [2]\to {\mathcal I}[2]\) by NEWLINE\[NEWLINE[C,x]\mapsto (g-1)x-\alpha \in\text{Jac} C.NEWLINE\]NEWLINE The rational homology class of \(\Theta_\alpha\) of \(\Theta_\alpha\) is independent of \(\alpha\) and is the pullback of a class in \(H^2({\mathcal J} [2],\mathbb{Q}]\) denoted by \(\theta\). There is a canonical projection \({\mathcal C}^2_g \to{\mathcal M}_g\) and a commutative square NEWLINE\[NEWLINE\begin{matrix} {\mathcal C}_g^2 & @>\delta>> & {\mathcal J}\\ \downarrow & & \downarrow\\ {\mathcal M}_g & \to & {\mathcal A}_g \end{matrix}NEWLINE\]NEWLINE where \(\delta\) is the difference map defined by \(\delta:[C,x,y] \mapsto[x]- [y]\in\text{Jac} C\). The diagonal copy of \({\mathcal C}_g\) in \({\mathcal C}^2_g\) is a divisor and thus has a class \(\Delta\) in \(H^2({\mathcal C}^2_g \mathbb{Q})\). For \(j=1,2\) denote the first Chern class of the relative cotangent bundle of the \(j\)th projection \(p_j:{\mathcal C}_g^2 \to{\mathcal C}_g\) by \(\psi_j\) (that is \(\psi_j= p_j^*(\omega))\). Then the authors prove the following due to Morita:NEWLINENEWLINENEWLINEFor all \(g\geq 1\): \(\delta^* (\varphi)= \Delta+(\psi_1 +\psi_2/2\in H^1( {\mathcal C}^2_g, \mathbb{Q}).\)NEWLINENEWLINENEWLINEThe authors prove two more expressions for the classes \(\omega\) and \(\lambda\) involving the orbifold fundamental group of \({\mathcal A}_g\). NEWLINENEWLINENEWLINEFinally they get some other relations coming from the theory of Weierstrass points.