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A Torelli-type theorem for stable curves. (English) Zbl 1038.14500
Mabuchi, T. (ed.) et al., Geometry and analysis on complex manifolds. Festschrift for Professor S. Kobayashi’s 60th birthday. Singapore: World Scientific (ISBN 981-02-2067-7/hbk). 75-95 (1994).
Let \(C\) be a stable curve of genus \(g\geq 2\) with singular points \(\{p_1,p_2, \dots, p_m\}\) and \(\omega_C\) the dualizing sheaf of \(C\). The authors consider the residue map \(\text{Res}_i: H^0(C, \omega_C^{\otimes\mu}) \to\mathbb{C}\) for each \(p_i\) \((i=1,2,\dots,m)\) and for a positive integer \(\mu\), defined by \(\text{Res}_i (\tau)= (\tau/ \xi_i) |_{p_i}\), where \(\xi_i=dz^{\otimes \mu}/z^\mu+(-1)^\mu d w^{\otimes\mu}/w^\mu\) on an open neighbourhood \(U_i=\{(z,w)\in \mathbb{C}^2 \mid zw=0,| z|<1,| w|<1\}\) of \(p_i\) in \(C\). These residue maps define a linear map \(\lambda: H^0(C, \omega_C^{ \otimes \mu})\to \mathbb{C}^m\) by \(\lambda (\tau)=(\text{Res}_1 (\tau), \text{Res}_2 (\tau),\dots, \text{Res}_m (\tau))\) and the map arises as a direct sum \(H^0(C,\omega_C^{\otimes\mu})= \text{ker}\, \lambda \oplus H^0(C,\omega_C^{\otimes\mu})/ \text{ker}\, \lambda\). The decomposition defines a norm in \(H^0(C, \omega_C^{\otimes\mu})\) in a natural manner. The authors show that if there is a linear isometry between \(H^0(C,\omega_C^{\otimes \mu})\) and \(H^0(C', \omega_{ C'}^{\otimes\mu})\) for \(\mu\geq 2g+1\), then \(C\) and \(C'\) are isomorphic up to permutations of glueing of components of regular points.
For the entire collection see [Zbl 0867.00037].
14C34 Torelli problem