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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].
##### MSC:
 14C34 Torelli problem