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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Hyperbolic $2$-spheres with conical singularities, accessory parameters and Kähler metrics on ${\mathcal{M}}_{0,n}$


Authors: Leon Takhtajan and Peter Zograf
Journal: Trans. Amer. Math. Soc. 355 (2003), 1857-1867
MSC (2000): Primary 14H15; Secondary 30F45, 81T40
Published electronically: December 9, 2002
MathSciNet review: 1953529
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Abstract: We show that the real-valued function $S_\alpha$ on the moduli space ${\mathcal{M}}_{0,n}$ of pointed rational curves, defined as the critical value of the Liouville action functional on a hyperbolic $2$-sphere with $n\geq 3$ conical singularities of arbitrary orders $\alpha=\{\alpha_1,\dots, \alpha_n\}$, generates accessory parameters of the associated Fuchsian differential equation as their common antiderivative. We introduce a family of Kähler metrics on ${\mathcal{M}}_{0,n}$ parameterized by the set of orders $\alpha$, explicitly relate accessory parameters to these metrics, and prove that the functions $S_\alpha$ are their Kähler potentials.


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Additional Information

Leon Takhtajan
Affiliation: Department of Mathematics, SUNY at Stony Brook, Stony Brook, New York 11794-3651
Email: leontak@math.sunysb.edu

Peter Zograf
Affiliation: Steklov Mathematical Institute, St. Petersburg, 191011 Russia
Email: zograf@pdmi.ras.ru

DOI: http://dx.doi.org/10.1090/S0002-9947-02-03243-9
PII: S 0002-9947(02)03243-9
Keywords: Fuchsian differential equations, accessory parameters, Liouville action, Weil-Petersson metric
Received by editor(s): March 12, 2002
Published electronically: December 9, 2002
Additional Notes: Research of the first author was partially supported by the NSF grant DMS-9802574
Article copyright: © Copyright 2002 American Mathematical Society