Degenerating Riemann surfaces and the Quillen metric

Dennis Eriksson

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arXiv · 1106.3092

Degenerating Riemann surfaces and the Quillen metric

Dennis Eriksson

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Abstract

The degeneration of the Quillen metric for a one-parameter family of Riemann surfaces has been studied by Bismut-Bost and Yoshikawa. In this article we propose a more geometric point of view using Deligne's Riemann-Roch theorem. We obtain an interpretation of the singular part of the metric as a discriminant and the continuous part as a degeneration of the metric on Deligne products, which gives an asymptotic development involving the monodromy eigenvalues. This generalizes the results of Bismut-Bost and is a version of Yoshikawa's results on the degeneration of the Quillen metric for general degenerations with isolated singularities in the central fiber.

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Dennis Eriksson. 2011-06-15. Degenerating Riemann surfaces and the Quillen metric. https://arxiv.org/abs/1106.3092

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