Kirchhoff’s theorem for Prym varieties

We prove an analogue of Kirchhoff’s matrix tree theorem for computing the volume of the tropical Prym variety for double covers of metric graphs. We interpret the formula in terms of a semi-canonical decomposition of the tropical Prym variety, via a careful study of the tropical Abel–Prym map. In pa...

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Bibliographic Details
Published in:Forum of mathematics. Sigma 2022-01, Vol.10, Article e11
Main Authors: Len, Yoav, Zakharov, Dmitry
Format: Article
Language:English
Subjects:
Algebra
Algebraic and Complex Geometry
Decomposition
Graphs
Theorems
Trees
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Summary:We prove an analogue of Kirchhoff’s matrix tree theorem for computing the volume of the tropical Prym variety for double covers of metric graphs. We interpret the formula in terms of a semi-canonical decomposition of the tropical Prym variety, via a careful study of the tropical Abel–Prym map. In particular, we show that the map is harmonic, determine its degree at every cell of the decomposition and prove that its global degree is $2^{g-1}$ . Along the way, we use the Ihara zeta function to provide a new proof of the analogous result for finite graphs. As a counterpart, the appendix by Sebastian Casalaina-Martin shows that the degree of the algebraic Abel–Prym map is $2^{g-1}$ as well.
ISSN:2050-5094
2050-5094
DOI:10.1017/fms.2021.75