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 inForum of mathematics. Sigma Vol. 10
Main Authors Len, Yoav, Zakharov, Dmitry
Format Journal Article
LanguageEnglish
Published Cambridge, UK Cambridge University Press 01.01.2022
Subjects
Algebra
Algebraic and Complex Geometry
Decomposition
Graphs
Theorems
Trees
14T05
14H40
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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.
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ISSN:2050-5094
2050-5094
DOI:10.1017/fms.2021.75