Kirchhoff's theorem for Prym varieties

Yoav Len, Dmitry Zakharov

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


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. 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 as well.

Original languageEnglish
Article numbere11 1-54
JournalForum of Mathematics, Sigma
StatePublished - Feb 16 2022


  • 2020 Mathematics Subject Classification 14T05 14H40


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