Cohomology, Volumes, and Theta Characteristics on Metric Graphs

Abstract

This thesis develops tropical and combinatorial analogues of several classical constructions in algebraic geometry, with metric graphs as the central objects of study. We first establish a tropical de Rham theorem for rational polyhedral spaces: the natural map from Lagerberg superform cohomology to the singular tropical cohomology of Itenberg, Katzarkov, Mikhalkin, and Zharkov is an isomorphism of bigraded rings, compatible with Poincare duality on both sides. This extends work of Jell, Shaw, and Smacka, who proved the isomorphism of vector spaces, by equipping singular tropical cohomology with an explicit ring structure via cup and cap products. We then apply this machinery to principally polarized tropical abelian varieties. A polarization determines a canonical constant (1,1)-superform whose cohomology class is the tropical analogue of the first Chern class of a polarization line bundle. For the Jacobian of a metric graph, the pullback of this form along the Abel-Jacobi map recovers the Zhang canonical measure on the graph. We also prove a tropical Poincare formula expressing the class of the d-th effective locus as the (g-d)-th power of the theta divisor class divided by (g-d) factorial, and compute the volume of the d-th effective locus to be the binomial coefficient g choose d. Next we study the Voronoi polytope of a metric graph inside the first homology with real coefficients, and its zonotopal subdivision by spanning trees. We solve the cell membership problem for this subdivision by introducing (f,q)-optimal spanning trees, and apply the solution to identify, for each 2-torsion class in the first homology with Z/2Z coefficients, an explicit break divisor with Zharkov's tropical theta characteristic in the degree g-1 Picard group. We further characterize break divisors as the unique minimizers of a quadratic energy functional built from the Moore-Penrose pseudoinverse of the graph Laplacian, under a positive resistance curvature hypothesis. Finally, we present joint work on integral aspects of Fourier duality for abelian varieties. Using Pappas's integral Grothendieck-Riemann-Roch theorem, we construct integral lifts of the classical Fourier-Beauville transforms on Chow groups and prove that Beauville's decomposition, the inversion formula, and the sl_2-action all hold after inverting the factorial of 2g+d+1.