Localization of weighted Goldman volume on $\mathrm{SL}(3,\R)$-Hitchin Component

Abstract

We compute Goldman volume on the moduli space of convex real projective structures on a closed oriented surface of genus at least two, with exponential weight defined by the shifted Blaschke area. For every positive parameter, the integral converges and is a finite sum of inverse powers of that parameter. Its coefficients are intersection numbers involving the bundle of regular cubic differentials on the moduli stack of stable curves. The proof establishes absolute convergence and vanishing signed boundary integrals of the transgression under simultaneous plumbing degeneration. The pushforward of Goldman measure by the shifted Blaschke area therefore has a polynomial density on $[0,\infty)$, and its sublevel volumes are polynomial in the threshold.