Inner geometry of complex surfaces: a valuative approach

Given a complex analytic germ $(X, 0)$ in $(\mathbb C^n, 0)$, the standard Hermitian metric of $\mathbb C^n$ induces a natural arc-length metric on $(X, 0)$, called the inner metric. We study the inner metric structure of the germ of an isolated complex surface singularity $(X,0)$ by means of an infinite family of numerical analytic invariants, called inner rates. Our main result is a formula for the Laplacian of the inner rate function on a space of valuations, the non-archimedean link of $(X,0)$. We deduce in particular that the global data consisting of the topology of $(X,0)$, together with the configuration of a generic hyperplane section and of the polar curve of a generic plane projection of $(X,0)$, completely determine all the inner rates on $(X,0)$, and hence the local metric structure of the germ. Several other applications of our formula are discussed in the paper.Comment: Proposition 5.3 strengthened, exposition improved, some typos corrected, references updated. 42 pages and 10 figures. To appear in Geometry & Topolog

Spectral Measures for G2G_2

Spectral measures provide invariants for braided subfactors via fusion modules. In this paper we study joint spectral measures associated to the rank two Lie group $G_2$, including the McKay graphs for the irreducible representations of $G_2$ and its maximal torus, and fusion modules associated to all known $G_2$ modular invariants.Comment: 36 pages, 40 figures; correction to Sections 5.4 and 5.5, minor improvements to expositio