Abstract
Let S be a connected closed oriented surface of genus g. Given a triangulation (resp. quadrangulation) of S, define the index of each of its vertices to be the number of edges originating from this vertex minus 6 (resp. minus 4). Call the set of integers recording the non-zero indices the profile of the triangulation (resp. quadrangulation). If κ is a profile for triangulations (resp. quadrangulations) of S, for any m∈ℤ >0 , denote by 𝒯(κ,m) (resp. 𝒬(κ,m)) the set of (equivalence classes of) triangulations (resp. quadrangulations) with profile κ which contain at most m triangles (resp. squares). In this paper, we will show that if κ is a profile for triangulations (resp. for quadrangulations) of S such that none of the indices in κ is divisible by 6 (resp. by 4), then 𝒯(κ,m)∼c 3 (κ)m 2g+|κ|-2 (resp. 𝒬(κ,m)∼c 4 (κ)m 2g+|κ|-2 ), where c 3 (κ)∈ℚ·(3π) 2g+|κ|-2 and c 4 (κ)∈ℚ·π 2g+|κ|-2 . The key ingredient of the proof is a result of J. Kollár [24] on the link between the curvature of the Hodge metric on vector subbundles of a variation of Hodge structure over algebraic varieties, and Chern classes of their extensions. By the same method, we also obtain the rationality (up to some power of π) of the Masur-Veech volume of arithmetic affine submanifolds of translation surfaces that are transverse to the kernel foliation.
Highlights
Abstract . — Let S be a connected closed oriented surface of genus g
Remark 1.2 – In [36], Thurston studied triangulations of the sphere where the valency of every vertex is at most 6. He relates the asymptotics of the number of such triangulations with the volume of the moduli space of pointed genus zero curves with respect to some complex hyperbolic metric
M is absolutely rigid if it is transverse to the kernel foliation of ΩMg(k). Examples of such affine submanifolds include strata of translation surfaces having a single singularity, double covers of quadratic differentials which only have zeros of odd order, and closed orbits generated by square-tiled surfaces
Summary
Translation surfaces are pairs (X, ω) where X is a compact Riemann surface and ω a non-zero holomorphic 1-form on X. M is absolutely rigid if it is transverse to the kernel foliation of ΩMg(k) Examples of such affine submanifolds include strata of translation surfaces having a single singularity (minimal strata), double covers of quadratic differentials which only have zeros of odd order, and closed orbits generated by square-tiled surfaces (this list is not exhaustive). That is, the linear submanifolds are supposed to be polarized and absolutely rigid (cf Section 2.5), we show that up to a rational constant, the Masur-Veech volume form is pointwise equal to some power of the curvature form of the natural metric on the tautological line bundle. We thank the referees for the careful reading and their useful comments
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