2020/07/08 by Vincent Koziarz, Koziarz, Vincent, Duc-Manh Nguyen +1
Mathematics · #14D07 #14D23 #51M15 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT)
paper · pdf · doi:10.48550/arxiv.2007.04185
openalex publication_date 2020/07/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let S be a connected closed oriented surface of genus g. Given a\ntriangulation (resp. quadrangulation) of S, define the index of each of its\nvertices to be the number of edges originating from this vertex minus 6\n(resp. minus 4). Call the set of integers recording the non-zero indices the\nprofile of the triangulation (resp. quadrangulation). If \κ is a profile\nfor triangulations (resp. quadrangulations) of S, for any m\∈\n\ℤ>0, denote by mathscrT(\κ,m) (resp.\n mathscrQ(\κ,m)) the set of (equivalence classes of) triangulations\n(resp. quadrangulations) with profile \κ which contain at most m\ntriangles (resp. squares). In this paper, we will show that if \κ is a\nprofile for triangulations (resp. for quadrangulations) of S such that none\nof the indices in \κ is divisible by 6 (resp. by 4), then\n mathscrT(\κ,m)\∼ c3(\κ)m2g+|\κ|-2 (resp.\n mathscrQ(\κ,m) \∼ c4(\κ)m2g+|\κ|-2), where c3(\κ)\n\∈ \ℚ\⋅(\√(3)\π)2g+|\κ|-2 and c4(\κ)\∈\n\ℚ\⋅\π2g+|\κ|-2. The key ingredient of the proof is a\nresult of J. Koll 'ar on the link between the curvature of the Hogde metric on\nvector subbundles of a variation of Hodge structure over algebraic varieties,\nand Chern classes of their extensions. By the same method, we also obtain the\nrationality (up to some power of \π) of the Masur-Veech volume of arithmetic\naffine submanifolds of translation surfaces that are transverse to the kernel\nfoliation.\n