arXiv subjects

Alessandro Giacchetto

Publications and source records attributed to Alessandro Giacchetto.

21 records · Page 2Linked to original sources

arXiv2021

An intersection-theoretic proof of the Harer-Zagier formula

We provide an intersection-theoretic formula for the Euler characteristic of the moduli space of smooth curves. This formula reads purely in terms of Hodge integrals and, as a corollary, the standard calculus of tautological classes gives a new short proof of the Harer-Zagier formula. Our result is based on the Gauss-Bonnet formula, and on the observation that a certain parametrisation of the $Ω$-class - the Chern class of the universal $r$-th root of the twisted log canonical bundle - provides the Chern class of the log tangent bundle to the moduli space of smooth curves. Being $Ω$-classes by now employed in many enumerative problems, mostly recently found and at times surprisingly different from each other, we dedicate some work to produce an extensive list of their general properties: extending existing ones, finding new ones, and writing down some only known to the experts.

math.AG↗

arXiv2021

On the Kontsevich geometry of the combinatorial Teichmüller space

For bordered surfaces S, we develop a complete parallel between the geometry of the combinatorial Teichmüller space $T_S^{comb}$ equipped with Kontsevich symplectic form $ω_K$, and then the usual Weil-Petersson geometry of Teichmüller space $T_S$. The basis for this is an identification of $T_S^{comb}$ with a space of measured foliations with transverse boundary conditions. We equip $T_S^{comb}$ with an analog of the Fenchel-Nielsen coordinates (defined similarly as Dehn-Thurston coordinates) and show they are Darboux for $ω_K$ (analog of Wolpert formula). We then set up the geometric recursion of Andersen-Borot-Orantin to produce mapping class group invariants functions on $T_S^{comb}$ whose integration with respect to Kontsevich volume form satisfy topological recursion. Further we establish an analog of Mirzakhani-McShane identities, and provide applications to the study of the enumeration of multicurves with respect to combinatorial lengths and Masur-Veech volumes. The formalism allows us to provide uniform and completely geometric proofs of Witten's conjecture/Kontsevich theorem and Norbury's topological recursion for lattice point count in the combinatorial moduli space, parallel to Mirzakhani's proof of her recursion for Weil-Petersson volumes. We strengthen results of Mondello and Do on the convergence of hyperbolic geometry to combinatorial geometry along the rescaling flow, allowing us to flow systematically natural constructions on the usual Teichmüller space to their combinatorial analogue, such as a new derivation of the piecewise linear structure of $T_S^{comb}$ originally obtained in the work of Penner, as the limit under the flow of the smooth structure of $T_S$.

math.DG↗

arXiv2020

Topological recursion for Masur-Veech volumes

We study the Masur-Veech volumes $MV_{g,n}$ of the principal stratum of the moduli space of quadratic differentials of unit area on curves of genus $g$ with $n$ punctures. We show that the volumes $MV_{g,n}$ are the constant terms of a family of polynomials in $n$ variables governed by the topological recursion/Virasoro constraints. This is equivalent to a formula giving these polynomials as a sum over stable graphs, and retrieves a result of \cite{Delecroix} proved by combinatorial arguments. Our method is different: it relies on the geometric recursion and its application to statistics of hyperbolic lengths of multicurves developed in \cite{GRpaper}. We also obtain an expression of the area Siegel--Veech constants in terms of hyperbolic geometry. The topological recursion allows numerical computations of Masur--Veech volumes, and thus of area Siegel--Veech constants, for low $g$ and $n$, which leads us to propose conjectural formulas for low $g$ but all $n$. We also relate our polynomials to the asymptotic counting of square-tiled surfaces with large boundaries.

math.GT↗