Combinatorial and algebro-geometric cohomology classes on the moduli spaces of curves

1994/06/30 by Enrico Arbarello, Arbarello, Enrico, Maurizio Cornalba +1 · 9 citations
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.alg-geom/9406008

openalex publication_date 1994/06/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Based on the combinatorial description of the moduli spaces of curves provided by Strebel differentials, Witten and Kontsevich have introduced combinatorial cohomology classes W_{(m0,m1,m2,…),n}, and conjectured that these can be expressed in terms of Mumford-Morita-Miller classes. It is argued that this link should be provided by a theorem of Di Francesco, Itzykson and Zuber which relates the derivatives of the Witten-Kontsevich partition function with respect to one set of variables to the derivatives with respect to the other set of variables. Two things are shown. First of all that this works in complex codimension 1. Secondly that in all the cases when it has been possible to make the Di Francesco, Itzykson and Zuber correpondence explicit this translates into identities of the type ∫_W_{(m0,m1,m2,…),n}∏ψ_i^d_i =∫_\calM_g,n X_{(m0,m1,m2,…),n}∏ψ_i^d_i where the X_{(m0,m1,m2,…),n} are explicit polynomials in the algebro-geometric classes and the ψ_i are the Chern classes of the point bundles, for any choice of d₁,…,d_n.

Cited by

• Getzler-Kapranov complexes and moduli stacks of curves
• The Witten equation, mirror symmetry and quantum singularity theory
• The Chern character of the Verlinde bundle over the moduli space of stable curves
• Log canonical models for the moduli space of pointed stable rational curves
• Calculating cohomology groups of moduli spaces of curves via algebraic geometry
• Logarithmic series and Hodge integrals in the tautological ring (with an appendix by D. Zagier)
• Cusps and the family hyperbolic metric
• Pullbacks of κ classes on M_0,n
• The fourth tautological group of M_g,n and relations with the cohomology

Related

• Cohomology of moduli spaces
• Hodge and Prym tau functions, Jenkins-Strebel differentials and combinatorial model of \mathcal M_g,n
• Characteristic classes of \ai-algebras
• Polynomial relations among kappa classes on the moduli space of curves
• Weight 11 Compactly Supported Cohomology of Moduli Spaces of Curves in excess four