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Masur-Veech volumes and intersection theory on moduli spaces of abelian differentials. (English) Zbl 1446.14015
Une différentielle abélienne est une paire \((X,\omega)\) où \(X\) est une surface de Riemann de genre \(g\) et \(\omega\) une section du fibré canonique \(K_{X}\). Pour chaque partition \(\alpha:=\left\{\alpha_{1},\dots,\alpha_{n}\right\}\) de \(2g-2\) avec \(\alpha_{i}\geq 0\), la strate \(\Omega\mathcal{M}_{g}(\alpha)\) paramètre les différentielles abéliennes telles que \(\mathrm{div}(\omega)=\sum \alpha_{i}P_{i}\).
Le lieu des différentielles abéliennes de volume \(1\) dans une strate possède un volume naturel fini, dit de Masur-Veech. Ce résultat a des conséquences fondamentales sur l’étude de la dynamique sur les strates. De nombreux efforts ont donc été produits pour calculer ces volumes. On pourra par exemple consulter [E. Goujard, Ann. Inst. Fourier 66, No. 6, 2203–2251 (2016; Zbl 1368.30020)] pour une vue de certaines stratégies et des volumes de strates de petites dimensions.
Dans cet article, ces volumes sont reliés à la théorie de l’intersection sur la compactification de la variété d’incidence des strates. Le théorème 1.1 donne une égalité entre le volume de la strate et l’intégrale d’un produit explicite du fibré en droites universel et le fibré en droites cotangent vertical associé aux zéros. De plus, le théorème 1.2 donne une relation de récurrence entre les volumes. Il est intéressant de noter que ces résultats s’étendent à chaque composante connexe des strates. Ces résultats permettent de déduire de belles formules pour les constantes de Siegel-Veech pour les connexions de selles et pour l’aire (voir les théorèmes 1.3 et 1.4). Enfin les auteurs prouvent dans les théorèmes 1.5 et 1.6 les conjectures importantes de [A. Eskin and A. Zorich, Arnold Math. J. 1, No. 4, 481–488 (2015; Zbl 1342.32012)] sur l’asymptote des volumes lorsque le genre tend vers l’infini.

MSC:
14H15 Families, moduli of curves (analytic)
30F30 Differentials on Riemann surfaces
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
14N10 Enumerative problems (combinatorial problems) in algebraic geometry
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
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