Intersection theory on moduli spaces of curves ... - User Web Pages

32 1. A gentle introduction to moduli spaces of curvesThe following is a brief selection of the early results concerning Weil–Petersson volumes. 6Wolpert [58, 59] proved that V 0,4 (0) = 2π 2 , V 1,1 (0) = π212 and V g,n(0) = q(2π 2 ) 3g−3+nfor some rational number q. This last fact is a corollary of Theorem 1.26, from which wededuce that q = ∫ M g,nκ 3g−3+n1.Penner [50] proved that V 1,2 (0) = π44 .Zograf [63, 62] proved that V 0,5 (0) = 10π 4 and that V 0,n (0) = (2π2 ) n−3a n , where a 3 = 1and for n ≥ 4,a n = 1 2n−3∑k=1k(n − k − 2)n − 1Näätänen and Nakanishi [40, 41] proved that(n−3)!( )( )n − 4 nak − 1 k + 1 k+2 a n−k .V 0,4 (L 1 , L 2 , L 3 , L 4 ) = 1 2 (L2 1 + L2 2 + L2 3 + L2 4 + 4π2 ) and V 1,1 (L 1 ) = 148 (L2 1 + 4π2 ).Much of the early work in this direction was rooted in algebraic geometry, where there is noanalogue to the length of a boundary component. Hence, the results often concerned the constantV g,n (0) rather than the more general volume function V g,n (L). However, Näätänen andNakanishi’s work suggests that this latter problem yields nice results, at least for moduli spacesof complex dimension one. In fact, they showed that V 1,1 (L 1 ) and V 0,4 (L 1 , L 2 , L 3 , L 4 ) are bothpolynomials in the boundary lengths. That this is the case for all of the Weil–Petersson volumesV g,n (L) was proven by Mirzakhani in two distinct ways [33, 34]. The remainder of this sectionis dedicated to giving the essential ideas, results and proofs involved in Mirzakhani’s work.Mirzakhani’s recursionOne of the main obstacles in calculating the volume of the moduli space is the fact that theFenchel–Nielsen coordinates do not behave nicely under the action of the mapping class group.In particular, there is no concrete description for a fundamental domain of M g,n (L) in T g,n (L)for general values of g and n. Mirzakhani had the idea of unfolding the integral required tocalculate the volume of the moduli space to a cover over the moduli space. In general, considera covering π : X 1 → X 2 , let dv 2 be a volume form on X 2 , and let dv 1 = π ∗ dv 2 be the pull-back6 When comparing these results with the original sources, there may be some discrepancy due to two issues. First,there are distinct normalisations of the Weil–Petersson symplectic form which differ by a factor of two. We have scaledthe results, where appropriate, to correspond with the Weil–Petersson symplectic form defined earlier. Second, onemust treat the special cases of V 1,1 (L 1 ) and V 2,0 with some care. This is due to the fact that every point on M 1,1 (L 1 ) andM 2,0 is an orbifold point, generically with orbifold group Z 2 . As a result, the statement of certain theorems holds trueonly if one considers V 1,1 (L 1 ) and V 2,0 as orbifold volumes — in other words, half of the true volumes. The upshot isthat one should not be alarmed if results concerning Weil–Petersson volumes from different sources differ by a factorwhich is a power of two.