The Gromov norm of the Kaehler class and the Maslov index

Abstract

Let D be a Hermitian symmetric space of the non-compact type, ! its Kaehler form. Fora geodesic triangle in D, we compute explicitly the integral R � !, generalizing previous results (see (D-T)). As a consequence, if X is a manifold which admits D as universal cover, we calculate the Gromov norm of (!) 2 H2(X, R). The formula for R � ! is extended to ideal triangles. Precise estimates are given and triangles for which the bound is achieved are studied. For tube-type domains we show the connection of these integrals with the Maslov index we introduced in a previous paper (see (C-O)). 0. Introduction. Let M be a Hermitian symmetric space of the non-compact type, which for simplicity, we assume to be irreducible. Let G be the neutral com- ponent of the group of biholomorphic automorphisms of M. The space M admits a natural (G-invariant) Kaehler form !. This real differential form of degree 2 is closed and hence can be integrated along any 2-cycle, in particular geodesic triangles (to mean triangles the sides of which are geodesic segments). When M is of type I,II or III (in E. Cartan's classification), the integral R � ! (the symplectic area of the geodesic triangle �) was computed in (D-T). By using their techniques, we give the result in the general case. It turns out that these quantities have an upper bound, and with the appropriate normalization, the bound depends only on the rank r of M. We extend these computations to ideal triangles, and we prove (new) sharp estimates for the areas. In particular, we determine precisely the triangles for which the upper bound is achieved. This turns out be of great geometric significance, as the summits of such an extremal triangle are contained in the image of a tight holomorphic totally geodesic imbedding of the complex unit disc into M (Theorem 4.7). Generally speaking, our study of the integrals R � ! requires the fine structure of Hermitian symmetric spaces : special role played by the tube-type case, behaviour of geodesics at infinity and structure of G-orbits in the boundary, use of partial Cayley transforms. This study is also related to a previous work (see (C-O)) where we extended the notion of Maslov index to the Shilov boundary S of a Hermitian symmetric space of tube-type. The Maslov index is (up to a factor �) nothing but the symplectic area of ideal triangles with summits in S, and in some sense the present work can be understood as a continuation of (C-O). The computation of the integrals was used in (D-T) to calculate the Gromov norm of the Kaehler class of a compact Hermitian locally symmetric manifold X = \M, where M is of type I and a discrete, torsion-free, co-compact subgroup of the group G. They observed that it has a nice topological corollary. Let S be a Riemann surface of genus g > 1 and f : S → X a continuous map. Then