1991 Mathematics Subject Classification: Primary 53C55, Secondary 14M25 53C25 58F05 A (symplectic) toric variety X, of real dimension In, is completely determined by its moment polytope A C K n . Recently Guillemin gave an explicit combinatorial way of constructing "toric" Kahler metrics on X, using only data on A. In this paper, differential geometric properties of these metrics are investigated using Guillemin's construction. In particular, a nice combinatorial formula for the scalar curvature R is given, and the Euler-Lagrange condition for such "toric" metrics being extremal (in the sense of Calabi) is proven to be R being an affine function on A C 1". A construction, due to Calabi, of a 1-parameter family of extremal Kahler metrics of non-constant scalar curvature on CP 2 jjCP 2 is recast very simply and explicitly using Guillemin's approach. Finally, we present a curious combinatorial identity for convex polytopes A C R n that follows from the wellknown relation between the total integral of the scalar curvature of a Kahler metric and the wedge product of the first Chern class of the underlying complex manifold with a suitable power of the Kahler class.

A theorem of Delzant states that any symplectic manifold (M, ω) of dimension 2n, equipped with an effective Hamiltonian action of the standard n-torus T n = R n /2πZ n , is a smooth projective toric variety completely determined (as a Hamiltonian T n -space) by the image of the moment map φ : M → R n , a convex polytope P = φ(M ) ⊂ R n . In this paper we show, using symplectic (action-angle) coordinates on P ×T n , how all ω-compatible toric complex structures on M can be effectively parametrized by smooth functions on P . We also discuss some topics suited for application of this symplectic coordinates approach to Kähler toric geometry, namely: explicit construction of extremal Kähler metrics, spectral properties of toric manifolds and combinatorics of polytopes.

A theorem of E. Lerman and S. Tolman, generalizing a result of T. Delzant, states that compact symplectic toric orbifolds are classified by their moment polytopes, together with a positive integer label attached to each of their facets. In this paper we use this result, and the existence of "global" action-angle coordinates, to give an effective parametrization of all compatible toric complex structures on a compact symplectic toric orbifold, by means of smooth functions on the corresponding moment polytope. This is equivalent to parametrizing all toric Kähler metrics and generalizes an analogous result for toric manifolds.A simple explicit description of interesting families of extremal Kähler metrics, arising from recent work of R. Bryant, is given as an application of the approach in this paper. The fact that in dimension four these metrics are selfdual and conformally Einstein is also discussed. This gives rise in particular to a one parameter family of self-dual Einstein metrics connecting the well known Eguchi-Hanson and Taub-NUT metrics.

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Let M M be either S 2 × S 2 S^2\times S^2 or the one point blow-up C P 2 # C P ¯ 2 {\mathbb {C}}P^2\#\overline {{\mathbb {C}}P}^2 of C P 2 {\mathbb {C}}P^2 . In both cases M M carries a family of symplectic forms ω λ \omega _{\lambda } , where λ > − 1 \lambda > -1 determines the cohomology class [ ω λ ] [\omega _\lambda ] . This paper calculates the rational (co)homology of the group G λ G_\lambda of symplectomorphisms of ( M , ω λ ) (M,\omega _\lambda ) as well as the rational homotopy type of its classifying space B G λ BG_\lambda . It turns out that each group G λ G_\lambda contains a finite collection K k , k = 0 , … , ℓ = ℓ ( λ ) K_k, k = 0,\dots ,\ell = \ell (\lambda ) , of finite dimensional Lie subgroups that generate its homotopy. We show that these subgroups “asymptotically commute", i.e. all the higher Whitehead products that they generate vanish as λ → ∞ \lambda \to \infty . However, for each fixed λ \lambda there is essentially one nonvanishing product that gives rise to a “jumping generator" w λ w_\lambda in H ∗ ( G λ ) H^*(G_\lambda ) and to a single relation in the rational cohomology ring H ∗ ( B G λ ) H^*(BG_\lambda ) . An analog of this generator w λ w_\lambda was also seen by Kronheimer in his study of families of symplectic forms on 4 4 -manifolds using Seiberg–Witten theory. Our methods involve a close study of the space of ω λ \omega _\lambda -compatible almost complex structures on M M .