Complex Monge-Ampere and Symplectic Manifolds

Curvature of vector bundles associated to holomorphic fibrations

“…0). The positivity of D then follows from a computation in [27], but we will give the short argument here too. Let Φ = i∂∂φ where the ∂∂-operator acts on t 1 and the z-variables, the remaining t-variables being fixed.…”

“…Now consider our space X above and let U = {|Re t| < 1} be a strip, and consider functions ψ that depend only on Re t. Then 4C(ψ) =ψ − |∂ zψ | 2 φ if we use dots to denote derivatives with respect to Re t. The link between Theorem 1.2 and the papers cited above lies in the fact that, by the results in [27], the right-hand side here is the geodesic curvature of the path in K(L) determined by ψ.…”

“…plays a crucial role in the recent work on variations of Kähler metrics on compact manifolds; see [27], [23], [11], [12], [26] and [8], to cite just a few. Fixing a line bundle L on Z, these papers consider the space K(L) of all Kähler metrics whose Kähler form is cohomologous to the Chern class of L. This means precisely that the Kähler form can be written…”

The Monge-Ampère operator and geodesics in the space of Kähler potentials

“…To prove the theorem we first apply the observation of Donaldson [10], Mabuchi [19] and Semmes [24], which shows that solving the geodesic equation on H is equivalent to solving the degenerate Monge-Ampère equation…”

2D gravitational Mabuchi action on Riemann surfaces with boundaries

“…Well-known examples of such gravitational actions are the Liouville [1], Mabuchi and Aubin-Yau actions [2][3][4][5], as well as the cosmological constant action S c [g 0 , g] = µ 0 d 2 x( √ g − √ g 0 ) = µ 0 (A − A 0 ). While the Liouville action is formulated entirely in terms of g 0 and the conformal factor σ (defined as g = e 2σ g 0 ), the Mabuchi and Aubin-Yau actions crucially involve also directly the Kähler potential φ.…”