Ju l 2 00 7 Higher Energies in Kähler Geometry I

Let X →֒ PN be a smooth complex projective variety of dimension n. Let λ be an algebraic one parameter subgroup ofG := SL(N+1,C). Let 0 ≤ l ≤ n+1. We associate to the coefficients Fl(λ) of the normalized weight of λ on the mth Hilbert point of X new energies Fω,l(φ). The (logarithmic) asymptotics of Fω,l(φ) along the potential deduced from λ is the weight Fl(λ). Fω,l(φ) reduces to the Aubin energy when l = 0 and the K-Energy map of Mabuchi when l = 1. When l ≥ 2 Fω,l(φ) coincides (modulo lower order terms) with the functional Eω,l−1(φ) introduced by X.X. Chen and G.Tian. §0 The Standard Energies of Kähler Geometry Recall that Mabuchi’s K-energy map (see [15] ) is given by νω(φ) := − 1 V ∫ 1