Twisted balanced metrics

We introduce the notion of twisted balanced metrics. These metrics are induced from specific projective embeddings and can be understood as zeros of a certain moment map. We prove that on a polarized manifold, twisted constant scalar curvature metrics are limits of twisted balanced metrics, extending a result of S.K. Donaldson and T. Mabuchi. Let M be a smooth projective manifold of complex dimension n. Let L be an ample line bundle on M , thus giving a polarization of the considered manifold. In that paper, we consider an extra data T , a twisting, where T is a line bundle on M . Let hT be a smooth hermitian metric on T and denote its curvature 12α. Let hL be a smooth hermitian metric on L whose curvature ω is a Kähler form. We are interested in the following twisted constant scalar curvature equation, Scal(ω)− Λωα = Cα (1) where Cα is a topological constant equal to 4nπ (c1(M)−2c1(T ))·c1(L)([M ]) c1(L)([M ]) . A solution to Equation (1) is said to be an α-twisted constant scalar curvature Kähler metric (α-twisted cscK metric in short). This equation was introduced by J. Fine in [Fi1, Fi2] and studied recently by J. Stoppa in order to understand the behavior of K-stability under deformations of polarizations [St1, St2]. We believe that it has others applications, since it appears naturally in various problems of complex geometry as we shall see later. Let now introduce some notations. Let Aut(M) be the group of holomorphic automorphisms of M . Then, the group of Âut(M,L) of holomorphic automorphisms of (M,L) is formed of couples (κ, κ̂) where κ is a biholomorphism of M and κ̂ is a biholomorphim of the bundle πL : L → M covering κ, i.e πL ◦ κ̂ = κ ◦ πL. The kernel of the projection on the first factor Âut(M,L) Aut(M) is composed of the trivial automorphisms C∗ and we will denote Aut(M,L) = Âut(M,L)/C∗. The following two conditions will appear naturally in the sequel : (C1) The Lie algebra Lie(Aut(M,L)) is trivial and T is semi-positive, α is a pointwise semi-positive (1, 1)-form on M . (C2) T is ample and α is a positive (1, 1)-form on M . 1 ha l-0 03 33 73 9, v er si on 1 23 O ct 2 00 8 Author manuscript, published in "Lie Groups : New research (2009) 15"