| Let L → X be an ample holomorphic line bundle over a compact Kahler manifold (X, o 0) with c1(L) represented by the Kahler form o0. Given a semi-positive real (1, 1) form eta representing --c1( KX ⊗ L), one can ask whether there exists a Kahler metric o ∈ c1( L) that solves the equation Ric(o) -- o = eta. We study this problem by twisting the Kahler-Ricci flow by eta, that is evolve along the flow w&d2;t = ot + eta -- Ric (ot) starting at o0 . We prove that such a metric exists provided wnt≥Kwn 0 for some K > 0 and all t ≥ 0. We also study a twisted version of Futaki's invariant, which we show is well-defined if eta is annihilated under the infinitesimal action of eta(X ), in particular eta is Aut0( X) invariant. Finally, using Chens epsilon-geodesics instead, we give another proof of the convexity of Lw along geodesics, which plays a central in Berman's proof of the uniqueness of critical points of Fw . |
