Let $L \to X$ be an ample holomorphic line bundle over a compact K"{a}hler manifold $(X,omega_{0})$ so that $c_1(L)$ is represented by the K"{a}hler form $\omega_0$. Given a semi-positive real $(1,1)$ form $\eta$ representing $-c_{1}(K_{X}\otimes L)$, one can ask whether there exists a K"ahler metric $\omega\in c_{1}(L)$ that solves the equation $Ric(\omega) -\omega=\eta$. We study this problem by twisting the K"ahler-Ricci flow by $eta$ , that is evolve along the flow $\dot{\omega_t}=\omega_{t}+\eta -Ric(\omega_{t})$ starting at $\omega_{0}$. We prove that such a metric exists provided $\omega_{t}^{n}\geq K \omega^{n}_{0}$ for some $K>0$ and all $t \geq 0$. 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 $\eta(X)$. Finally, using Chens $\epsilon$-geodesics instead, we give another proof of the convexity of $\mathcal{L}_{\omega}$ along geodesics, which plays a central to Berman's proof of the uniqueness of critical points of $\mathcal{F}_{\omega}$.
Read
