Lower bounds on the modified K-energy and complex deformations

Let (X,L)(X,L) be a polarized Kähler manifold that admits an extremal metric in c1(L)c1(L). We show that on a nearby polarized deformation (X′,L′)(X′,L′) that preserves the symmetry induced by the extremal vector field of (X,L)(X,L), the modified K-energy is bounded from below. This generalizes a result of Chen, Székelyhidi and Tosatti [8], [35] and [38] to extremal metrics. Our proof also extends a convexity inequality on the space of Kähler potentials due to X.X. Chen [7] to the extremal metric setup. As an application, we compute explicit polarized 4-points blow-ups of CP1×CP1CP1×CP1 that carry no extremal metric but with modified K-energy bounded from below.