Lower bounds on the modified K-energy and complex deformations

2013/02/04 by Andrew Clarke, Clarke, Andrew, Carl Tipler +1
Mathematics · #53C55 #Differential Geometry (math.DG) #FOS: Mathematics #math.DG #msc:53C55

paper · pdf · doi:10.48550/arxiv.1302.0556

22 pages; Last Version, to be published in Advances in Maths

arxiv created 2013/11/16 · arxiv updated 2013/11/19

Abstract

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

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