Expanding Kähler-Ricci solitons coming out of Kähler cones

2016/07/12 by Conlon, Ronan J., Deruelle, Alix · 3 citations
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1607.03546

Abstract

We give necessary and sufficient conditions for a Kähler equivariant resolution of a Kähler cone, with the resolution satisfying one of a number of auxiliary conditions, to admit a unique asymptotically conical (AC) expanding gradient Kähler-Ricci soliton. In particular, it follows that for any n∈ℕ₀ and for any negative line bundle L over a compact Kähler manifold D, the total space of the vector bundle L^{⊕ (n+1)} admits a unique AC expanding gradient Kähler-Ricci soliton with soliton vector field a positive multiple of the Euler vector field if and only if c₁(K_D⊗(L^*)^{⊗ (n+1)})>0. This generalises the examples already known in the literature. We further prove a general uniqueness result and show that the space of certain AC expanding gradient Kähler-Ricci solitons on ℂⁿ with positive curvature operator on (1, 1)-forms is path-connected.

Cited by

• Minimal Degrees, Volume Growth, and Curvature Decay on Complete Kähler Manifolds
• On curvature estimates for four-dimensional gradient Ricci solitons
• Curvature pinching of asymptotically conical gradient expanding Ricci solitons

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