Expanding Kähler-Ricci solitons coming out of Kähler cones. (English) Zbl 1510.53083

Summary: 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 \(n \in \mathbb{N}_0\) and for any negative line bundle \(L\) over a compact Kähler manifold \(D\), the total space of the vector bundle \(L^{\oplus (n+1)}\) 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 \(c_1 \bigg( K_D \oplus{(L^\ast)}^{\oplus (n+1)} \biggr) > 0\). This generalizes 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 \(\mathbb{C}^n\) with positive curvature operator on \((1, 1)\)-forms is path-connected.


53C55 Global differential geometry of Hermitian and Kählerian manifolds
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53E30 Flows related to complex manifolds (e.g., Kähler-Ricci flows, Chern-Ricci flows)
58D17 Manifolds of metrics (especially Riemannian)
Full Text: DOI arXiv Euclid


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