Journal of Differential Geometry

Expanding Kähler–Ricci solitons coming out of Kähler cones

Ronan J. Conlon and Alix Deruelle

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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 $c_1 \Bigl ( K_D \oplus {(L^\ast)}^{\oplus (n+1)} \Bigr ) \gt 0$. 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.

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J. Differential Geom., Volume 115, Number 2 (2020), 303-365.

Received: 26 September 2016
First available in Project Euclid: 19 May 2020

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Conlon, Ronan J.; Deruelle, Alix. Expanding Kähler–Ricci solitons coming out of Kähler cones. J. Differential Geom. 115 (2020), no. 2, 303--365. doi:10.4310/jdg/1589853627.

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