An Aubin continuity path for asymptotically conical toric shrinking gradient Kähler-Ricci solitons: openness and a solution for $t=0$

We show that any toric asymptotically conical shrinking gradient Kähler-Ricci soliton on an anti-canonically polarised resolution of a Kähler cone satisfies a complex Monge-Ampère equation. We then set up an Aubin continuity path to solve the resulting equation and show that it has a solution at the initial value of the path parameter in the toric case. This we do by implementing another continuity method. Finally, we prove openness of the initial value of the path parameter independent of the toricity.

💡 Research Summary

The paper addresses the construction of complete non‑compact shrinking gradient Kähler‑Ricci solitons on asymptotically conical (AC) toric manifolds. The authors first show that any toric AC shrinking gradient soliton on a quasi‑projective resolution of a Kähler cone can be expressed as a solution of a complex Monge‑Ampère equation. This is encapsulated in Theorem A, which provides a precise geometric set‑up: a toric Kähler cone (C_{0}) with Reeb vector in the torus Lie algebra, a toric‑equivariant resolution (\pi:M\to C_{0}) with (-K_{M}) (\pi)-ample, and a distinguished holomorphic vector field (JX) uniquely determined by the weighted volume functional on the moment polytope. The theorem also supplies a background Kähler form (\omega) on (M) that is asymptotic to the cone metric and a Hamiltonian potential (f) for (X). Every shrinking soliton then has the form (\omega+i\partial\bar\partial\varphi) with (\varphi) satisfying
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