The partial 𝐶⁰-estimate along the continuity method

Abstract

Abstract. We prove that the partial C 0 -estimate holds for metrics along Aubin's continuity method for finding Kähler-Einstein metrics, confirming a special case of a conjecture due to Tian. We use the method developed in recent work of Chen-Donaldson-Sun on the analogous problem for conical Kähler-Einstein metrics.

Kähler–Einstein metrics along the smooth continuity method

“…This was established in [42] in the case when v = 0, using the method in Chen-DonaldsonSun [19], and it was shown by Phong-Song-Sturm [36] for Kähler-Ricci solitons (i.e.…”

“…We can write p = lim p k for p k ∈ M k , such that for a sufficiently small r > 0 the balls B d k (p k , r), scaled to unit size are very close in the Gromov-Hausdorff sense to the unit ball in a cone C n−1 × C γ , for large k. As discussed in [42], based on the ideas in [19], this implies that we have biholomor-…”

“…We briefly describe the argument for the sake of completeness. Just as in the case T < 1, we have embeddings F t : M → P N using suitable orthonormal bases for H 0 (K −m M ) with respect to the metric ω t , for some large m. The partial C 0 -estimate is still valid, in the Kähler-Einstein case by [42] based on the method in [20], and in the soliton case due to Jiang-Wang-Zhu [31]. It follows that as before, up to increasing m and choosing a sequence t k → 1 we have the algebraic convergence F t k (M ) → W ∈ P N to a normal Q-Fano variety, homeomorphic to the Gromov-Hausdorff limit (Z, d Z ) of the sequence (M, ω t k ).…”

“…In particular we obtain a new proof of the result of Chen-Donaldson-Sun [20], without using metrics with conical singularities. At the same time our arguments are analogous to those in [20], using also the adaptation of some of those ideas to the smooth continuity method in [42]. A key advantage of the smooth continuity path is that it allows one to work in a G-equivariant setting.…”

“…We give some examples of the applications of our results to toric manifolds and other manifolds of large symmetry group in Section 4. In Section 5 we discuss how to adapt the methods of [42] and [36] to obtain the partial C 0 -estimates along the continuity method for solitons. A crucial point is the reductivity of the automorphism group of the limiting variety.…”