The Kähler-Ricci flow, Ricci-flat metrics and collapsing limits

Abstract

We investigate the Kähler-Ricci flow on holomorphic fiber spaces whose generic fiber is a Calabi-Yau manifold. We establish uniform metric convergence to a metric on the base, away from the singular fibers, and show that the rescaled metrics on the fibers converge to Ricci-flat Kähler metrics. This strengthens previous work of Song-Tian and others. We obtain analogous results for degenerations of Ricci-flat Kähler metrics.

On a twisted conical Kähler–Ricci flow

“…Items (2.1) and (2.2) in Theorem 1.2 can be seen as generalizations of Chen-Wang [5] and Shen [24] to the volume collapsing case. Of course, Theorem 1.2 are also generalizations of results on the Kähler-Ricci flow [29,30,9,35] to the conical setting.…”

“…In this section, we will prove C 0 -convergence away from cone divisor D. The strategy used here is taken from Tosatti-Weinkove-Yang [35].…”

“…h is an uniform upper bound for bisectional curvature of χ on Y \ D ′ and we have used tr ω(t) f * χ is uniformly bounded by Lemma 3.1 (note that χ and χ * is uniformly equivalent). Then one can apply arguments in [35,Lemma 3.4] to conclude (5.10).…”

“…Proof of Claim. We make use of an argument in [35] to conclude. Suppose there exists a sequence (x k , t k ) ∈ (X \ D) × [1, ∞) with t k → ∞ as k → ∞ such that…”

The Anomaly Flow over Riemann Surfaces

“…The metrics on the base will converge smoothly to a metric ωΣ which can be associated to the vanishing spinorial pair (Σ, ϕ). Theorem 1.1 can be compared to the phenomenon of collapsing in the Kähler-Ricci flow, as pioneered by Song-Tian [46,47] and further explored by several others [53,18,48,55,22,59]. In this case, there is a general theory of collapsing of Calabi-Yau fibrations over Kähler manifolds B.…”