Regularizing properties of complex Monge–Ampère flows II: Hermitian manifolds

Abstract

Abstract. We prove that a general complex Monge-Ampère flow on a Hermitian manifold can be run from an arbitrary initial condition with zero Lelong number at all points. Using this property, we confirm a conjecture of Tosatti-Weinkove: the ChernRicci flow performs a canonical surgical contraction. Finally, we study a generalization of the Chern-Ricci flow on compact Hermitian manifolds, namely the twisted ChernRicci flow. IntroductionLet (X, g, J) be a compact Hermitian manifold of complex dimension n, that is a compact complex manifold such that J is compatible with the Riemannian metric g. Recently a number of geometric flows have been introduced to study the structure of Hermitian manifolds. Some flows which do preserve the Hermitian property have been proposed by Streets-Tian [ST10, ST11, ST13], Liu-Yang [LY12] and also anomaly flows due to Phong-Picard-Zhang [PPZ15, PPZ16a, PPZ16b] which moreover preserve the conformally balanced condition of Hermitian metrics. Another such flow, namely the Chern-Ricci flow, was introduced by Gill [Gil11] and has been further developed by Tosatti-Weinkove in [TW15]. The Chern-Ricci flow is written aswhere Ric(ω) is the Chern-Ricci form which is defined locally by Assume that there exists a holomorphic map between compact Hermitian manifolds π : X → Y blowing down an exceptional divisor E on X to one point y 0 ∈ Y . In addition, assume that there exists a smooth function ρ on X such thatwith T < +∞. Tosatti and Weinkove proved:Observe that ω T induces a singular metric ω ′ on Y which is smooth in Y \ {y 0 }. In this note, we confirm this conjecture 1 . An essential ingredient of its proof is to prove that the Monge-Ampère flow corresponding to the Chern-Ricci flow can be run from a rough data. By generalizing a result of Székelyhidi-Tosatti [SzTo11], Nie [Nie14] has proved this property for compact Hermitian manifolds of vanishing first Bott-Chern class and continous initial data. In this paper, we generalize the previous results of Nie [Nie14] and the author [Tô16] by considering the following complex Monge-Ampère flow:where (θ t ) t∈[0,T ] is a family of Hermitian forms with θ 0 = ω and F is a smooth function on R × X × R. Theorem A. Let ϕ 0 be a ω-psh function with zero Lelong number at all points. Let (t, z, s) → F (t, z, s) be a smooth function on [0, T ] × X × R such that ∂F/∂s ≥ 0 and ∂F/∂t is bounded from below. Then there exists a family of smooth strictly is continuous. This family is moreover unique if ∂F/∂t is bounded and ∂F/∂s ≥ 0.The following stability result is a straighforward extension of [Tô16, Theorem 4.3, 4.4].1 After this paper was completed, the author learned that Xiaolan Nie proved the first statement of the conjecture for complex surfaces (cf. [Nie17]). She also proved that the Chern-Ricci flow can be run from a bounded data. The author would like to thank Xiaolan Nie for sending her preprint. REGULARIZING PROPERTIES OF COMPLEX MONGE-AMPÈRE FLOWS II 3Theorem B. Let ϕ 0 , ϕ 0,j be ω-psh functions with zero Lelong number at all points, such that ϕ 0,...

Pluripotential Chern-Ricci Flows

“…This result can be seen as a generalization of the main result of [TW15,Tô18]. It encompasses the case of smooth parabolic Monge-Ampère equations on mildly singular compact hermitian varieties, as well as more degenerate settings, hermitian analogues of the main results of [ST17,BG13].…”

“…The weak Chern-Ricci flow smoothes out the initial current in the sense that the flow becomes smooth on the nonsingular part of Y once t > 0 and the evolving metrics always admit bounded local potentials for any t ∈ [0, T max ). In particular, the smoothing property of the Chern-Ricci flow holds when Y is a compact complex manifold (see Section 4 or [TW15,Tô18]).…”

Continuous solutions to Monge–Ampère equations on Hermitian manifolds for measures dominated by capacity

“…In the context of parabolic equations the pluripotential estimates are also useful. For example, To [41] (independently, Nie [31] in particular cases) used results in [11,32] to prove a conjecture by Tosatti and Weinkove [38]. The geometric applications of pluripotential theory on Hermitian manifolds are discussed at length in surveys by Dinew [9,10].…”

Weak solutions of the Chern–Ricci flow on compact complex surfaces

“…After this note was completed, the author learned that related results were proved by Tat Dat Tô in [17]. The author would like to thank Tat Dat Tô for sending his preprint.…”