Non-Kähler Calabi-Yau geometry and pluriclosed flow

Abstract

Hermitian, pluriclosed metrics with vanishing Bismut-Ricci form give a natural extension of Calabi-Yau metrics to the setting of complex, non-Kähler manifolds, and arise independently in mathematical physics. We reinterpret this condition in terms of the Hermitian-Einstein equation on an associated holomorphic Courant algebroid, and thus refer to solutions as Bismut Hermitian-Einstein. This implies Mumford-Takemoto slope stability obstructions, and using these we exhibit infinitely many topologically distinct complex manifolds in every dimension with vanishing first Chern class which do not admit Bismut Hermitian-Einstein metrics. This reformulation also leads to a new description of pluriclosed flow in terms of Hermitian metrics on holomorphic Courant algebroids, implying new global existence results, in particular on all complex non-Kähler surfaces of Kodaira dimension κ ≥ 0. On complex manifolds which admit Bismut-flat metrics we show global existence and convergence of pluriclosed flow to a Bismut-flat metric, which in turn gives a classification of generalized Kähler structures on these spaces.

Harmonic metrics for the Hull-Strominger system and stability

“…An open question which we have not been able to solve is to establish a more clear relation between the Dorfman bracket [, ] on Q and the orthogonal connection D G , in a way that our new stability condition is formulated more naturally in terms of the triple (Q, , , [, ]). Even though our picture is mostly conjectural, we expect that this stability condition, along with the notion of harmonic metric that we introduce, will lead to new obstructions to the existence of solutions in future studies, similarly as in [16].…”

Pluriclosed flow and Hermitian-symplectic structures

“…Meanwhile, they [17,18,19] prove it is a strictly parabolic systems under the pluriclosed assumption and give the short time existence and some basic regularity results. More results about regularity and long-time existence can be found in [6,9,13,14,15].…”

“…For any selection of (2, 0) form ϕ 0 , it is a flow of Hermitian-symplectic forms with initial data Ω 0 = ϕ 0 + ω 0 + φ0 . The idea of flowing a (2, 0) form by the (2, 0)-part of Bismut-Ricci form along the pluriclosed flow first appears in [13] (can also see [6,7]). They consider the system consisting of pluriclosed flow and an evolution equation of (2, 0) form…”

On the pluriclosed flow on Oeljeklaus–Toma manifolds

“…, ω |t=0 = ω . The pluriclosed flow was deeply studied in literature, see for instance [3,5,6,9,12,19,22,23,24,25,26,27] and the references therein.…”