New curvature flows in complex geometry

Abstract

The Anomaly flow is a flow of (2, 2)-forms on a 3-fold which was originally motivated by string theory and the need to preserve the conformally balanced property of a Hermitian metric in the absence of a ∂∂-Lemma. It has revealed itself since to be a remarkable higher order extension of the Ricci flow. It has also led to several other curvature flows which may be interesting from the point of view of both non-Kähler geometry and the theory of non-linear partial differential equations. This is a survey of what is known at the present time about these new curvature flows.

Kähler–Ricci flow on blowups along submanifolds

“…On the complex aspect, the Ricci flow preserves the Kähler condition ( [1]) and is reduced to a scalar equation with Monge-Ampère type, which after suitable normalization converges to a solution of the Calabi conjecture ( [28,1]). The non-Kähler analogue of Ricci flow also generates much interest recently, among them are the Chern-Ricci flow ( [25]), the Anomaly flow ( [11]) and etc, and we refer to [10] for a survey on the recent development of non-Kähler geometric flows. The analytic minimal model program, laid out in [20], predicts how the Kähler-Ricci flow behaves on a projective variety.…”

Solutions to the Hull–Strominger System with Torus Symmetry

“…[1]) and Goldstein and Prokushkin showed in [26] that for a Ricci-flat base and an appropriate choice of the principal torus bundle, the total space has trivial canonical bundle and admits a balanced metric. Starting from the result of Goldstein and Prokushkin, Fu and Yau showed that the Hull-Strominger system on some principal torus fibrations on K3 manifolds can be reduced to a complex Monge-Ampère type equation for a scalar function on the base, and solved it by means of hard analytical techniques (see also [46,50]).…”

“…Since then, and the work by Li and Yau [41], the successive studies of different analytical and geometrical aspects of the Hull-Strominger system have had an important influence to non-Kähler complex geometry (see for instance [15,24,50]). Up to now the biggest pool of solutions is provided by the choice of ∇ given by the Chern connection [7,8,[16][17][18][19]23,43,44,[46][47][48][49][50], which includes the first solutions found by Fu, Li, Tseng, and Yau.…”

“…Since then, and the work by Li and Yau [41], the successive studies of different analytical and geometrical aspects of the Hull-Strominger system have had an important influence to non-Kähler complex geometry (see for instance [15,24,50]). Up to now the biggest pool of solutions is provided by the choice of ∇ given by the Chern connection [7,8,[16][17][18][19]23,43,44,[46][47][48][49][50], which includes the first solutions found by Fu, Li, Tseng, and Yau. More recently, new examples of solutions of the Hull-Strominger system on non-Kähler torus bundles over K3 surfaces originally considered by Fu and Yau, with the property that the connection ∇ is Hermitian-Yang-Mills have been constructed in [25].…”

The Hull-Strominger system and the Anomaly flow on a class of solvmanifolds

“…For other relevant results on the Anomaly flow, we refer the reader to [4,12,14,37,39]; while, for some results on the Hermitian curvature flow in the homogeneous setting we refer the reader to [3,10,28,36,35,42,43,46,50].…”