Hermitian Curvature Flow on Compact Homogeneous Spaces
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Hermitian Curvature Flow on Compact Homogeneous Spaces Francesco Panelli1 · Fabio Podestà1 Received: 29 March 2019 © Mathematica Josephina, Inc. 2019
Abstract We study a version of the Hermitian curvature flow on compact homogeneous complex manifolds. We prove that the solution has a finite extinction time T > 0 and we analyze its behavior when t → T . We also determine the invariant static metrics and we study the convergence of the normalized flow to one of them. Keywords Lie group actions · Homogeneous complex manifolds · Hermitian curvature flow Mathematics Subject Classification 53C25 · 53C30
1 Introduction Given a Hermitian manifold (M, J , h), it is well known that there exists a family of metric connections leaving the complex structure J parallel (see [6]). Among these, the Chern connection is particularly interesting and provides different Ricci tensors which can be used to define several meaningful parabolic metric flows preserving the Hermitian condition and generalizing the classical Ricci flow in the non-Kähler setting. In [7], Gill introduced an Hermitian flow on a compact complex manifold involving the first Chern–Ricci tensor, namely, the one whose associated 2-form represents the first Chern class of M (see also [14] for further related results). In [13], Streets and Tian introduced a family of Hermitian curvature flows (HCFs) involving the second Ricci tensor S together with an arbitrary symmetric Hermitian form Q(T ) which is quadratic in the torsion T of the Chern connection:
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Fabio Podestà [email protected] Francesco Panelli [email protected]
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Dipartimento di Matematica e Informatica ”Ulisse Dini”, Università di Firenze, V.le Morgagni 67/A, 50100 Firenze, Italy
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F. Panelli, F. Podestà
h t = −S + Q(T ). For any admissible Q(T ), the corresponding flow is strongly parabolic and the shorttime existence of the solution is established. This family includes geometrically interesting flows as for instance the pluriclosed flow that was previously introduced in [12] and preserves the pluriclosed condition ∂∂ω = 0. More recently Ustinovskiy focused on a particular choice of Q(T ) obtaining another remarkable flow, which we will call HCFU for brevity, with several geometrically relevant features (see [15]). In particular, Ustinovskiy proves that the HCFU on a compact Hermitian manifold preserves Griffiths non-negativity of the Chern curvature, generalizing the classical result that Kähler–Ricci flow preserves the positivity of the bisectional holomorphic curvature (see e.g., [10]). In [17], the author could prove stronger results showing that the HCFU preserves several natural curvature positivity conditions besides Griffiths positivity. In [16], Ustinovskiy focuses on complex homogeneous manifolds and proves that the finite-dimensional space of induced metrics (which are not necessarily invariant) is preserved by the HCFU . Given a connected complex Lie group G acting transitively, effectively, and holomorphically on a complex manifold M, a simple observation shows that M d
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