Generalized Kähler geometry and the pluriclosed flow

Streets, Jeffrey; Тиан, Ганг

doi:10.1016/j.nuclphysb.2012.01.008

Cited by 67 publications

(69 citation statements)

References 17 publications

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“…We conclude this section with a remark about generalized Kähler structures. Recall that a generalized Kähler manifold is a Riemannian manifold

(M^{2 n}, g)

together with two

g

‐compatible complex structures

J_{+}, J_{-}

satisfying

d_{+}^{c} ω_{+} = - d_{-}^{c} ω_{-} = : H; d H = 0,

(see, for example, ). Here,

d_{\pm}^{c} = \sqrt{- 1} false({\bar{\partial}}_{\pm} - \partial_{\pm} false) = {(- 1)}^{r} J_{\pm} d_{\pm} J_{\pm}

r

‐forms, with

false(J α false) false(\cdot, \dots, \cdot false) = {(- 1)}^{r} α false(J \cdot, \dots, J \cdot false)

for an

r

‐form

α

.…”

Section: Almost‐abelian Solvmanifoldsmentioning

confidence: 99%

“…(see, for example, [15,32]). Here, d c ± = √ −1(∂ ± − ∂ ± ) = (−1) r J ± d ± J ± on r-forms, with (Jα)(·, .…”

Section: The Lemma Now Follows Since As(a) + S(a)a + a T S(a) = S(aamentioning

confidence: 99%

“…In recent years, the pluriclosed flow has been an active subject of study, and already many regularity and convergence results have been established , as well as connections with generalized Kähler geometry . Remarkably, still not much is known in the homogeneous case.…”

Section: Introductionmentioning

confidence: 99%

“…If ω 0 is not Kähler, a key point in considering this connection instead of the Levi-Civita connection of g is that the complex structure and the pluriclosed condition are preserved along the flow. In recent years, the pluriclosed flow has been an active subject of study, and already many regularity and convergence results have been established [29,33], as well as connections with generalized Kähler geometry [32]. Remarkably, still not much is known in the homogeneous case.…”

Section: Introductionmentioning

confidence: 99%

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The long‐time behavior of the homogeneous pluriclosed flow

Arroyo

Lafuente

2019

Proc. London Math. Soc.

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We study the asymptotic behavior of the pluriclosed flow in the case of left‐invariant Hermitian structures on Lie groups. We prove that solutions on 2‐step nilpotent Lie groups and on almost‐abelian Lie groups converge, after a suitable normalization, to self‐similar solutions of the flow. Given that the spaces are solvmanifolds, an unexpected feature is that some of the limits are shrinking solitons. We also exhibit the first example of a homogeneous manifold on which a geometric flow has some solutions with finite extinction time and some that exist for all positive times.

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“…We conclude this section with a remark about generalized Kähler structures. Recall that a generalized Kähler manifold is a Riemannian manifold

(M^{2 n}, g)

together with two

g

‐compatible complex structures

J_{+}, J_{-}

satisfying

d_{+}^{c} ω_{+} = - d_{-}^{c} ω_{-} = : H; d H = 0,

(see, for example, ). Here,

d_{\pm}^{c} = \sqrt{- 1} false({\bar{\partial}}_{\pm} - \partial_{\pm} false) = {(- 1)}^{r} J_{\pm} d_{\pm} J_{\pm}

r

‐forms, with

false(J α false) false(\cdot, \dots, \cdot false) = {(- 1)}^{r} α false(J \cdot, \dots, J \cdot false)

for an

r

‐form

α

.…”

Section: Almost‐abelian Solvmanifoldsmentioning

confidence: 99%

“…(see, for example, [15,32]). Here, d c ± = √ −1(∂ ± − ∂ ± ) = (−1) r J ± d ± J ± on r-forms, with (Jα)(·, .…”

Section: The Lemma Now Follows Since As(a) + S(a)a + a T S(a) = S(aamentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The long‐time behavior of the homogeneous pluriclosed flow

Arroyo

Lafuente

2019

Proc. London Math. Soc.

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“…It evolves an initial pluriclosed form ω 0 via ∂ t ω t = −(ρ B ) 1,1 , where ρ B is the Ricci-form of the Bismut connection. Some regularity results involving the PCF are known [43,45,48], the PCF preserves the generalized Kähler condition, and is a powerful tool in generalized geometry [3,42,43,47]. In the case of homogeneous Hermitian structures, the PCF on Lie groups was initially studied in [17], where it is proved that the flow on 2-step nilpotent SKT Lie groups has always a long-time solution (see also [40]).…”

Section: Introductionmentioning

confidence: 99%

Hermitian curvature flow on unimodular Lie groups and static invariant metrics

Lafuente

Pujia

Vezzoni

2020

Trans. Amer. Math. Soc.

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We investigate the Hermitian curvature flow (HCF) of left-invariant metrics on complex unimodular Lie groups. We show that in this setting the flow is governed by the Ricciflow type equation ∂tgt = −Ric 1,1 (gt). The solution gt always exist for all positive times, and (1 + t) −1 gt converges as t → ∞ in Cheeger-Gromov sense to a non-flat left-invariant soliton (Ḡ,ḡ). Moreover, up to homotheties on each of these groups there exists at most one left-invariant soliton solution, which is a static Hermitian metric if and only if the group is semisimple. In particular, compact quotients of complex semisimple Lie groups yield examples of compact non-Kähler manifolds with static Hermitian metrics. We also investigate the existence of static metrics on nilpotent Lie groups and we generalize a result in [16] for the pluriclosed flow. In the last part of the paper we study HCF on Lie groups with abelian complex structures.

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Classification of Generalized Kähler‐Ricci Solitons on Complex Surfaces

Streets

Ustinovskiy

2020

Comm Pure Appl Math

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Using toric geometry we give an explicit construction of the compact steady solitons for pluriclosed flow first constructed in by the first author in 2019. This construction also reveals that these solitons are generalized Kähler in two distinct ways, with vanishing and nonvanishing Poisson structure. This gives the first examples of generalized Kähler structures with nonvanishing Poisson structure on nonstandard Hopf surfaces, completing the existence question for such structures. Moreover, this gives a complete answer to the existence question for generalized Kähler-Ricci solitons on compact complex surfaces. In the setting of generalized Kähler geometry with vanishing Poisson structure, we show that these solitons are unique. We show that these solitons are global attractors for the generalized Kähler-Ricci flow among metrics with maximal symmetry.

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Generalized Kähler geometry and the pluriclosed flow

Cited by 67 publications

References 17 publications

The long‐time behavior of the homogeneous pluriclosed flow

The long‐time behavior of the homogeneous pluriclosed flow

Hermitian curvature flow on unimodular Lie groups and static invariant metrics

Classification of Generalized Kähler‐Ricci Solitons on Complex Surfaces

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