Odd sphere bundles, symplectic manifolds, and their intersection theory

Tanaka, Hiro; Tseng, Li-Sheng

doi:10.4310/cjm.2018.v6.n3.a1

Cited by 7 publications

(6 citation statements)

References 17 publications

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“…To show this, we recall a result of Tanaka-Tseng [5] that the primitive TTY algebra is A ∞quasi-isomorphic to the cone algebra C * (M ) = Ω * (M ) ⊕ θ Ω * (M ) where dθ = ω. Furthermore, when ω is integral and we can consider a circle bundle X over M whose Euler class is ω, then the de Rham DGA of the circle bundle Ω * (X) is quasi-isomorphic to both the C * (M ) algebra and the primitive TTY algebra.…”

Section: Examples and Properties Of Symplectically Flat Bundlesmentioning

confidence: 99%

Symplectic Flatness and Twisted Primitive Cohomology

Tseng

Zhou

2022

J Geom Anal

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We introduce the notion of symplectic flatness for connections and fibre bundles over symplectic manifolds. Given an $$A_\infty $$ A ∞ -algebra, we present a flatness condition that enables the twisting of the differential complex associated with the $$A_\infty $$ A ∞ -algebra. The symplectic flatness condition arises from twisting the $$A_\infty $$ A ∞ -algebra of differential forms constructed by Tsai, Tseng and Yau. When the symplectic manifold is equipped with a compatible metric, the symplectic flat connections represent a special subclass of Yang–Mills connections. We further study the cohomologies of the twisted differential complex and give a simple vanishing theorem for them.

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Section: Examples and Properties Of Symplectically Flat Bundlesmentioning

confidence: 99%

Symplectic Flatness and Twisted Primitive Cohomology

Tseng

Zhou

2022

J Geom Anal

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“…It is worthwhile to point out that the second functional also has an interpretation as the normed square of the cone curvature. The cone algebra of interest here consists of elements [7] for a discussion of this cone algebra in the context of symplectic geometry). This algebra is quasi-isomorphic to the algebra of differential forms of the prequantum circle bundle X, Ω * (X), when ω is integral class.…”

Section: Introductionmentioning

confidence: 99%

Symplectically Flat Connections and Their Functionals

Tseng¹

2022

Preprint

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We continue our study of symplectically flat bundles. We broaden the notion of symplectically flat connections on symplectic manifolds to ζ-flat connections on smooth manifolds.These connections on principal bundles can be represented by maps from an extension of the base's fundamental group to the structure group. We introduce functionals with zeroes being symplectically flat connections and study their critical points. Such functionals lead to novel geometric flows. We also describe some characteristic classes of the ζ-flat bundles.

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“…Such structures have recently become of interest in the study of Calabi-Yau A ∞ algebras used in topological conformal field theory [Cos07]. In particular, it was shown in [TT18] that the Calabi-Yau A ∞ algebras studied in [TTY16] are equivalent to the standard de Rham differential graded algebra on certain odd dimensional sphere bundles. For this among other reasons, we feel that a concerted study of Sasaki geometry on sphere bundles is warranted.…”

Section: Introductionmentioning

confidence: 99%

Sasakian Geometry on Sphere Bundles

Boyer¹,

Tønnesen-Friedman²

2020

Preprint

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The purpose of this paper is to study the Sasakian geometry on odd dimensional sphere bundles over a smooth projective algebraic variety N with the ultimate, but probably unachievable goal of understanding the existence and non-existence of extremal and constant scalar curvature Sasaki metrics. We apply the fiber join construction of Yamazaki [Yam99] for K-contact manifolds to the Sasaki case. This construction depends on the choice of d + 1 integral Kähler classes [ω j ] on N that are not necessarily colinear in the Kähler cone. We show that the colinear case is equivalent to a subclass of a different join construction orginally described in [BG00a, BGO07], applied to the spherical case by the authors in [BTF14, BTF16] when d = 1, and known as cone decomposable [BHLTF18]. The non-colinear case gives rise to infinite families of new inequivalent cone indecomposable Sasaki contact CR on certain sphere bundles. We prove that the Sasaki cone for some of these structures contains an open set of extremal Sasaki metrics and, for certain specialized cases, the regular ray within this cone is shown to have constant scalar curvature. We also compute the cohomology groups of all such sphere bundles over a product of Riemann surfaces.

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Odd sphere bundles, symplectic manifolds, and their intersection theory

Cited by 7 publications

References 17 publications

Symplectic Flatness and Twisted Primitive Cohomology

Symplectic Flatness and Twisted Primitive Cohomology

Symplectically Flat Connections and Their Functionals

Sasakian Geometry on Sphere Bundles

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