Some differential complexes within and beyond parabolic geometry

Eastwood, Michael; Gover, A. Rod; Neusser, Katharina

doi:10.2969/aspm/08210013

Cited by 13 publications

(27 citation statements)

References 10 publications

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“…This complex was found in the four-dimensional case by Smith in 1976 [14]. In higher dimensions, besides [19], it was also independently found by Eastwood and Seshadri [4] (see also [3,5]) who were motivated by the hyperelliptic complex of Rumin in contact geometry [16].…”

Section: Filtered Cohomologies 31 Elliptic Complexes and Associated supporting

confidence: 57%

Cohomology and Hodge theory on symplectic manifolds: III

Tsai¹,

Tseng²,

Yau³

2016

J. Differential Geom.

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We introduce filtered cohomologies of differential forms on symplectic manifolds. They generalize and include the cohomologies discussed in Paper I and II as a subset. The filtered cohomologies are finite-dimensional and can be associated with differential elliptic complexes. Algebraically, we show that the filtered cohomologies give a two-sided resolution of Lefschetz maps, and thereby, they are directly related to the kernels and cokernels of the Lefschetz maps. We also introduce a novel, non-associative product operation on differential forms for symplectic manifolds. This product generates an A ∞ -algebra structure on forms that underlies the filtered cohomologies and gives them a ring structure. As an application, we demonstrate how the ring structure of the filtered cohomologies can distinguish different symplectic four-manifolds in the context of a circle times a fibered three-manifold.

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Section: Filtered Cohomologies 31 Elliptic Complexes and Associated supporting

confidence: 57%

Cohomology and Hodge theory on symplectic manifolds: III

Tsai¹,

Tseng²,

Yau³

2016

J. Differential Geom.

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“…To see that the complex is exact, if suffices to recall that the spectral sequence { ⋆ E p,q i ,∂ i } converges to the Dolbeault cohomology. This resolution may be thought of as a Dolbeault analog of the Rumin complex [8,29]. A similar argument gives (6.27) for p > 0.…”

Section: Appendix a Examplesmentioning

confidence: 69%

Characteristic cohomology of the infinitesimal period relation

Robles

2016

Asian Journal of Mathematics

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The infinitesimal period relation (also known as Griffiths' transversality) is the system of partial differential equations constraining variations of Hodge structure. This paper presents a study of the characteristic cohomology associated with that system of PDE.Date: May 9, 2018. 2010 Mathematics Subject Classification. 14D07, 32G20. 58A15, 58A17. Key words and phrases. Variation of Hodge structure, infinitesimal period relation (Griffiths' transversality), characteristic cohomology, flag domain.(1) The notationĎ for G C /P comes from Hodge theory: we think ofĎ as the compact dual of a period domain (or, more generally, a Mumford-Tate domain).

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“…The Rumin operator was also a key tool in the construction of a convolution product on smooth and generalized translation invariant valuations on affine space [5,6], the product of smooth valuations on manifolds [3,14] (note that the operator Q played some role in this case), and the operations of push-forward and pull-back of smooth valuations [2]. Furthermore, the Rumin-de Rham complex appears naturally in the study of invariant differential operators on parabolic geometries [9]. More precisely, the symplectic group G = Sp 2n+2 R acts on the unit sphere S 2n+1 , and the stabilizer is a parabolic subgroup P .…”

Section: Introductionmentioning

confidence: 99%