arXiv math/04030672004-03math.SGmath.DG

The Lefschetz property, formality and blowing up in symplectic geometry

Gil R. Cavalcanti

备注：: 19 pages. Minor changes in the text, typos corrected. To appear in Trans. Amer. Math. Soc
发表信息：: Trans. Amer. Math. Soc. 359 (2007), 333--348.

英文摘要：

In this paper we study the behaviour of the Lefschetz property under the blow-up construction. We show that it is possible to reduce the dimension of the kernel of the Lefschetz map if we blow up along a suitable submanifold satisfying the Lefschetz property. We use that, together with results about Massey products, to construct nonformal (simply connected) symplectic manifolds satisfying the Lefschetz property.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Formality and the Lefschetz property in symplectic and cosymplectic geometry

We review topological properties of Kähler and symplectic manifolds, and of their odd-dimensional counterparts, coKähler and cosymplectic manifolds. We focus on formality, Lefschetz property and parity of Betti numbers, also distinguishing the simply-connected case (in the Kähler/symplectic situation) and the $b_1=1$ case (in the coKähler/cosymplectic situation).

2015-04-09math.ATmath.DGmath.SG

Symplectic resolutions, Lefschetz property and formality

We introduce a method to resolve a symplectic orbifold into a smooth symplectic manifold. Then we study how the formality and the Lefschetz property of the symplectic resolution are compared with that of the symplectic orbifold. We also study the formality of the symplectic blow-up of a symplectic orbifold along symplectic submanifolds disjoint from the orbifold singularities. This allows us to construct the first example of a simply connected compact symplectic manifold of dimension 8 which satisfies the Lefschetz property but is not formal, therefore giving a counter-example to a conjecture …

2007-10-03math.ATmath.DGmath.SG

Minimal models, formality and hard Lefschetz properties of solvmanifolds with local systems

For a simply connected solvable Lie group G with a cocompact discrete subgroup Γ, we consider the space of differential forms on the solvmanifold G/Γ with values in certain flat bundle so that this space has a structure of a differential graded algebra(DGA). We construct Sullivan's minimal model of this DGA. This result is an extension of Nomizu's theorem for ordinary coefficients in the nilpotent case. By using this result, we refine Hasegawa's result of formality of nilmanifolds and Benson-Gordon's result of hard Lefschetz properties of nilmanifolds.

2012-08-09math.CVmath.GRmath.GT

Formality of Donaldson submanifolds

We introduce the concept of s-formal minimal model as an extension of formality. We prove that any orientable compact manifold M, of dimension 2n or (2n-1), is formal if and only if M is (n-1)-formal. The formality and the hard Lefschetz property are studied for the symplectic manifolds constructed by Donaldson with asymptotically holomorphic techniques. This study permits us to show an example of a Donaldson symplectic manifold of dimension eight which is formal simply connected and does not satisfy the hard Lefschetz theorem.

2007-05-21math.ATmath.SG

Equivariant formality of transversely symplectic foliations and Frobenius manifolds

Consider the Hamiltonian action of a compact connected Lie group on a transversely symplectic foliation which satisfies the transverse hard Lefschetz property. We establish an equivariant formality theorem and an equivariant symplectic $dδ$-lemma in this setting. As an application, we show that if the foliation is also Riemannian, then there exists a natural formal Frobenius manifold structure on the equivariant basic cohomology of the foliation.

2018-08-13math.SG

Formality of canonical symplectic complexes and Frobenius manifolds

It is shown that the de Rham complex of a symplectic manifold $M$ satisfying the hard Lefschetz condition is formal. Moreover, it is shown that the differential Gerstenhaber-Batalin-Vilkoviski algebra associated to such a symplectic structure gives rise, along the lines explained in the papers of Barannikov and Kontsevich [alg-geom/9710032] and Manin [math/9801006], to the structure of a Frobenius manifold on the de Rham cohomology of $M$.

1998-05-19math.SG

更多相关论文收起更多论文

Asymptotically holomorphic theory for symplectic orbifolds

We extend Donaldson's asymptotically holomorphic techniques to symplectic orbifolds. More precisely, given a symplectic orbifold such that the symplectic form defines an integer cohomology class, we prove that there exist sections of large tensor powers of the prequantizable line bundle such that their zero sets are symplectic suborbifolds. We then derive a Lefschetz hyperplane theorem for these suborbifolds, that computes their real cohomology up to middle dimension. We also get the hard Lefschetz and formality properties for them, when the ambient manifold satisfies those properties.

2022-02-18math.ATmath.DGmath.SG