Real homotopy theory of K�hler manifolds

Deligné, Pierre; Griffiths, Phillip; Morgan, John; Sullivan, Dennis

doi:10.1007/bf01389853

Cited by 745 publications

(953 citation statements)

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“…There are many such varieties that are not smooth. The main result of [4] follows from Theorem 5. The following interesting corollary of Theorem 5 and a result from [9] was pointed out to me by S. Halperin.…”

Section: Theorem 3 ( Cf [12] When V Is Smooth) If (V X) Is a Pointmentioning

confidence: 93%

Mixed Hodge structures on homotopy groups

Hain¹

1986

Bull. Amer. Math. Soc.

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In [3] Deligne defined mixed Hodge structures (M.H.S.'s) and showed that the cohomology of every algebraic variety over C has a natural M.H.S. Morgan [12], using Sullivan's minimal models, showed that the rational homotopy Lie algebra and rational homotopy type of every smooth variety have natural M.H.S.'s. In this note we announce an extension of mixed Hodge theory to arbitrary varieties and homotopy fibers of morphisms between varieties. The latter is a major step in extending asymptotic Hodge theory to homotopy groups and periods of iterated integrals. The bar construction and KuoTsai Chen's iterated integrals [1] provide the link between Hodge theory and homotopy groups. Some of the results announced have been distributed in preprint form [7]. Proofs of the results stated will be published elsewhere.Because the higher homotopy groups of a non-nilpotent topological space are inaccessible to rational homotopy theory, we make the following definition. The homotopy Lie algebra of a pointed topological space (X, x) is the graded Lie algebra 0.(X, x) where 0o(^> z) is the Malcev Lie algebra associated with 7Ti(X, x) and, when k > 1,The class of nilpotent spaces includes simply connected spaces and topological groups. There is a Hurewicz homomorphism If (V, x) is simply connected, then the M.H.S. on 7Tfc(V, x) does not depend on the basepoint x. However, if V is not simply connected, this is not the case.

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Section: Theorem 3 ( Cf [12] When V Is Smooth) If (V X) Is a Pointmentioning

confidence: 93%

Mixed Hodge structures on homotopy groups

Hain¹

1986

Bull. Amer. Math. Soc.

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“…A free G.A. means a tensor product of a polynomial algebra of even dimensional elements and an exterior algebra of odd dimensional elements, and we denote by A (*,,..., x") the free algebra generated by (x,,... Sullivan [2,6] showed that rational homotopy types are in 1-1 correspondence with minimal models of D.G.A. via the ^-polynomial differential forms.…”

Section: Q-mentioning

confidence: 99%

“…Rational homotopy types are homotopy types of localized spaces at zero. In [2,6] Sullivan constructed the theory of minimal models which algebraically describes rational homotopy types.…”

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confidence: 99%

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Graded algebras having a unique rational homotopy type

Shiga¹,

Yagita²

1982

Proc. Amer. Math. Soc.

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“…those compact 2n-dimensional symplectic manifolds (M, κ) for which the maps [κ] p : H n−p (M, R) → H n+p (M, R) , 0 p n are isomorphisms. A classical result states that, if (M, κ, J) is a compact Kähler manifold, then (M, κ) satisfies the HLC (see [7]), and ∧ * (M ) is a formal DGA; moreover, HLC symplectic manifolds possess some of the cohomological properties of a Kähler manifold (e.g. the odd Betti numbers b 2p+1 (M ) are even, b p (M ) b p+2 (M ) , 0 p < n−1).…”

Section: Introductionmentioning

confidence: 99%