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Symplectic resolutions, Lefschetz property and formality.

Cavalcanti, Gil R. and Fernández, Marisa and Muñoz, Vicente (2008) Symplectic resolutions, Lefschetz property and formality. Advances in Mathematics, 218 (2). pp. 576-599. ISSN 0001-8708

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Abstract

We introduce a method to resolve a symplectic orbifold (M, omega) into a smooth symplectic manifold ((M) over tilde,(omega) over tilde). Then we study how the formality and the Lefschetz property of ((M) over tilde,(omega) over tilde) are compared with that of (M, omega). We also study the formality of the symplectic blow-up of (M, omega) 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 of Babenko and Taimanov.

Item Type:Article
Uncontrolled Keywords:Symplectic resolutions; Symplectic blow-ups; Lefschetz property; Formality
Subjects:Sciences > Mathematics > Geometry
ID Code:17084
References:

I.K. Babenko, I.A. Taimanov, On non-formal simply connected symplectic manifolds, Siberian Math. J. 41 (2000)204–217.

G.R. Cavalcanti, The Lefschetz property, formality and blowing up in symplectic geometry, Trans. Amer. Math. Soc.359 (2007) 333–348.

P. Deligne, P. Griffiths, J. Morgan, D. Sullivan, Real homotopy theory of Kähler manifolds, Invent. Math. 29(1975)245–274.

M. Fernández, V. Muñoz, Formality of Donaldson submanifolds, Math. Z. 250 (2005) 149–175.

M. Fernández, V. Muñoz, An 8-dimensional non-formal simply connected symplectic manifold, Ann. of Math. (2),in press.

P. Griffiths, J.W. Morgan, Rational Homotopy Theory and Differential Forms, Progr. Math., vol. 16, Birkhäuser Boston, Boston, MA, 1981.

S. Halperin, Lectures on minimal models, Mém. Soc. Math. France 230 (1983).

H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II, Ann. of Math. (2) 79 (1964) 109–203,205–326.

D. McDuff, Examples of symplectic simply connected manifolds with no Kähler structure, J.Differential Geom.20(1984) 267–277.

D. McDuff, D. Salamon, Introduction to Symplectic Geometry, second ed., Oxford Math. Monogr., Clarendon,Oxford, 1998.

S.A. Merkulov, Formality of canonical symplectic complexes and Frobenius manifolds, Int. Math. Res. Not. 14 (1998) 727–733.

T.J. Miller, J. Neisendorfer, Formal and conformal spaces, Illinois. J. Math. 22 (1978) 565–580.

K. Niederkrüger, F. Pasquotto, Resolution of symplectic cyclic orbifold singularities, preprint math.SG/0707.4141.

D. Sullivan, Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. 47 (1978) 269–331.

W.P. Thurston, Three-dimensional Geometry and Topology,vol. 1, Princeton Math. Ser., vol. 35, Princeton Univ. Press, Princeton, NJ, 1997.

A. Tralle, J. Oprea, Symplectic Manifolds with no Kähler Structure, Lecture Notes in Math., vol. 1661, Springer–Verlag, Berlin, 1997.

Deposited On:13 Nov 2012 09:53
Last Modified:07 Feb 2014 09:41

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