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Symplectic topology on algebraic 3-folds. (English) Zbl 0810.53021

This paper contributes to the classification of symplectic structures on real 6-manifolds. If \(V\) is a differentiable manifold, the set of all symplectic structures on \(V\) is an open set in the space of closed 2- forms on \(V\). The quotient \(\mathcal W\) of this open set by the group of orientation preserving diffeomorphisms is the moduli space of symplectic structures on \(V\).
The main aim of the author is to distinguish different components of \(\mathcal W\). He calls two symplectic structures on \(V\) deformation equivalent iff they are in the same component of \(\mathcal W\). He focuses on 6-manifolds, because their smooth classification is easy (in contrast to the 4-dimensional case). More precisely, he works with three-dimensional complex Kähler manifolds. Any two Kähler forms on a complex manifold are deformation equivalent (as symplectic forms).
As his main tool to distinguish different components of \(\mathcal W\) he uses a Donaldson-type invariant which was introduced by the author in [Topological sigma models and Donaldson type Gromov invariants (Preprint)]. He proves the following theorem:
Let \(V\) be a nonminimal algebraic surface and \(W\) be a minimal one. The blow up of \(k \geq 0\) distinct points in \(V\) (resp. \(W\)) will be denoted by \(V_ k\) (resp. \(W_ k\)). Let \(\Sigma\) be a Riemannian surface. If the first Pontryagin class of \(W_ k\) does not vanish, then \(V_ k \times \Sigma\) and \(W_ k \times \Sigma\) are not deformation equivalent.
Combining this with the results of Z. Chen [Math. Ann. 277, 141-164 (1987; Zbl 0609.14023)] and U. Persson [Compos. Math. 43, 3-58 (1981; Zbl 0479.14018)] on the geography of simple connected minimal surfaces of general type, he obtaines the following existence result:
For any \(n\) there exists a simply connected minimal algebraic surface of general type \(W\) such that \(W \times \Sigma\) admits at least \(n\) distinct deformation classes of symplectic structures (which can not be distinguished by “classical” invariants).
It is interesting to apply his results on the Barlow surface \(B\), which is a minimal surface of general type with \(c_ 1^ 2 = 1\). This surface is homeomorphic to the blow up \(V\) of \(\mathbb{P}_ 2\) at 8 points in general position, whereas from Donaldson’s theory it is known that \(V\) and \(B\) are not diffeomorphic [see D. Kotschick, Invent. Math. 95, No. 3, 591-600 (1989; Zbl 0691.57008) and C. Okonek and A. van de Ven, ibid., 601-614 (1989; Zbl 0691.57007)]. On the other hand, \(V \times \Sigma\) and \(B \times \Sigma\) are diffeomorphic. But their exoticness can still be detected by symplectic topology: They are not deformation equivalent as symplectic manifolds.

MSC:

53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53C55 Global differential geometry of Hermitian and Kählerian manifolds
14J30 \(3\)-folds
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