## 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

### Citations:

Zbl 0609.14023; Zbl 0479.14018; Zbl 0691.57008; Zbl 0691.57007
Full Text: