Nonisotopic symplectic tori in the fiber class of elliptic surfaces. (English) Zbl 1078.57025

For a complex surface, there are at most finitely many smooth complex curves (up to smooth isotopy) in a given homology class. However, there is no analogous finiteness for symplectic two-dimensional surfaces. Namely, R. Fintushel and R. Stern [J. Differ. Geom. 52, No. 2, 203–222 (1999; Zbl 0981.53085)] gave examples of 4-dimensional simply connected symplectic manifolds \(M\) and homology classes \(\alpha\in H_2(M)\) that can be realized by infinitely many mutually non-isotopic symplectic tori. However, it was not clear whether this can happen for primitive classes \(\alpha\).
The author constructs an infinite family of mutually non-isotopic symplectic tori realizing an arbitary multiple \(q[F]\) of the homology class \([F]\) of the fiber \(F\) of an elliptic surface \(E(n), n\geq 3\). Compare also with the author’s paper [Proc. Am. Math. Soc. 133, No. 8, 2477–2481 (2005; Zbl 1073.57016)].


57R17 Symplectic and contact topology in high or arbitrary dimension
53D35 Global theory of symplectic and contact manifolds
57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
57R57 Applications of global analysis to structures on manifolds
57R95 Realizing cycles by submanifolds
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