## 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)].

### MSC:

 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

### Keywords:

symplectic tori; elliptic surfaces

### Citations:

Zbl 0981.53085; Zbl 1073.57016
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