Symplectic forms and surfaces of negative square. (English) Zbl 1120.53052

Let \((M, \omega)\) be a symplectic \(4\)-manifold and let \(C^2\) be an embedded symplectic surface realizing the homology class \(e\). If \(e\cdot e\geq 0,\) then for all \(t>0\) the class \([\omega]+t PD(e)\) contains symplectic forms given by the inflation construction of F. Lalonde and D. McDuff [Math. Res. Lett. 3, No. 6, 769–778 (1996; Zbl 0874.57019)].
The authors introduce the analogue of the inflation technique when \(e\cdot e=-k<0\) and use it to obtain new symplectic forms. Namely, put \(a=\omega(e)\), put \(h=k+1\) if \(C\) is a sphere and \(k\) is odd, and put \(h=k\) in all other cases. The authors show that for all \(t\in [0, \frac{2 a}{h})\) there exists a symplectic form realizing \([\omega]+t PD(e).\)
They remark that, in a sense, this result is best possible. This holds since for any triple \((g, k, a)\) with \(g\geq 0, k>0, a>0\) and any \(\varepsilon>0\) there exist symplectic \(4\)-manifolds containing symplectic surfaces \(C\) of genus \(g\) with \(e\cdot e=-k, a=\omega(e)\) such that \([\omega]+(\frac{2a}{h}+\varepsilon) PD(e)\) is not realizable by a symplectic form.
Let \((M,J)\) be a Kähler surface and \(H^{1,1}_J\) be the real part of the \((1,1)\)-subspace of \(H^2(M,\mathbb C)\) determined by \(J.\) The positive cone of \(H^{1,1}_J\) is the subset of \(H^{1,1}_J\) formed by all classes that have positive squares and pair positively with the class of the Kähler form. The authors pose a question if the following symplectic Nakai-Moishezon criterion is true: every class in the positive cone of \(H^{1,1}_J\) can be realized by a symplectic form provided that the class is positive on all \((-1)\)-spheres.


53D35 Global theory of symplectic and contact manifolds
32Q15 Kähler manifolds
32Q65 Pseudoholomorphic curves
57R17 Symplectic and contact topology in high or arbitrary dimension


Zbl 0874.57019
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