## 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.

### MSC:

 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
Full Text: