## Constructing symplectic forms on 4–manifolds which vanish on circles.(English)Zbl 1054.57027

Given a smooth, closed, oriented 4-manifold $$X$$ and $$\alpha \in H_2(X,\mathbb{Z})$$ such that $$\alpha.\alpha > 0$$, a closed 2-form $$\omega$$ is constructed, Poincaré dual to $$\alpha$$, which is symplectic on the complement of a finite set of unknotted circles. The number of circles, counted with sign, is given by $$d = (c_1(s)^2 -3\sigma(X) -2\chi(X))/4$$, where $$s$$ is a certain spin$$^\mathbb C$$-structure naturally associated to $$\omega$$.

### MSC:

 57R17 Symplectic and contact topology in high or arbitrary dimension 57M50 General geometric structures on low-dimensional manifolds 32Q60 Almost complex manifolds

### Keywords:

symplectic; 4-manifold; spin$$^\mathbb C$$; almost complex; harmonic
Full Text:

### References:

 [1] C Bohr, Embedded surfaces and almost complex structures, Proc. Amer. Math. Soc. 128 (2000) 2147 · Zbl 0954.53022 [2] F Ding, H Geiges, A Legendrian surgery presentation of contact 3-manifolds, Math. Proc. Cambridge Philos. Soc. 136 (2004) 583 · Zbl 1069.57015 [3] F Ding, H Geiges, A I Stipsicz, Surgery diagrams for contact 3-manifolds, Turkish J. Math. 28 (2004) 41 · Zbl 1077.53071 [4] Y Eliashberg, Classification of overtwisted contact structures on 3-manifolds, Invent. Math. 98 (1989) 623 · Zbl 0684.57012 [5] Y Eliashberg, Topological characterization of Stein manifolds of dimension $$>2$$, Internat. J. Math. 1 (1990) 29 · Zbl 0699.58002 [6] D T Gay, Explicit concave fillings of contact three-manifolds, Math. Proc. Cambridge Philos. Soc. 133 (2002) 431 · Zbl 1012.57039 [7] D T Gay, Open books and configurations of symplectic surfaces, Algebr. Geom. Topol. 3 (2003) 569 · Zbl 1035.57015 [8] N Goodman, Contact structures and open books, PhD thesis, University of Texas, Austin (2003) [9] R E Gompf, Handlebody construction of Stein surfaces, Ann. of Math. $$(2)$$ 148 (1998) 619 · Zbl 0919.57012 [10] R E Gompf, $$\mathrm{Spin}^c$$-structures and homotopy equivalences, Geom. Topol. 1 (1997) 41 · Zbl 0886.57021 [11] F Hirzebruch, H Hopf, Felder von Flächenelementen in 4-dimensionalen Mannigfaltigkeiten, Math. Ann. 136 (1958) 156 · Zbl 0088.39403 [12] K Honda, An openness theorem for harmonic 2-forms on 4-manifolds, Illinois J. Math. 44 (2000) 479 · Zbl 0970.58001 [13] K Honda, Local properties of self-dual harmonic 2-forms on a 4-manifold, J. Reine Angew. Math. 577 (2004) 105 · Zbl 1065.53066 [14] K Honda, On the classification of tight contact structures I, Geom. Topol. 4 (2000) 309 · Zbl 0980.57010 [15] G Mikhalkin, $$J$$-holomorphic curves in almost complex surfaces do not always minimize the genus, Proc. Amer. Math. Soc. 125 (1997) 1831 · Zbl 0866.57025 [16] R Scott, Closed self-dual two-forms on four-dimensional handlebodies, PhD thesis, Harvard University (2003) [17] C H Taubes, The structure of pseudo-holomorphic subvarieties for a degenerate almost complex structure and symplectic form on $$S^1\times B^3$$, Geom. Topol. 2 (1998) 221 · Zbl 0908.53013 [18] A Weinstein, Contact surgery and symplectic handlebodies, Hokkaido Math. J. 20 (1991) 241 · Zbl 0737.57012
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.