Barlow surface

In mathematics, a Barlow surface is one of the complex surfaces discovered by Rebecca Barlow (1984, 1985). They are simply connected surfaces of general type with p_g = 0. They are homeomorphic but not diffeomorphic to a projective plane blown up in 8 points. The Hodge diamond for the Barlow surfaces is:

		1
	0		0
0		9		0
	0		0
		1

Detection of exotic smooth structures

According to Freedman's classification, a simply connected oriented closed topological 4-manifold with intersection form $[+1]\oplus 8[-1]$ is homeomorphic to $\mathbb {C} P^{2}\#8{\overline {\mathbb {C} P^{2}}}$. Any such smooth manifold with a non-vanishing Seiberg–Witten invariant has an exotic smooth structure since $\mathbb {C} P^{2}\#8{\overline {\mathbb {C} P^{2}}}$ with the induced smooth structure allows a Riemannian metric of positive scalar curvature (psc metric) and therefore has only vanishing Seiberg–Witten invariants.

See also

• Hodge theory

References

• Barlow, Rebecca (1984), "Some new surfaces with $p_{g}=0$", Duke Mathematical Journal, 51 (4): 889–904, doi:10.1215/S0012-7094-84-05139-1, ISSN 0012-7094, MR 0771386
• Barlow, Rebecca (1985), "A simply connected surface of general type with $p_{g}=0$", Inventiones Mathematicae, 79 (2): 293–301, doi:10.1007/BF01388974, ISSN 0020-9910, MR 0778128
• Barth, Wolf P.; Hulek, Klaus; Peters, Chris A.M.; Van de Ven, Antonius (2004), Compact Complex Surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., vol. 4, Springer-Verlag, Berlin, ISBN 978-3-540-00832-3, MR 2030225
• Kotschick, Dieter (1989), "On manifolds homeomorphic to $\mathbb {CP} ^{2}\#8{\overline {\mathbb {CP} }}^{2}$", Inventiones Mathematicae, 95 (3): 591–600, doi:10.1007/BF01393892, ISSN 0020-9910, MR 0979367