# Positive scalar curvature in dimension 4

Let $M^n$ be a compact simply connected spin manifold.

Gromov, Lawson, and Stolz proved that if $n\geq 5$, then $M$ admits a metric of positive scalar curvature iff $\alpha(M)=0$.

Question: What happens in dimension 4?

1. Are there compact simply connected spin manifolds of dimension 4 which have a metric of positive scalar curvature and $\alpha(M)\neq 0$?
2. Are there compact simply connected spin manifolds of dimension 4 which have $\alpha(M)=0$ but no metric of positive scalar curvature?
3. What are necessary and/or sufficient conditions for a compact simply connected spin 4-manifold to have a metric with positive scalar curvature?

$$1.$$ Lichnerowicz proved that if $$M$$ is a closed spin manifold which admits a positive scalar curvature metric, then $$\hat{A}(M) = 0$$. In dimensions $$4k$$, $$\alpha(M) = 2\hat{A}(M)$$, so it follows that $$\alpha(M) = 0$$. In fact, Hitchin proved that $$\alpha(M) = 0$$, regardless of dimension.
$$2.$$ In dimension four, $$\hat{A}(M) = -\frac{1}{24}p_1(M) = -\frac{1}{8}\tau(M)$$, so $$\alpha(M) = 0$$ if and only if $$\tau(M) = 0$$. If $$M$$ is a simply connected, closed, spin four-manifold with signature zero, then by Freedman's classification, there is an integer $$k \geq 0$$ such that $$M$$ is homeomorphic to $$k(S^2\times S^2)$$, the connected sum of $$k$$ copies of $$S^2\times S^2$$; note, the connected sum of zero copies of $$S^2\times S^2$$ is defined to be $$S^4$$. If $$M$$ is diffeomorphic to $$k(S^2\times S^2)$$, then $$M$$ admits positive scalar curvature metrics. Said another way, the topological manifolds $$k(S^2\times S^2)$$ all admit positive scalar curvature for their standard smooth structure.
A closed four-manifold can potentially admit a countably infinite number of smooth structures, and the existence of a positive scalar curvature metric depends on which smooth structure you choose. So it is possible that for some choice of $$k$$ and some choice of smooth structure on $$k(S^2\times S^2)$$, there does not admit a positive scalar curvature metric. Such examples exist. In Hirzebruch Surfaces: Degenerations, Related Braid Monodromy, Galois Covers (MR 1720873), Teicher constructs examples of simply connected general-type complex surfaces which are spin and have signature zero; see Theorem $$5.8$$. These are homeomorphic to $$k(S^2\times S^2)$$ for some $$k$$, but general type surfaces do not admit positive scalar curvature metrics.
$$3.$$ In dimension four, unlike in other dimensions, there are gauge theoretic obstructions to the existence of positive scalar curvature metrics. In particular, on a closed smooth four-manifold with $$b^+ \geq 2$$, the existence of a positive scalar curvature metric implies that all of its Seiberg-Witten invariants vanish. This is a necessary condition, but it is not sufficient. Note that these invariants are usually too complicated to calculate, except in ideal situations such as when the four-manifold in question is the underlying four-manifold of a Kähler surface.
• I think that the link http://www.ams.org.proxy.library.stonybrook.edu/books/conm/241/3642/conm241-3642.pdf only works for people from this particular university. Would http://www.ams.org/books/conm/241/3642/conm241-3642.pdf be the correct link? Although looking at links from ams.org/books/conm/241 (which is accessible to me), it seems that links to individual articles are behind paywall. Commented Aug 17, 2019 at 7:03