Curvature and smooth topology in dimension four (Q2765849)

It is well know that for a Riemannian manifold of dimension four, the curvature tensor consists of the self-dual and the anti-self-dual Weyl curvatures \(W_{\pm}\), the scalar curvature \(S\), and the trace-free part \(r^{\circ}\) of the Ricci curvature.NEWLINENEWLINENEWLINERemarkable Seiberg-Witten theory leads to intimately interplay between Riemannian geometry and smooth topology in dimension four. In particular the Seiberg-Witten equations give a lower bound for the \(L^2\)-norm of the scalar curvature, and similar estimates for other parts of the curvature tensor. The present paper is devoted to applications of Seiberg-Witten invariants in the theory of Einstein manifolds. More precisely, let \({\chi}(M)\) and \({\tau}(M)\) be the signature and the Euler characteristic of \(M\) respectively. One knows that a necessary condition for the existence of an Einstein metric on a compact oriented 4-manifold is, that the Hitchin-Thorpe inequality \(2{\chi}(M){\geq} 3|{\tau}(M)|\) must hold, while there is valid NEWLINE\[NEWLINE (2{\chi}{\pm}3{\tau})(M)= \frac{1}{4{\pi}^2}{\int}_M\left(\frac{s^2}{24} +2|W_{\pm}|^2- \frac{|r^{\circ}|^2}{2}\right) d{\mu}.NEWLINE\]NEWLINE Using the estimations of various parts of the curvature tensor derived from the Seiberg-Witten invariants, the author obtains:NEWLINENEWLINENEWLINEIf \((X, J_X)\) is a compact complex surface with \(b_+>1\) and \((M, J_M)\) is obtained from \(X\) by blowing up \(k\) points, then the smooth compact 4-manifold \(M\) does not admit any Einstein metric if \(k>C^2_1(X)/3\), where \(b_+\) is the number of positive eigenvalues of the Hodge star duality operator of \(X\), and \(C_1\) is the first Chern class of \(X\).NEWLINENEWLINEFor the entire collection see [Zbl 0973.00041].