Higher type adjunction inequalities for Donaldson invariants (Q2719038)

The paper under review presents a result parallel to one obtained by Ozsváth and Szabó using the Seiberg-Witten invariants [\textit{P. Ozsváth} and \textit{Z. Szabó}, Higher type adjunction inequalities in Seiberg-Witten theory, math.DG/0005268]. The algebraic structure of the Fukaya-Floer homology ring of a surface \(\Sigma\) times a circle is obtained using the Donaldson invariants. Let \(X\) denote a closed oriented 4-manifold with \(b_2^+(X) > 1\) and \(\Sigma \subset X\) an embedded surface of genus \(g\) with \(\Sigma^2 = 0\) and \(\Sigma\) representing an odd homology class. Let \(b \in C{(x^2-4)} \otimes \wedge^*H_1(\Sigma)\) for \(x\in H_0(\Sigma)\). If \(K\) is a basic class for \(D_X^{\omega}(b \cdot)\) [\textit{P. B. Kronheimer} and \textit{T. S. Mrowka}, J. Differ. Geom. 95, No. 3, 573-734 (1995; Zbl 0842.57022)], then NEWLINE\[NEWLINE|K \cdot \Sigma|+ \Sigma^2 + d(b) \leq 2g -2. NEWLINE\]NEWLINE Note that \(\Sigma^2 > 0\) can be reduced to \(\tilde{\Sigma} = \Sigma - E_1 - \cdots - E_n \subset X \# n \overline{CP^2}\) by Fintushel and Stern's blow-up formula [\textit{R. Fintushel} and \textit{R. J. Stern}, Ann. Math. (2) 143, No. 3, 529-546 (1996; Zbl 0869.57019)]. The method used in the paper under review is standard in gauge theory. One decomposes \(X = (X \smallsetminus N(\Sigma))\cup_Y N(\Sigma)\), where \(N(\Sigma)\) is a tubular neighborhood of \(\Sigma\) with \(\partial N(\Sigma) = Y = \Sigma \times S^1\), and the relative Donaldson invariant lies in the (effective) Fukaya-Floer homology. Analyzing the pairing of relative Donaldson invariants, one gains some vanishing result for the constraint on the degree of \(b\). Section 2 gives a refinement of the author's previous work and describes the Artin ring structure on \(HFF^*_g\). Section 3 computes the Floer homology of \(\Sigma \times S^1\) and compares it with the ring structure obtained in section 2. Understanding the degree of the homogeneous components of the relations in the Artinian decomposition of \(HFF^*_g\) plays a key role in the proofs of the claimed results. The statements and formulae in section 2, section 3 and section 4 are only valid if the genus of the surface \(\Sigma\) is bigger than or equal to 2. For \(g(\Sigma) =1\), \(HF^*(\Sigma \times S^1)\) is generated by two nontrivial elements with degree differ by 4, and many formulae are needed to be adjusted. In particular, no Artinian decomposition makes sense for \(HFF^*_1\).