Minimal genus for 4-manifolds with \(b^{+} = 1\) (Q2797308)

Let \(X\) be a smooth, closed, connected, oriented \(4\)-manifold. For \(A \in H_2(X;\mathbb{Z})/\mathrm{Tor}\), we define \(mg_{X}(A)\) to be the minimal genus of a connected smooth surface in \(X\) representing \(A\). Let \(\Lambda^i = H^i(X;\mathbb{Z})/\mathrm{Tor}\) for \(i = 0, 1, 2, 3,4\), and \(\Lambda=\oplus_{i=0}^4 \Lambda^i\). For the skew-symmetric bilinear form \(T : \Lambda^1 \times \Lambda^1 \to \Lambda^2\), let \(\tilde{b}_1(\Lambda)\) be the rank of \(T\), \(\tilde{\chi} = 2 + b_2(\Lambda)-2\tilde{b}_1(\Lambda)\). Let \(\sigma(\Lambda)\) be the signature of the symmetric bilinear form \(\Gamma : \Lambda^2 \times \Lambda^2 \to \Lambda^4\) \(\cong \mathbb{Z}\). If a class \(c \in \Lambda^2\) satisfies either (I) \(c \cdot c > \sigma(\Lambda)\) or (II) \(c \cdot c \geq 2 \tilde{\chi}(\Lambda) + 3 \sigma (\Lambda)\) and \(c\) pairs non-trivially with \(\mathrm{Im } T\) when \(T\) is non-trivial, then \(c\) is called an adjunction class. For an adjunction class \(c\) and any \(A \in \Lambda^2\), let \(h_c(A) = 1 + (A \cdot A - | c \cdot A|)/2\) if \(A \not= 0\), \(=0\) if \(A = 0\), and let \(h(A)\) be the minimum of \(h_c(A)\) over all adjunction classes \(c\) of \(\Lambda\). For \(X\) with \(b^+(X)=1\), where \(b^+(X)\) is the dimension of the maximal positive definite subspace for \(\Gamma\), the main results of this paper are (1) \(mg_{X}(A) \geq h(A)\) for any \(A \in \Lambda^2\) with \(A \cdot A > 0\), (2) if \(2 \tilde{\chi}(\Lambda) + 3 \sigma(\Lambda) \geq 0\), then \(h(A) \geq 0\) for any \(A \in \Lambda^2\) with \(A \cdot A > 0\) or \(A \cdot A =0\) and \(A\) is primitive, furthermore, there exists a smooth, closed, connected, oriented \(4\)-manifold \(X_{\Lambda}\) such that \(H^*(X_{\Lambda}; \mathbb{Z})/\mathrm{Tor}\) is isomorphic to \(\Lambda\) and, \(h(A) = mg_{X_{\Lambda}}(A)\) whenever \(h(A) \geq 0\).