Surface bundles: Some interesting examples (Q2719818)

Two constructions for surface bundles \(X\rightarrow \Sigma _g\) with nonzero signature are given. The first construction is given by the following result: For all sufficiently large \(g\) there exists a 4-manifold \(X\) which carries a surface bundle structure \(f:X\rightarrow \Sigma _g\) with fiber genus 5 and has positive signature. The second construction is provided by: There exist smooth algebraic surfaces \(X_n\) that have signature \(8n(n-1)(n+1)/3\) and admit two smooth fibrations \(X_n\rightarrow B\) and \(X_n\rightarrow \overline B\) so that the pair \((g,h)\) given by the base genus and fiber genus of the fibration are \((3,3n^3-n^2+1)\) and \((2n^2+1,3n)\) respectively.