Simply connected \(4\)-manifolds near the Bogomolov-Miyaoka-Yau line (Q1574675)

I. The geography of compact complex surfaces (of general type) is the pair of integer lattices corresponding to minimal models of existing complex surfaces with coordinates \((x, y) = (\chi_h, c_1^2)\), where \(\chi_h\) is the holomorphic Euler characteristic and \(c_1^2\) is the square of the first Chern class of the complex surface. The same geography pictures can be extended for minimal symplectic 4-manifolds and irreducible 4-manifolds \(X\) via the relations \[ \chi_h(X) = (e(X) + \sigma (X))/2,\quad c_1^2(X) = 3\sigma (X) + 2e(X), \] where \(e(X)\) is its Euler characteristic and \(\sigma (X)\) is its signature. II. The basic questions in the geography theory are (a) (Existence) Which lattice points \((m,n)\) occur for complex surfaces, minimal symplectic 4-manifolds or irreducible 4-manifolds ? (b) (Uniqueness) How many inequivalent complex surfaces, minimal symplectic 4-manifolds, irreducible 4-manifolds (up to deformations, symplectic isotopy and diffeomorphisms, smooth structures respectively) exist for a single lattice point? (c) (Territory) What is the domain in \(Z \times Z\) representing complex surfaces, minimal symplectic manifolds and irreducible 4-manifolds ? There are several important drawings in the geography: (i) the Noether line, (ii) the Bogomolov-Miyaoka-Yau line \(y=9x\) (BMY-line), (iii) the \(11/8\)th line. III. Some known results: (1) Donaldson showed first examples of simply connected irreducible 4-manifolds without smooth structure, or with many inequivalent smooth structures; (2) Gompf and Mrowka showed first examples of minimal symplectic 4-manifolds without complex structures; (3) Szabó showed first examples of irreducible simply connected 4-manifolds without symplectic structures; (4) McMullen and Taubes showed first examples of irreducible smooth 4-manifolds with two inequivalent symplectic structures. IV. Basic Constructions: (a) the rational blowdown by Fintushel-Stern allows to move the lattice point vertically up by one unit; (b) the fiber sum along self-intersection zero torus allows to move the lattice point horizontally by \(+\) one unit. Fintushel and Stern, using (a) and (b), provided examples for the above mentioned results. V. The paper under review is to answer question (a) for minimal symplectic 4-manifolds with lattice points nearby the BMY-line \((y = 9x)\). The author first constructed some complex surfaces on the BMY-line. Let \(H(1)\) be a 5-fold cyclic branched cover of \(\Sigma_2 \times \Sigma_2 \# 3 \overline{CP^2}\) along five complex curves, where \(\Sigma_2\) is a complex curve of genus 2, and \(H(n)\) be an \(n\)-fold cover of \(H(1)\). The surfaces \(\{H(n)\}_{n \in Z}\) all lie on the BMY-line. Then the author constructed the fiber the sum of \(H(n)\) with a Lefschetz fibration \(X\), constructed the normal connected sum \(X_d\) of the resulting 4-manifold from the fiber sum with \((CP^2 \# h\overline{CP^2}, \widetilde{C}_d)\), where \(\widetilde{C}_d\) is a complex curve with genus \(n+1\), \(\widetilde{C}_d \cdot \widetilde{C}_d = n+1\) and \(n+1 = (d-1)(d-2)/2\). The symplecticity is preserved by Gompf's result, and the topological calculations are \[ \begin{aligned} \chi_h(X_d) &= 13d^2 -39d + 16 + \chi_h(X),\\ c_1^2(X_d) &= 116d^2 - 351d+130 + c_1^2(X). \end{aligned} \] For sufficiently large \(d\), \(X_d\) is the required symplectic 4-manifold which lies in between the BMY-line \(y=9x\) and the line \(y=8.9x\).