This paper addresses the construction of new symplectic 4-manifolds with non-negative signatures and the smallest Euler characteristics.
The classical geography problem in algebraic geometry, initially posed by Persson, seeks to determine which ordered pairs of positive integers can be realized as \((\chi_h(M),c_1^2(M))\) for some minimal complex surface \(M\) of general type. Here, \(\chi_h(M)\) represents the holomorphic Euler characteristic of \(M\), while \(c_1^2(M)\) denotes the square of the first Chern class of \(M\). The related botany problem, significantly more challenging, aims to classify all minimal complex surfaces with a given pair of invariants \((\chi_h(M),c_1^2(M))\).
The symplectic geography problem, first posed by McCarthy and Wolfson in 1994, investigates which ordered pairs of integers can be realized as \((\chi_h(M),c_1^2(M))\) for some minimal symplectic 4-manifold \(M\). Recent years have seen steady progress on this problem, with a complete solution achieved for simply connected minimal symplectic 4-manifolds with negative signatures (cf. Akhmedov et al. 2010; Akhmedov and Park 2010; Park and Szabó 2000). However, the symplectic botany problem, involving the classification of minimal symplectic 4-manifolds with a given pair of invariants \((\chi_h(M),c_1^2(M))\), remains significantly more challenging.
The work presented in this paper builds upon several previous results concerning the geography of symplectic 4-manifolds (see, for example, the works by A. Akhmedov and B. D. Park [Invent. Math. 181, No. 3, 577–603 (2010; Zbl 1206.57029); Canad. Math. Bull. 64, No. 2, 418–428 (2021; Zbl 1470.57041)] and references therein). It is motivated by the works of D. Cartwright et al. [J. Algebraic Geom. 26, No. 4, 655–689 (2017; Zbl 1375.14120)], and G. Prasad and S.-K. Yeung [Invent. Math. 168, No. 2, 321–370 (2007; Zbl 1253.14034)]. Specifically, the authors construct:

1.

infinitely many irreducible symplectic and infinitely many non-symplectic 4-manifolds that are homeomorphic but not diffeomorphic to \((2n-1)\mathbb{CP}^2\#(2n-1)\overline{\mathbb{CP}}^2\) for each integer \(n \geq 9\) (Theorem 1.1);

2.

families of simply connected irreducible nonspin symplectic and non-symplectic 4-manifolds that have the smallest Euler characteristics among all known simply connected 4-manifolds with positive signatures and with more than one smooth structure (Theorem 1.2);

3.

a new complex surface with positive signature which arises, from Hirzebruch’s line-arrangement surfaces, as a complex ball quotient.

In particular, these manifolds represent new points on the geography chart, meaning that no such manifolds with the given invariants and properties were known before this work. However, the main symplectic geography problem remains open, as the existence of exotic \(\mathbb{CP}^2\) is still unknown.
If \(X\) is a complex surface of general type with \(c_1^2(X)= 9 \chi_h(X)\), then the universal cover of \(X\) is biholomorphic to the open unit 4-ball \(B^4\) in \(\mathbb C^2\). \(X\) is thus a quotient of \(B^4\) in \(\mathbb C^2\) by an infinite discrete group and is therefore called a complex ball quotient. These are fundamental in the construction of the symplectic 4-manifolds given in the main theorems, serving as the building blocks used in the proofs of the theorems. Sections 2 and 3 are dedicated to obtaining such building blocks, including fake projective planes, normal covers of Cartwright-Steger surfaces, and surfaces obtained from product symplectic 4-manifolds \(\Sigma_g \times \Sigma_h\), with \(g \geq 1\) and \( h \geq 0\), undergoing Luttinger surgeries. Finally, in Sections 4 and 5, using the constructions from the previous sections, the authors prove the main results