Geography of simply connected nonspin symplectic 4-manifolds with positive signature

Abstract

We construct new families of closed simply connected nonspin irreducible symplectic 4-manifolds with positive signature that are interesting with respect to the geography problem. Note that e(M) and σ (M) are in turn completely determined by χ h (M) and c 2 1 (M), that is, e(M) = 12χ h (M) − c 2 1 (M) and σ (M) = c 2 1 (M) − 8χ h (M). When M is a complex surface, χ h (M) is the holomorphic Euler characteristic of M while c 2 1 (M) is the square of the first Chern class of M. The classical "geography problem" in algebraic geometry, originally posed by Persson [1981], asks which ordered pairs of positive integers can be realized as the pair (χ h (M), c 2 1 (M)) for some minimal complex surface M of general type. The related "botany problem", which is a lot more difficult, asks for the classification of all minimal complex surfaces with a given pair of invariants (χ h , c 2 1). The symplectic geography problem, first posed in [McCarthy and Wolfson 1994], asks which ordered pairs of integers can be realized as (χ h (M), c 2 1 (M)) for some minimal symplectic 4-manifold M. There has been steady progress on the symplectic geography problem in recent years and the problem has been completely solved for simply connected minimal symplectic 4-manifolds with negative signature

On the geography of simply connected nonspin symplectic 4-manifolds with nonnegative signature

“…In terms of the symplectic geography problem, the work in [3,6] concluded that there exists an irreducible symplectic 4-manifold and infinitely many irreducible non-symplectic 4-manifolds with odd intersection form that realize the following coordinates (χ, c 2 1 ) when 0 ≤ c 2 1 < 8χ. A similar results for the nonnegative signature case were obtained in [7,4]. We would like to remark that throughout this paper, we consider the geography problem for non-spin symplectic and smooth 4-manifolds.…”

“…The next theorem will be used to produce an infinite family of pairwise nondiffeomorphic, but homeomorphic simply connected 4-manifolds. The following corollary follows from the above Theorems, and proof can be found in [4]. Corollary 3.8.…”

“…It can be shown that p is branched exactly in D given by (4). Since D has simple normal crossings, the total space S is smooth.…”

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In [7,4], the first author and his collaborators constructed the irreducible symplectic 4-manifolds that are homeomorphic but not diffeomorphic to (2n − 1)CP 2 #(2n − 1)CP 2 for each integer n ≥ 25, and the families of simply connected irreducible nonspin symplectic 4manifolds with positive signature that are interesting with respect to the symplectic geography problem. In this paper, we improve the main results in [7,4].

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Reducible smooth structures on 4-manifolds with zero signature

“…It was shown in [2] that for every integer q 25, there exists an irreducible symplectic 4manifold Y (q) that is homeomorphic but not diffeomorphic to (2q − 1)(CP 2 #CP 2 ). Moreover, it was shown in [1] that each Y (q) contains a symplectic torus T of self-intersection 0 such that π 1 (Y (q) \ T ) = 0. Similarly, it was shown in [3] that for every integer p 138, there exists an irreducible symplectic 4-manifold Z(p) that is homeomorphic but not diffeomorphic to (2p − 1)(S 2 × S 2 ) and contains a symplectic torus T of self-intersection 0 such that π 1 (Z(p) \ T ) = 0.…”