Non-holomorphic surface bundles and Lefschetz fibrations

Abstract

We show how certain stabilizations produce infinitely many closed oriented four-manifolds, which are the total spaces of genus g surface bundles (resp. Lefschetz fibrations) over genus h surfaces and have non-zero signature, but do not admit complex structures with either orientations, for "most" (resp. all) possible values of g ≥ 3 and h ≥ 2 (resp. g ≥ 2 and h ≥ 0).

Nonholomorphic Lefschetz fibrations with (−1)-sections

“…The fundamental group of the total space X Un of a genus-g Lefschetz fibration in the family in Theorem 1.2 is π 1 (X Un ) = Z⊕Z n . From the work of [30] (see also [6]), the 4-manifold X Un does not carry any complex structure with either orientation. For g ≥ 22, non-holomorphic Lefschetz fibrations with the same property of Theorem 1.2 were constructed in [21] based on the technique of this paper.…”

Small Lefschetz fibrations and exotic 4-manifolds

“…The relations (9) and (16) 2 = 1, which shows that a 1 can be generated by a 2 , b 2 . Thus π 1 (X) is a free abelian group of rank 2 generated by a 2 and b 2 (also by a 2 and b 1 ).…”

“…. , 8, we obtain genus-2 Lefschetz fibrations of the types (12,9), (14,8), (16,7), (18,6), (20,5), (22,4), (24,3), (26,2). They are all minimal by Proposition 6, and once we show that X i is simply-connected, we can once use Theorem 13 again to conclude that X i is an exotic 3CP 2 #(11 + i)CP 2 , for i = 1, .…”

“…With a small variation of the above constructions as in [40,30,9], we can also produce minimal genus-2 Lefschetz fibration with any prescribed abelian group of rank at most 2: Let i For each fixed i = 1, 2, 3, infinitely many of these symplectic fibrations have total spaces which cannot be a complex surface. Here is a quick argument: by the Enrique-Kodaira classification, a complex surface with odd first betti number is either of type VII or elliptic.…”

“…We are thus left with the possible values (8,6), (10,5), (12,4) when b + = 1 and (10, 10), (12,9), (14,8), (16, 7), (18,6), (20,5), (22,4), (24,3), (26,2) when b + = 3. In all these cases s > 0, so the presence of a reducible fiber implies that there exists a homology class with an odd intersection number.…”