Complex Ball Quotients and New Symplectic 4-manifolds with Nonnegative Signatures

2021/02/10 by Anar Akhmedov, Akhmedov, Anar, Sümeyra Sakallı +3
Mathematics · #Geometric and Algebraic Topology #Algebraic Geometry and Number Theory #Advanced Combinatorial Mathematics

paper · pdf · doi:10.48550/arxiv.2102.05265

Abstract

We present the various constructions of new symplectic 4-manifolds with non-negative signatures using the complex surfaces on the BMY line c₁² = 9χ_h, the Cartwright-Steger surfaces, the quotients of Hirzebruch's certain line-arrangement surfaces, along with the exotic symplectic 4-manifolds constructed in \citeAP2, AS. In particular, our constructions yield to (i) an irreducible symplectic and infinitely many non-symplectic 4-manifolds that are homeomorphic but not diffeomorphic to (2n-1)CP²#(2n-1)CP² for each integer n ≥ 9, (ii) the families of simply connected irreducible nonspin symplectic 4-manifolds that have the smallest Euler characteristics among the all known simply connected 4-manifolds with positive signatures and with more than one smooth structure. We also construct a complex surface with positive signature from the Hirzebruch's line-arrangement surfaces, which is a ball quotient.

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