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

Authors

Anar Akhmedov, University of MinnesotaFollow
Sümeyra Sakalli, University of Arkansas, FayettevilleFollow
Sai-Kee Yeung, Purdue UniversityFollow

Document Type

Article

Publication Date

2-2024

Keywords

complex ball quotients; exotic smooth structures; symplectic surgeries

Abstract

We construct new symplectic 4-manifolds with non-negative signatures and with the smallest Euler characteristics, using fake projective planes, Cartwright– Steger surfaces and their normal covers and product symplectic 4-manifolds Σ_g × Σ_h, where g ≥ 1 and h ≥ 0, along with exotic symplectic 4-manifolds constructed in [7, 12]. In particular, our constructions yield to (1) infinitely many 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, (2) infinite families of simply connected irreducible nonspin symplectic and such infinite families of non-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.

Citation

Akhmedov, A., Sakalli, S., & Yeung, S. (2024). Complex Ball Quotients and New Symplectic 4-manifolds with Nonnegative Signatures. Taiwanese Journal of Mathematics, 28 (1), 29-53. https://doi.org/10.11650/tjm/230905