2024/06/21 by Zahraa Khaled, Khaled, Zahraa, Andrei Teleman +1
Mathematics · #32J15 #32Q57 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #Complex Variables (math.CV) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2406.15158
openalex publication_date 2024/06/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove that any Inoue surface admits a unique holomorphic connection. Using this result we show that two Inoue surfaces S=H×ℂ/G, S'=H×ℂ/G' are biholomorphic if and only if G, G' are conjugate in the group of affine transformations of H×ℂ. This result allows us to prove explicit classification theorems for Inoue surfaces: Let M be the set of \rm SL(3,ℤ)-matrices M with a real eigenvalue α>1 and two non-real eigenvalues, and N^± the set of \rm GL(2,ℤ)-matrices N with a real eigenvalue α>1 and det(N)=± 1. We prove that: For any \rm GL(3,ℤ)-similarity class \mathfrakM∈ M/∼, there exists exactly two biholomorphism classes of type I Inoue surfaces. For any \rm GL(2,ℤ) similarity class \mathfrakN=[N]∈ N+/∼ and positive integer r∈ℕ^*, we have a finite set of deformation classes of type II Inoue surfaces. This set is parameterised by the quotient of ℤ2/(I2-N)ℤ2+rℤ2 by an action of the "positive centraliser" Z+_\rm GL(2,ℤ)(N) of N in \rm GL(2,ℤ). The set of biholomorphism types corresponding to a deformation class, endowed with its natural topology, can be identified with either ℂ^* or ℂ. For any \rm GL(2,ℤ)-similarity class \mathfrakN=[N]∈ N-/∼ and positive integer r∈ℕ^*, we have a finite set of biholomorphism classes of type III Inoue surfaces. This set is parameterised by the quotient of ℤ2/(I2+N)ℤ2+rℤ2 by an action of Z+_\rm GL(2,ℤ)(N). In both cases the group Z+_\rm GL(2,ℤ)(N) is infinite cyclic (see section 5).