>>81
>A long \mathbb C^2 without holomorphic functions
>Luka Boc Thaler, Franc Forstneric

むずいが、まずは
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A long C^2 without holomorphic functions
＜結果＞
AI による概要
"A long \(\mathbb{C}^{2}\) without holomorphic functions" refers to a specific type of complex manifold construction in complex geometry, pioneered by papers from Fornaess, Fornæs, & Wold, which creates a space that looks like \(\mathbb{C}^{2}\) (extendable by biholomorphic copies of \(\mathbb{C}^{2}\)) but lacks any non-constant holomorphic or plurisubharmonic functions, contrasting with standard \(\mathbb{C}^{2}\) and solving open problems about distinguishing such spaces using biholomorphic invariants like the "stable core". 
Key Concepts: 
・Long \(\mathbb{C}^{2}\) (or \(\mathbb{C}^{n}\)): A complex manifold \(X\) that can be exhausted by an increasing sequence of biholomorphic copies of \(\mathbb{C}^{n}\) (i.e., \(X\cong \cup _{i=1}^{\infty }\Phi _{i}(\mathbb{C}^{n})\)), meaning it "looks like" \(\mathbb{C}^{n}\) from within but might have global differences.
・Holomorphic Functions: Functions that are complex-differentiable (e.g., \(f(z)\) in one variable, or plurisubharmonic functions in higher dimensions).
・Bi holomorphic Invariants: Properties that remain the same under biholomorphic maps (essentially, structure-preserving maps). 
The Significance of the Result: 
1.Distinguishing Spaces: Before these constructions, it was hard to tell some long \(\mathbb{C}^{n}\)'s apart.
By introducing the stable core, researchers showed there are uncountably many non-equivalent long \(\mathbb{C}^{2}\)'s.
2.Counterexamples: This work provides non-Stein (a type of complex domain) long \(\mathbb{C}^{2}\)'s, answering long-standing questions about the behavior of complex manifolds.
3.No Non-Trivial Functions: These specific long \(\mathbb{C}^{2}\)'s are so "thin" globally that they can't support non-constant holomorphic or plurisubharmonic functions, unlike the standard \(\mathbb{C}^{2}\). 
In essence, the phrase describes a complex space that behaves locally like the familiar complex plane \(\mathbb{C}^{2}\) but possesses exotic global properties, specifically the absence of standard holomorphic functions, challenging our intuition about such spaces.. 
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