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A note on o-minimal flows and the Ax-Lindemann-Weierstrass theorem for semi-abelian varieties over \(\mathbb{C}\). (English) Zbl 1402.14060

Summary: In this short note we present an elementary proof of Theorem 1.2 from [E. Ullmo and A. Yafaev, Q. J. Math. 68, No. 2, 359–367 (2017; Zbl 1386.14165)], and also the Ax-Lindemann-Weierstrass theorem for abelian and semi-abelian varieties. The proof uses ideas of Pila, Ullmo, Yafaev, Zannier (see, e.g., [J. Pila and U. Zannier, Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 19, No. 2, 149–162 (2008; Zbl 1164.11029)]) and is based on basic properties of sets definable in o-minimal structures. It does not use the Pila-Wilkie counting theorem.

MSC:

14K12 Subvarieties of abelian varieties
03C64 Model theory of ordered structures; o-minimality
11J81 Transcendence (general theory)
14P15 Real-analytic and semi-analytic sets
32B15 Analytic subsets of affine space
03C98 Applications of model theory
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References:

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