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On homomorphic sections of a certain Kodaira surface revisited. (English) Zbl 1222.32028
Ohnita, Yoshihiro (ed.) et al., Riemann surfaces, harmonic maps and visualization. Proceedings of the 16th Osaka City University International Academic Symposium, Osaka, Japan, December 15–20, 2008. Osaka: Osaka Municipal Universities Press (ISBN 978-4-901409-69-8/hbk). OCAMI Studies 3, 149-160 (2010).
Summary: In [Kodai Math. J. 32, No. 3, 450–470 (2009; Zbl 1192.14030)] we analysed the number of holomorphic sections of a holomorphic family of genus two surfaces whose complex structure was originally studied by G. Riera [Duke Math. J. 44, 291–304 (1977; Zbl 0361.32014)]. Our aim was to give a precise estimation of the number of holomorphic sections, whose finiteness was alreadey known as the Mordell conjecture in the function field case. In this note, we review the present state of our results and discuss them more carefully than in [the authors, loc. cit.]. We give a simple proof for a result about the order of the local monodromy around a puncture, which is crucial for our sections to be extendable at the puncture.
For the entire collection see [Zbl 1203.00026].
32G15 Moduli of Riemann surfaces, Teichm├╝ller theory (complex-analytic aspects in several variables)
30F10 Compact Riemann surfaces and uniformization
32H02 Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables
55R05 Fiber spaces in algebraic topology