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Affine connections on complex compact surfaces and Riccati distributions

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Abstract

Let M be a complex surface. We show that there is a one-to-one correspondence between torsion-free affine connections on M and Riccati distributions on \(\mathbb {P}(TM)\). Furthermore, if M is compact, then this correspondence induces a one-to-one correspondence between affine structures on M and Riccati foliations on \(\mathbb {P}(TM)\). As applications of this result, we classify the regular k-webs on compact complex surfaces for \(k\ge 3\), and we also get a new proof of the classification of regular pencils of foliations on compact complex surfaces.

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References

  1. Barth, W., Peters, C., Van de Ven, A.: Compact Complex Surfaces. Springer, Berlin (1984)

    Book  MATH  Google Scholar 

  2. Beauville, A.: Complex Manifolds with Split Tangent Bundle, Vol. en mémoire de M.Schneider, de Gruyter, à paraître

  3. Bombieri, E., Mumford, D.: Enriques’ Classification of Surfaces in Char, p, II. In: Complex Analysis and Algebraic Geometry, Iwanami Shoten, pp. 23–42 (1977)

  4. Dumitrescu, S.: Connexions affines et projectives sur les surfaces complexes compactes. Math. Zeit. 264, 301–316 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Inoue, M.: On surfaces of class \(V II_0\). Invent. Math. 24, 269–310 (1974)

    Article  MathSciNet  Google Scholar 

  6. Inoue, M., Kobayashi, S., Ochiai, T.: Holomorphic affine connections on compact complex surfaces. J. Fac. Sci. Univ. Tokyo 27, 888 (1980)

    MathSciNet  MATH  Google Scholar 

  7. Falla Luza, M., Loray, F.: Projective structures and neighborhood of rational curves. https://hal.archives-ouvertes.fr/hal-01567347v2

  8. Gunning, R.: Lectures on Riemann Surfaces. Princeton Mathematical Notes, Princeton (1966)

    MATH  Google Scholar 

  9. Kato, M.: On characteristic forms on complex manifolds. J. Algebra 138, 424–439 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  10. Klingler, B.: Structures affines et projectives sur les surfaces complexes. Ann. Inst. Fourier Grenoble 48(2), 441–477 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kobayashi, S., Ochiai, T.: Holomorphic projective structures on compact complex surfaces. Math. Ann. 249, 75–94 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kobayashi, S., Ochiai, T.: Holomorphic projective structures on compact complex surfaces. II. Math. Ann. 255, 519–521 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  13. Kodaira, K.: On the structure of compact complex analytic surfaces I, II, III. Am. J. Math. 86, 751–798 (1964)

    Article  MATH  Google Scholar 

  14. Lins Neto, A.: Curvature of pencil of foliations. Astérisque 296, 167–190 (2004)

    MathSciNet  MATH  Google Scholar 

  15. Maehara, K.: On elliptic surfaces whose first Betti numbers are odd. Intl. Symp. Alg. Geom. Kyoto 2, 565–574 (1977)

    MATH  Google Scholar 

  16. Molzon, R., Pinney, M.K.: The Schwarzian derivate for maps between manifolds with complex projective connections. In: Transactions of the American Mathematical Society, vol. 348, no. 8 (1996)

  17. Pereira, J.V., Pirio, L.: An invitation to web geometry. Publicações Matemáticas do IMPA. Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro (2009)

  18. Puchuri, L.: Classification of flat pencils of foliations on compact complex surfaces. Annales de l’Institut Fourier. Tome 70(5), 2191–2214 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  19. Robert, G.: Poincaré maps and Bol’s Theorem. http://park.itc.u-tokyo.ac.jp/MSF/topology/conference/GHC/data/Robert.pdf

  20. Suwa, T.: Compact quotient spaces of \(\mathbb{C} ^2\) by affine transformation groups. J. Differ. Geom. 10, 239–252 (1975)

    Article  MATH  Google Scholar 

  21. Vitter, A.: Affine structures on compact complex manifolds. Invent. Math. 17, 231–244 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  22. Zhao, S.Y.: Groupes kleiniens birationnels en dimension deux. L’Université de Rennes 1 Thesis 2020. https://perso.univ-rennes1.fr/shengyuan.zhao/index.html/paper/main.pdf

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Acknowledgements

The author wishes to express his deepest gratitude to Frank Loray for lots of fruitful discussions about the content of this paper, and also for reading the drafts and suggesting improvements. The author also wishes to thank Cesar Hilario for the suggestions and comments on the manuscript. The author acknowledge financial support from CAPES/COFECUB (Ma 932/19 ”Feuilletages holomorphes et intération avec la géométrie” / process number 88887.356980/2019-00). The author is grateful to the Institut de Recherche en Mathématique de Rennes, IRMAR and the Université de Rennes 1 for their hospitality and support. The author is supported by FAPERJ (Grant number E-26/010.001143/2019).

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  1. UERJ, Universidade do Estado do Rio de Janeiro, Rua São Francisco Xavier, 524, Maracanã, 20550-900, Rio de Janeiro, Brazil

    Ruben Lizarbe

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  1. Ruben Lizarbe
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Correspondence to Ruben Lizarbe.

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Lizarbe, R. Affine connections on complex compact surfaces and Riccati distributions. manuscripta math. 172, 885–904 (2023). https://doi.org/10.1007/s00229-022-01435-6

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