no. 1
Holomorphic maps between moduli spaces
Antonakoudis, Stergios1; Aramayona, Javier2; Souto, Juan3
1 University of Cambridge Wilberforce Road Cambridge, CB3 0WB (UK)
2 Universidad Autónoma de Madrid & ICMAT Campus Cantoblanco UAM Nicolás Cabrera, 13-15 28049 Madrid (Spain)
3 Université de Rennes 1 Rue du Thabor 35000 Rennes (France)
Annales de l'Institut Fourier, Volume 68 (2018) no. 1, pp. 217-228.

We prove that every non-constant holomorphic map ℳ g,p →ℳ g ' ,p ' between moduli spaces of Riemann surfaces is a forgetful map, provided that g≥6 and g ' ≤2g-2.

Nous démontrons que toute application non-constante et holomorphe ℳ g,p →ℳ g ' ,p ' entre deux espaces de modules est une application d’oubli, à condition que g≥6 et g ' ≤2g-2.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/aif.3158
Classification: 57M50, 32H02
Keywords: Moduli spaces, holomorphic map, forgetful map
Mots-clés : Espaces de modules, application holomorphe, application d’oubli

Antonakoudis, Stergios 1; Aramayona, Javier 2; Souto, Juan 3

1 University of Cambridge Wilberforce Road Cambridge, CB3 0WB (UK)
2 Universidad Autónoma de Madrid & ICMAT Campus Cantoblanco UAM Nicolás Cabrera, 13-15 28049 Madrid (Spain)
3 Université de Rennes 1 Rue du Thabor 35000 Rennes (France)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
@article{AIF_2018__68_1_217_0,
     author = {Antonakoudis, Stergios and Aramayona, Javier and Souto, Juan},
     title = {Holomorphic maps between moduli spaces},
     journal = {Annales de l'Institut Fourier},
     pages = {217--228},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {68},
     number = {1},
     year = {2018},
     doi = {10.5802/aif.3158},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3158/}
}
TY  - JOUR
AU  - Antonakoudis, Stergios
AU  - Aramayona, Javier
AU  - Souto, Juan
TI  - Holomorphic maps between moduli spaces
JO  - Annales de l'Institut Fourier
PY  - 2018
SP  - 217
EP  - 228
VL  - 68
IS  - 1
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3158/
DO  - 10.5802/aif.3158
LA  - en
ID  - AIF_2018__68_1_217_0
ER  - 
%0 Journal Article
%A Antonakoudis, Stergios
%A Aramayona, Javier
%A Souto, Juan
%T Holomorphic maps between moduli spaces
%J Annales de l'Institut Fourier
%D 2018
%P 217-228
%V 68
%N 1
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3158/
%R 10.5802/aif.3158
%G en
%F AIF_2018__68_1_217_0
Antonakoudis, Stergios; Aramayona, Javier; Souto, Juan. Holomorphic maps between moduli spaces. Annales de l'Institut Fourier, Volume 68 (2018) no. 1, pp. 217-228. doi : 10.5802/aif.3158. https://aif.centre-mersenne.org/articles/10.5802/aif.3158/

[1] Ahlfors, Lars V. The complex analytic structure of the space of closed Riemann surfaces, Analytic functions (Princeton Mathematical Series), Volume 24, Princeton University Press, 1960, pp. 45-66 | MR | Zbl

[2] Aramayona, Javier; Leininger, Christopher J.; Souto, Juan Injections of mapping class groups, Geom. Topol., Volume 13 (2009) no. 5, pp. 2523-2541 | DOI | MR | Zbl

[3] Aramayona, Javier; Souto, Juan Homomorphisms between mapping class groups, Geom. Topol., Volume 16 (2012) no. 4, pp. 2285-2341 | DOI | MR | Zbl

[4] Aramayona, Javier; Souto, Juan Rigidity phenomena in the mapping class group, IRMA Lectures in Mathematics and Theoretical Physics, 27 (2016), pp. 131-165 | MR | Zbl

[5] Arbarello, Enrico; Cornalba, Maurizio; Griffiths, Pillip A. Geometry of algebraic curves. Volume II, Grundlehren der Mathematischen Wissenschaften, 268, Springer, 2011, xxx+963 pages (With a contribution by Joseph Daniel Harris) | DOI | MR | Zbl

[6] Ballmann, Werner Lectures on Kähler manifolds, ESI Lectures in Mathematics and Physics, European Mathematical Society, 2006, x+172 pages | DOI | MR | Zbl

[7] Bers, Lipman Fiber spaces over Teichmüller spaces, Acta. Math., Volume 130 (1973), pp. 89-126 | DOI | MR | Zbl

[8] Bers, Lipman Finite-dimensional Teichmüller spaces and generalizations, Bull. Am. Math. Soc., Volume 5 (1981) no. 2, pp. 131-172 | DOI | MR | Zbl

[9] Bruno, Andrea; Mella, Massimiliano The automorphism group of M ¯ 0,n , J. Eur. Math. Soc., Volume 15 (2013) no. 3, pp. 949-968 | DOI | MR | Zbl

[10] Deligne, Pierre; Mumford, David The irreducibility of the space of curves of given genus, Publ. Math., Inst. Hautes Étud. Sci. (1969) no. 36, pp. 75-109 | DOI | MR | Zbl

[11] Earle, Clifford J.; Kra, Irwin On holomorphic mappings between Teichmüller spaces, Contributions to analysis (a collection of papers dedicated to Lipman Bers), Academic Press, 1974, pp. 107-124 | MR | Zbl

[12] Earle, Clifford J.; Kra, Irwin On isometries between Teichmüller spaces, Duke Math. J., Volume 41 (1974), pp. 583-591 http://projecteuclid.org/euclid.dmj/1077310579 | DOI | MR | Zbl

[13] Eells, James Jr.; Sampson, Joseph H. Harmonic mappings of Riemannian manifolds, Am. J. Math., Volume 86 (1964), pp. 109-160 | DOI | MR | Zbl

[14] Farb, Benson; Margalit, Dan A primer on mapping class groups, Princeton Mathematical Series, 49, Princeton University Press, 2012, xiv+472 pages | MR | Zbl

[15] Gibney, Angela; Keel, Sean; Morrison, Ian Towards the ample cone of M ¯ g,n , J. Am. Math. Soc., Volume 15 (2002) no. 2, pp. 273-294 | DOI | MR | Zbl

[16] Hubbard, John Hamal Teichmüller theory and applications to geometry, topology, and dynamics. Vol. 1: Teichmüller theory,, Matrix Editions, 2006, xx+459 pages | MR | Zbl

[17] Imayoshi, Yoichi; Shiga, Hiroshige A finiteness theorem for holomorphic families of Riemann surfaces, Holomorphic functions and moduli, Vol. II (Berkeley, 1986) (Mathematical Sciences Research Institute Publications), Volume 11, Springer, 1988, pp. 207-219 | DOI | MR | Zbl

[18] Imayoshi, Yoichi; Taniguchi, Masahiko An introduction to Teichmüller spaces, Springer, 1992, xiv+279 pages (Translated and revised from the Japanese by the authors) | DOI | MR | Zbl

[19] Ji, Lizhen; Wolpert, Scott A. A cofinite universal space for proper actions for mapping class groups, In the tradition of Ahlfors-Bers. V (Contemporary Mathematics), Volume 510, American Mathematical Society, 2010, pp. 151-163 | DOI | MR | Zbl

[20] Looijenga, Eduard Smooth Deligne-Mumford compactifications by means of Prym level structures, J. Algebr. Geom., Volume 3 (1994) no. 2, pp. 283-293 | MR | Zbl

[21] McMullen, Curtis T. From dynamics on surfaces to rational points on curves, Bull. Am. Math. Soc., Volume 37 (2000) no. 2, pp. 119-140 | DOI | MR | Zbl

[22] McMullen, Curtis T. The moduli space of Riemann surfaces is Kähler hyperbolic, Ann. Math., Volume 151 (2000) no. 1, pp. 327-357 | DOI | MR | Zbl

[23] Nag, Subhashis The complex analytic theory of Teichmüller spaces, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1988, xiv+427 pages | MR | Zbl

[24] Royden, H. L. Automorphisms and isometries of Teichmüller space, Advances in the Theory of Riemann Surfaces (Stony Brook, 1969), Princeton University Press, 1971, pp. 369-383 | MR | Zbl

Cited by Sources: