Deformaciones de variedades abelianas con un grupo de automorfismos: Deformations of abelian varieties with an automorphism group

U. Guerrero-Valadez; H. Torres-López; A. G. Zamora

doi:10.56754/0719-0646.2702.343

DOI:

https://doi.org/10.56754/0719-0646.2702.343

Abstract

Given a polarized abelian variety with an automorphism group \(G\), we prove that the associated local moduli functor is pro-representable, the algebra that pro-represents it is formally smooth, and compute the dimension of this algebra as a function of the analytic action of the group. We present the explicit computations in the case of the action of the symmetric group \(S_3\) on the factors of the product \(E\times E\times E\) of an elliptic curve.

Resumen

Dada una variedad abeliana polarizada con un grupo de automorfismos \(G\), demostramos que el funtor de moduli local asociado es pro-representable; el álgebra que lo pro-representa es formalmente suave y calculamos la dimensión de esta álgebra en función de la acción analítica del grupo. Presentamos los cálculos explícitos del caso de la acción del grupo simétrico \(S_3\) sobre los factores del producto \(E\times E\times E\) de una curva elíptica.

Keywords

Moduli of polarized abelian varieties , automorphism group , local deformations

Mathematics Subject Classification:

14B12 , 14D15 , 14D22 , 14K04 , 14K10

U. Guerrero-Valadez U. A. Matemáticas, U. Autónoma de Zacatecas Calzada Solidaridad entronque Paseo a la Bufa, C.P. 98000, Zacatecas, Zac. México. https://orcid.org/0009-0008-3680-9270
H. Torres-López Secihti - U. A. Matemáticas, U. Autónoma de Zacatecas, Calzada Solidaridad entronque Paseo a la Bufa, C.P. 98000, Zacatecas, Zac. México. https://orcid.org/0000-0002-4574-5052
A. G. Zamora U. A. Matemáticas, U. Autónoma de Zacatecas, Calzada Solidaridad entronque Paseo a la Bufa, C.P. 98000, Zacatecas, Zac. México. https://orcid.org/0000-0002-3983-1990

Pages: 343–362
Date Published: 2025-10-15
In Press

A. Antón-Sancho, “Triality and automorphisms of principal bundles moduli spaces,” Adv. Geom., vol. 24, no. 3, pp. 421–435, 2024, doi: 10.1515/advgeom-2024-0013.

R. Auffarth, A. Carocca, y R. Rodríguez, “Counting polarizations on abelian varieties with group action,” 2024, arXiv:2412.01676.

H. Bennama y J. Bertin, “Remarques sur les varietés abéliennes avec un automorphisme d’ordre premier,” Manuscripta Math., vol. 94, no. 4, pp. 409–425, 1997, doi: 10.1007/BF02677863.

C. Birkenhake y H. Lange, Complex abelian varieties, 2nd ed., ser. Grundlehren der mathematischen Wissenschaften. Springer-Verlag, Berlin, 2004, vol. 302, doi: 10.1007/978-3-662-06307-1.

A. Carocca, S. Reyes-Carocca, y R. E. Rodríguez, “Abelian varieties and Riemann surfaces with generalized quaternion group action,” J. Pure Appl. Algebra, vol. 227, no. 11, 2023, Art. ID 107398, doi: 10.1016/j.jpaa.2023.107398.

B. Fantechi, L. Göttsche, L. Illusie, S. L. Kleiman, N. Nitsure, y A. Vistoli, Fundamental algebraic geometry, ser. Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2005, vol. 123, doi: 10.1090/surv/123.

W. Fulton y J. Harris, Representation theory, ser. Graduate Texts in Mathematics. Verlag, New York, 1991, vol. 129, doi: 10.1007/978-1-4612-0979-9. Springer-Verlag, New York, 1991, vol. 129, doi: 10.1007/978-1-4612-0979-9.

V. Gonzalez-Aguilera, J. M. Muñoz Porras, y A. G. Zamora, “On the irreducible components of the singular locus of Ag,” J. Algebra, vol. 240, no. 1, pp. 230–250, 2001, doi: 10.1006/jabr.2000.8707.

V. González-Aguilera, J. M. Muñoz Porras, y A. G. Zamora, “On the 0-dimensional irreducible components of the singular locus of Ag,” Arch. Math. (Basel), vol. 84, no. 4, pp. 298–303, 2005, doi: 10.1007/s00013-004-1193-x.

V. González-Aguilera, J. M. Munoz-Porras, y A. G. Zamora, “On the irreducible components of the singular locus of Ag. II,” Proc. Amer. Math. Soc., vol. 140, no. 2, pp. 479–492, 2012, doi: 10.1090/S0002-9939-2011-10933-X.

A. Grothendieck y J. P. Murre, The tame fundamental group of a formal neighbourhood of a divisor with normal crossings on a scheme, ser. Lecture Notes in Mathematics. Springer-Verlag, Berlin-New York, 1971, vol. 208.

R. Hartshorne, Deformation theory, ser. Graduate Texts in Mathematics. Springer, New York, 2010, vol. 257, doi: 10.1007/978-1-4419-1596-2.

K. Kodaira y D. C. Spencer, “On deformations of complex analytic structures. I, II,” Ann. of Math. (2), vol. 67, pp. 328–466, 1958, doi: 10.2307/1970009.

H. Lange, R. E. Rodríguez, y A. M. Rojas, “Polarizations on abelian subvarieties of principally polarized abelian varieties with dihedral group actions,” Math. Z., vol. 276, no. 1-2, pp. 397–420, 2014, doi: 10.1007/s00209-013-1206-1.

D. Lee y C. Ray, “Automorphisms of abelian varieties and principal polarizations,” Rend. Circ. Mat. Palermo (2), vol. 71, no. 1, pp. 483–494, 2022, doi: 10.1007/s12215-020-00590-7.

H. Matsumura, Commutative ring theory, ser. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1986, vol. 8.

D. Mumford, J. Fogarty, y F. Kirwan, Geometric invariant theory, 3rd ed., ser. Ergebnisse der Mathematik und ihrer Grenzgebiete (2). Springer-Verlag, Berlin, 1994, vol. 34.

D. Mumford, Abelian varieties, ser. Tata Institute of Fundamental Research Studies in Mathematics. Tata Institute of Fundamental Research, Bombay; by Oxford University Press, London, 1970, vol. 5.

F. Oort, “Finite group schemes, local moduli for abelian varieties, and lifting problems,” Compositio Math., vol. 23, pp. 265–296, 1971.

F. Oort, “Singularities of coarse moduli schemes,” in Séminaire d’Algèbre Paul Dubreil, 29ème année (Paris, 1975–1976), ser. Lecture Notes in Math. Springer, Berlin-New York, 1977, vol. 586, pp. 61–76.

S. Reyes-Carocca y R. E. Rodríguez, “On the connectedness of the singular locus of the moduli space of principally polarized abelian varieties,” Mosc. Math. J., vol. 18, no. 3, pp. 473–489, 2018, doi: 10.17323/1609-4514-2018-18-3-473-489.

M. Schlessinger, “Functors of Artin rings,” Trans. Amer. Math. Soc., vol. 130, pp. 208–222, 1968, doi: 10.2307/1994967.

E. Sernesi, Deformations of algebraic schemes, ser. Grundlehren der mathematischen Wissenschaften. Springer-Verlag, Berlin, 2006, vol. 334.

T. K. Srivastava, “Lifting automorphisms on Abelian varieties as derived autoequivalences,” Arch. Math. (Basel), vol. 116, no. 5, pp. 515–527, 2021, doi: 10.1007/s00013-020-01564-y.