Algebraic Geometry
Mixed Hodge structures and Weierstrass σ-function
Comptes Rendus. Mathématique, Volume 351 (2013) no. 13-14, pp. 551-555.

A σ-operator on a complexification VC of an R-vector space VR is an operator AEndC(VC) such that σ(A)=0, where σ(z) denotes the Weierstrass σ-function. In this paper, we define the notion of strongly pseudo-real σ-operator and prove that there is a one-to-one correspondence between real mixed Hodge structures and strongly pseudo-real σ-operators.

Un σ-opérateur sur la complexification VC dʼun espace vectoriel réel VR est un opérateur AEndC(VC) tel que σ(A)=0, où σ(z) est la fonction σ de Weierstrass. Dans cet article, nous introduisons la notion de σ-opérateur fortement pseudo-réel et démontrons quʼil y a une correspondance biunivoque entre les structures de Hodge mixtes réelles et les σ-opérateurs fortement pseudo-réels.

Published online:
DOI: 10.1016/j.crma.2013.07.015
Banaszak, Grzegorz 1; Milewski, Jan 2

1 Department of Mathematics and Computer Science, Adam Mickiewicz University, Poznań 61-614, Poland
2 Institute of Mathematics, Poznań University of Technology, ul. Piotrowo 3A, 60-965 Poznań, Poland
     author = {Banaszak, Grzegorz and Milewski, Jan},
     title = {Mixed {Hodge} structures and {Weierstrass} \protect\emph{\ensuremath{\sigma}}-function},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {551--555},
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Banaszak, Grzegorz; Milewski, Jan. Mixed Hodge structures and Weierstrass σ-function. Comptes Rendus. Mathématique, Volume 351 (2013) no. 13-14, pp. 551-555. doi : 10.1016/j.crma.2013.07.015.

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[2] Cattani, E.; Kaplan, A.; Schmid, W. Degeneration of Hodge structures, Ann. Math. (2), Volume 123 (1986) no. 3, pp. 457-535

[3] P. Deligne, Structures de Hodge mixtes réelles. The appendix to “On the SL(2)-orbits in Hodge theory” by E. Cattani and A. Kaplan, IHES pre-pub./M/82/58 (1982).

[4] Deligne, P. Structures de Hodge mixtes réelles, Proc. Symp. Pure Math., Volume 55 (1994), pp. 509-514

[5] Gordon, B. A survey of the Hodge conjecture for Abelian varieties (Lewis, J., ed.), A Survey of the Hodge Conjecture, 1999, pp. 297-356 (Appendix B)

[6] Peters, C.; Steenbrink, J. Mixed Hodge Structures, Ergeb. Math. Grenzgeb., vol. 52, Springer, 2008

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