Mixed Hodge structures and Weierstrass <i>σ</i>-function

Grzegorz Banaszak; Jan Milewski

doi:10.1016/j.crma.2013.07.015

Algebraic Geometry

Mixed Hodge structures and Weierstrass σ-function

Presented by: the Editorial Board

Grzegorz Banaszak ¹ ; Jan Milewski 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

Comptes Rendus. Mathématique, Volume 351 (2013) no. 13-14, pp. 551-555

Abstract (Original language)
Résumé

A σ-operator on a complexification $V_{C}$ of an $R$ -vector space $V_{R}$ is an operator $A \in {End}_{C} (V_{C})$ 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 $V_{C}$ dʼun espace vectoriel réel $V_{R}$ est un opérateur $A \in {End}_{C} (V_{C})$ 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.

Received: 2012-11-03
Accepted: 2013-07-18
Published online: 2013-07-01

DOI: 10.1016/j.crma.2013.07.015

Author's affiliations:

Grzegorz Banaszak ¹ ; Jan Milewski 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

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     author = {Grzegorz Banaszak and Jan Milewski},
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Grzegorz Banaszak; Jan Milewski. 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

References
Cited by

[1] G. Banaszak; J. Milewski Hodge structures and Weierstrass sigma-function, C. R. Acad. Sci. Paris, Ser. I, Volume 350 (2012), pp. 777-780

[2] E. Cattani; A. Kaplan; W. Schmid 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] P. Deligne Structures de Hodge mixtes réelles, Proc. Symp. Pure Math., Volume 55 (1994), pp. 509-514

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

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

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