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Hodge structures for orbifold cohomology


Author: Javier Fernandez
Journal: Proc. Amer. Math. Soc. 134 (2006), 2511-2520
MSC (2000): Primary 14F43, 14C30; Secondary 14J32
DOI: https://doi.org/10.1090/S0002-9939-06-08515-7
Published electronically: February 17, 2006
MathSciNet review: 2213728
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Abstract: We construct a polarized Hodge structure on the primitive part of Chen and Ruan's orbifold cohomology $ H_{orb}^k(X)$ for projective $ SL$-orbifolds $ X$ satisfying a ``Hard Lefschetz Condition''. Furthermore, the total cohomology $ \bigoplus_{p,q}H_{orb}^{p,q}(X)$ forms a mixed Hodge structure that is polarized by every element of the Kähler cone of $ X$. Using results of Cattani-Kaplan-Schmid this implies the existence of an abstract polarized variation of Hodge structure on the complexified Kähler cone of $ X$.

This construction should be considered as the analogue of the abstract polarized variation of Hodge structure that can be attached to the singular cohomology of a crepant resolution of $ X$, in light of the conjectural correspondence between the (quantum) orbifold cohomology and the (quantum) cohomology of a crepant resolution.


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Additional Information

Javier Fernandez
Affiliation: Department of Mathematics, University of Utah, Salt Lake City, Utah 84112–0090
Address at time of publication: Instituto Balseiro, Univerisdad Nacional de Cuyo – C.N.E.A., Bariloche, Río Negro, R8402AGP, República Argentina
Email: jfernand@ib.edu.ar

DOI: https://doi.org/10.1090/S0002-9939-06-08515-7
Keywords: Orbifold cohomology, polarized Hodge structure, Lefschetz package
Received by editor(s): May 31, 2004
Received by editor(s) in revised form: March 29, 2005
Published electronically: February 17, 2006
Communicated by: Michael Stillman
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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