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International Mathematics Research Papers (2008) Vol. 2008 : article ID rpn008, 77 pages, doi:10.1093/imrp/rpn008 published on October 7, 2008
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© The Author 2008. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oxfordjournals.org.

Geometry of Multiplicative Preprojective Algebra

Daisuke Yamakawa

Department of Mathematics, Faculty of Science, Kyoto University, Kyoto, 606-8502, Japan

Correspondence: Correspondence to be sent to: yamakawa{at}math.kyoto-u.ac.jp

Crawley-Boevey and Shaw recently introduced a certain multiplicative analogue of the deformed preprojective algebra, which they called a multiplicative preprojective algebra. In this paper, we study a moduli space of (semi)stable representations of such an algebra (the multiplicative quiver variety), which in fact has many similarities to the quiver variety. We show that there is a complex analytic isomorphism between the nilpotent subvariety of the quiver variety and that of the multiplicative quiver variety (which can be extended to a symplectomorphism between these tubular neighborhoods). We also show that when the quiver is star-shaped, the multiplicative quiver variety parameterizes Simpson's (poly)stable filtered local systems on a punctured Riemann sphere with prescribed filtration type, weight, and associated graded local systems around each puncture.



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