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Realization of an equivariant holomorphic Hermitian line bundle as a Quillen determinant bundle


Author: Indranil Biswas
Journal: Proc. Amer. Math. Soc. 143 (2015), 3183-3187
MSC (2010): Primary 58J52, 14H60
DOI: https://doi.org/10.1090/S0002-9939-2015-12512-9
Published electronically: February 5, 2015
MathSciNet review: 3336643
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Abstract: Let $M$ be an irreducible smooth complex projective variety equipped with an action of a compact Lie group $G$, and let $({\mathcal L} ,h)$ be a $G$–equivariant holomorphic Hermitian line bundle on $M$. Given a compact connected Riemann surface $X$, we construct a $G$–equivariant holomorphic Hermitian line bundle $(L ,H)$ on $X\times M$ (the action of $G$ on $X$ is trivial) such that the corresponding Quillen determinant line bundle $({\mathcal Q} , h_Q)$, which is a $G$–equivariant holomorphic Hermitian line bundle on $M$, is isomorphic to the given $G$–equivariant holomorphic Hermitian line bundle $({\mathcal L} ,h)$. This proves a conjecture by Dey and Mathai (2013).


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Additional Information

Indranil Biswas
Affiliation: School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India
MR Author ID: 340073
Email: indranil@math.tifr.res.in

Keywords: Quillen determinant, equivariant bundle, fiber integral
Received by editor(s): March 10, 2014
Published electronically: February 5, 2015
Communicated by: Varghese Mathai
Article copyright: © Copyright 2015 American Mathematical Society