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

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).