# American Institute of Mathematical Sciences

2011, 18: 91-96. doi: 10.3934/era.2011.18.91

## Deligne pairing and determinant bundle

 1 School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India 2 Fachbereich Mathematik und Informatik, Philipps-Universität Marburg, Lahnberge, Hans-Meerwein-Strasse, D-35032 Marburg, Germany 3 Graduate School of Mathematics, Kyushu University Fukuoka, 819-0395, Japan

Received  March 2011 Revised  June 2011 Published  July 2011

Let $X \rightarrow\ S$ be a smooth projective surjective morphism, where $X$ and $S$ are integral schemes over $\mathbb C$. Let $L_0\, L_1\, \cdots \, L_{n-1}\, L_{n}$ be line bundles over $X$. There is a natural isomorphism of the Deligne pairing 〈$L_0\, \cdots\, L_{n}$〉with the determinant line bundle $Det(\otimes_{i=0}^{n} (L_i- \mathcal O_{X}))$.
Citation: Indranil Biswas, Georg Schumacher, Lin Weng. Deligne pairing and determinant bundle. Electronic Research Announcements, 2011, 18: 91-96. doi: 10.3934/era.2011.18.91
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