Deligne pairing and determinant bundle

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

Deligne pairing and determinant bundle

Indranil Biswas ^1, , Georg Schumacher ^2, and Lin Weng ^3,

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

Abstract
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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 $Det(\otimes_{i=0}^{n} (L_i- \mathcal O_{X}))$.

Keywords: determinant line bundle, Deligne pairing, Quillen metric..

Mathematics Subject Classification: Primary: 14F05; Secondary: 14D06.

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

References:

[1]	J.-M. Bismut, H. Gillet and C. Soulé, Analytic torsion and holomorphic determinant bundles. I: Bott-Chern forms and analytic torsion, Commun. Math. Phys., 115 (1988), 49-78. doi: 10.1007/BF01238853.
[2]	P. Deligne, "Le Déterminant de la Cohomologie," Current Trends in Arithmetical Algebraic Geometry, 93-177, Contemp. Math. 67, Amer. Math. Soc., Providence, RI, 1987.
[3]	R. Elkik, Métriques sur les fibrés d'intersection, Duke Math. Jour., 61 (1990), 303-328. doi: 10.1215/S0012-7094-90-06113-7.
[4]	J. Franke, Chow categories, Algebraic Geometry (Berlin, 1988), Compositio Math., 76 (1990), 101-162.
[5]	C. Gasbarri, Heights and geometric invariant theory, Forum Math., 12 (2000), 135-153. doi: 10.1515/form.2000.001.
[6]	A. Fujiki and G. Schumacher, The moduli space of extremal compact Kähler manifolds and generalized Weil-Petersson metrics, Publ. Res. Inst. Math. Sci., 26 (1990), 101-183. doi: 10.2977/prims/1195171664.
[7]	F. Knudsen and D. Mumford, The projectivity of the moduli space of stable curves. I: Preliminaries on "det" and "Div," Math. Scand., 39 (1976), 19-55.
[8]	S. Kobayashi, "Differential Geometry of Complex Vector Bundles," Publications of the Mathematical Society of Japan, 15, Kanô Memorial Lectures, 5, Princeton University Press, Princeton, NJ, Iwanami Shoten, Tokyo, 1987.
[9]	T. Mabuchi and L. Weng, Kähler-Einstein metrics and Chow-Mumford stability, preprint, 1998.
[10]	D. H. Phong and J. Sturm, Scalar curvature, moment maps, and the Deligne pairing, Amer. Jour. Math., 126 (2004), 693-712. doi: 10.1353/ajm.2004.0019.
[11]	D. H. Phong, J. Ross and J. Sturm, Deligne pairings and the Knudsen-Mumford expansion, Jour. Diff. Geom., 78 (2008), 475-496.
[12]	D. G. Quillen, Determinants of Cauchy-Riemann operators on Riemann surfaces, (Russian), Funktsional. Anal. i Prilozhen, 19 (1985), 37-41.
[13]	S. Zhang, Heights and reductions of semi-stable varieties, Compos. Math., 104 (1996), 77-105.

show all references

References:

[1]	J.-M. Bismut, H. Gillet and C. Soulé, Analytic torsion and holomorphic determinant bundles. I: Bott-Chern forms and analytic torsion, Commun. Math. Phys., 115 (1988), 49-78. doi: 10.1007/BF01238853.
[2]	P. Deligne, "Le Déterminant de la Cohomologie," Current Trends in Arithmetical Algebraic Geometry, 93-177, Contemp. Math. 67, Amer. Math. Soc., Providence, RI, 1987.
[3]	R. Elkik, Métriques sur les fibrés d'intersection, Duke Math. Jour., 61 (1990), 303-328. doi: 10.1215/S0012-7094-90-06113-7.
[4]	J. Franke, Chow categories, Algebraic Geometry (Berlin, 1988), Compositio Math., 76 (1990), 101-162.
[5]	C. Gasbarri, Heights and geometric invariant theory, Forum Math., 12 (2000), 135-153. doi: 10.1515/form.2000.001.
[6]	A. Fujiki and G. Schumacher, The moduli space of extremal compact Kähler manifolds and generalized Weil-Petersson metrics, Publ. Res. Inst. Math. Sci., 26 (1990), 101-183. doi: 10.2977/prims/1195171664.
[7]	F. Knudsen and D. Mumford, The projectivity of the moduli space of stable curves. I: Preliminaries on "det" and "Div," Math. Scand., 39 (1976), 19-55.
[8]	S. Kobayashi, "Differential Geometry of Complex Vector Bundles," Publications of the Mathematical Society of Japan, 15, Kanô Memorial Lectures, 5, Princeton University Press, Princeton, NJ, Iwanami Shoten, Tokyo, 1987.
[9]	T. Mabuchi and L. Weng, Kähler-Einstein metrics and Chow-Mumford stability, preprint, 1998.
[10]	D. H. Phong and J. Sturm, Scalar curvature, moment maps, and the Deligne pairing, Amer. Jour. Math., 126 (2004), 693-712. doi: 10.1353/ajm.2004.0019.
[11]	D. H. Phong, J. Ross and J. Sturm, Deligne pairings and the Knudsen-Mumford expansion, Jour. Diff. Geom., 78 (2008), 475-496.
[12]	D. G. Quillen, Determinants of Cauchy-Riemann operators on Riemann surfaces, (Russian), Funktsional. Anal. i Prilozhen, 19 (1985), 37-41.
[13]	S. Zhang, Heights and reductions of semi-stable varieties, Compos. Math., 104 (1996), 77-105.

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