American Institute of Mathematical Sciences

Advanced
  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

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.  doi: 10.1007/BF01238853.  Google Scholar

[2]

P. Deligne, "Le Déterminant de la Cohomologie,", Current Trends in Arithmetical Algebraic Geometry, 67 (1987), 93.   Google Scholar

[3]

R. Elkik, Métriques sur les fibrés d'intersection,, Duke Math. Jour., 61 (1990), 303.  doi: 10.1215/S0012-7094-90-06113-7.  Google Scholar

[4]

J. Franke, Chow categories,, Algebraic Geometry (Berlin, 76 (1990), 101.   Google Scholar

[5]

C. Gasbarri, Heights and geometric invariant theory,, Forum Math., 12 (2000), 135.  doi: 10.1515/form.2000.001.  Google Scholar

[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.  doi: 10.2977/prims/1195171664.  Google Scholar

[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.   Google Scholar

[8]

S. Kobayashi, "Differential Geometry of Complex Vector Bundles,", Publications of the Mathematical Society of Japan, 15 (1987).   Google Scholar

[9]

T. Mabuchi and L. Weng, Kähler-Einstein metrics and Chow-Mumford stability,, preprint, (1998).   Google Scholar

[10]

D. H. Phong and J. Sturm, Scalar curvature, moment maps, and the Deligne pairing,, Amer. Jour. Math., 126 (2004), 693.  doi: 10.1353/ajm.2004.0019.  Google Scholar

[11]

D. H. Phong, J. Ross and J. Sturm, Deligne pairings and the Knudsen-Mumford expansion,, Jour. Diff. Geom., 78 (2008), 475.   Google Scholar

[12]

D. G. Quillen, Determinants of Cauchy-Riemann operators on Riemann surfaces,, (Russian), 19 (1985), 37.   Google Scholar

[13]

S. Zhang, Heights and reductions of semi-stable varieties,, Compos. Math., 104 (1996), 77.   Google Scholar

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.  doi: 10.1007/BF01238853.  Google Scholar

[2]

P. Deligne, "Le Déterminant de la Cohomologie,", Current Trends in Arithmetical Algebraic Geometry, 67 (1987), 93.   Google Scholar

[3]

R. Elkik, Métriques sur les fibrés d'intersection,, Duke Math. Jour., 61 (1990), 303.  doi: 10.1215/S0012-7094-90-06113-7.  Google Scholar

[4]

J. Franke, Chow categories,, Algebraic Geometry (Berlin, 76 (1990), 101.   Google Scholar

[5]

C. Gasbarri, Heights and geometric invariant theory,, Forum Math., 12 (2000), 135.  doi: 10.1515/form.2000.001.  Google Scholar

[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.  doi: 10.2977/prims/1195171664.  Google Scholar

[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.   Google Scholar

[8]

S. Kobayashi, "Differential Geometry of Complex Vector Bundles,", Publications of the Mathematical Society of Japan, 15 (1987).   Google Scholar

[9]

T. Mabuchi and L. Weng, Kähler-Einstein metrics and Chow-Mumford stability,, preprint, (1998).   Google Scholar

[10]

D. H. Phong and J. Sturm, Scalar curvature, moment maps, and the Deligne pairing,, Amer. Jour. Math., 126 (2004), 693.  doi: 10.1353/ajm.2004.0019.  Google Scholar

[11]

D. H. Phong, J. Ross and J. Sturm, Deligne pairings and the Knudsen-Mumford expansion,, Jour. Diff. Geom., 78 (2008), 475.   Google Scholar

[12]

D. G. Quillen, Determinants of Cauchy-Riemann operators on Riemann surfaces,, (Russian), 19 (1985), 37.   Google Scholar

[13]

S. Zhang, Heights and reductions of semi-stable varieties,, Compos. Math., 104 (1996), 77.   Google Scholar

[1]

Simon Scott. Relative zeta determinants and the geometry of the determinant line bundle. Electronic Research Announcements, 2001, 7: 8-16.

[2]

Katrin Grunert, Helge Holden, Xavier Raynaud. Lipschitz metric for the Camassa--Holm equation on the line. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2809-2827. doi: 10.3934/dcds.2013.33.2809
[3]

Yakov Varshavsky. A proof of a generalization of Deligne's conjecture. Electronic Research Announcements, 2005, 11: 78-88.

[4]

Florian Luca, Igor E. Shparlinski. On finite fields for pairing based cryptography. Advances in Mathematics of Communications, 2007, 1 (3) : 281-286. doi: 10.3934/amc.2007.1.281
[5]

Joshua Du, Jun Ji. An integral representation of the determinant of a matrix and its applications. Conference Publications, 2005, 2005 (Special) : 225-232. doi: 10.3934/proc.2005.2005.225
[6]

Hiroaki Yoshimura, Jerrold E. Marsden. Dirac cotangent bundle reduction. Journal of Geometric Mechanics, 2009, 1 (1) : 87-158. doi: 10.3934/jgm.2009.1.87
[7]

Pedro Teixeira. Dacorogna-Moser theorem on the Jacobian determinant equation with control of support. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 4071-4089. doi: 10.3934/dcds.2017173
[8]

V. M. Gundlach, Yu. Kifer. Expansiveness, specification, and equilibrium states for random bundle transformations. Discrete & Continuous Dynamical Systems - A, 2000, 6 (1) : 89-120. doi: 10.3934/dcds.2000.6.89
[9]

Oliver Butterley, Carlangelo Liverani. Robustly invariant sets in fiber contracting bundle flows. Journal of Modern Dynamics, 2013, 7 (2) : 255-267. doi: 10.3934/jmd.2013.7.255
[10]

Robert Skiba, Nils Waterstraat. The index bundle and multiparameter bifurcation for discrete dynamical systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5603-5629. doi: 10.3934/dcds.2017243
[11]

Ming Huang, Xi-Jun Liang, Yuan Lu, Li-Ping Pang. The bundle scheme for solving arbitrary eigenvalue optimizations. Journal of Industrial & Management Optimization, 2017, 13 (2) : 659-680. doi: 10.3934/jimo.2016039
[12]

Anton Petrunin. Metric minimizing surfaces. Electronic Research Announcements, 1999, 5: 47-54.

[13]

Valentin Afraimovich, Lev Glebsky, Rosendo Vazquez. Measures related to metric complexity. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1299-1309. doi: 10.3934/dcds.2010.28.1299
[14]

Vincenzo Recupero. Hysteresis operators in metric spaces. Discrete & Continuous Dynamical Systems - S, 2015, 8 (4) : 773-792. doi: 10.3934/dcdss.2015.8.773
[15]

Vladimir Georgiev, Eugene Stepanov. Metric cycles, curves and solenoids. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1443-1463. doi: 10.3934/dcds.2014.34.1443
[16]

Anton Petrunin. Correction to: Metric minimizing surfaces. Electronic Research Announcements, 2018, 25: 96-96. doi: 10.3934/era.2018.25.010
[17]

Baolin He. Entropy of diffeomorphisms of line. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4753-4766. doi: 10.3934/dcds.2017204
[18]

Xianfeng Ma, Ercai Chen. Pre-image variational principle for bundle random dynamical systems. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 957-972. doi: 10.3934/dcds.2009.23.957
[19]

Michał Jóźwikowski, Mikołaj Rotkiewicz. Bundle-theoretic methods for higher-order variational calculus. Journal of Geometric Mechanics, 2014, 6 (1) : 99-120. doi: 10.3934/jgm.2014.6.99
[20]

Chunming Tang, Jinbao Jian, Guoyin Li. A proximal-projection partial bundle method for convex constrained minimax problems. Journal of Industrial & Management Optimization, 2019, 15 (2) : 757-774. doi: 10.3934/jimo.2018069

2018 Impact Factor: 0.263

Tools

Metrics

  • PDF downloads (21)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences