# 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
##### References:

show all references

##### References:
 [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] Baolin He. Entropy of diffeomorphisms of line. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4753-4766. doi: 10.3934/dcds.2017204 [13] Anton Petrunin. Metric minimizing surfaces. Electronic Research Announcements, 1999, 5: 47-54. [14] 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 [15] Vincenzo Recupero. Hysteresis operators in metric spaces. Discrete & Continuous Dynamical Systems - S, 2015, 8 (4) : 773-792. doi: 10.3934/dcdss.2015.8.773 [16] 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 [17] Anton Petrunin. Correction to: Metric minimizing surfaces. Electronic Research Announcements, 2018, 25: 96-96. doi: 10.3934/era.2018.25.010 [18] Sergio Muñoz. Robust transitivity of maps of the real line. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1163-1177. doi: 10.3934/dcds.2015.35.1163 [19] Elena Beretta, Markus Grasmair, Monika Muszkieta, Otmar Scherzer. A variational algorithm for the detection of line segments. Inverse Problems & Imaging, 2014, 8 (2) : 389-408. doi: 10.3934/ipi.2014.8.389 [20] Adrian Tudorascu. On absolutely continuous curves of probabilities on the line. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5105-5124. doi: 10.3934/dcds.2019207

2018 Impact Factor: 0.263