

Proof Of The Effective Bogolomov Conjecture Over Function Fields Of Characteristic Zero by ZÜBEYIR ÇINKIR, Gaziantep Zirve University
September 14  16, 2011
TÜBİTAK  FEZA GÜRSEY INSTITUTE
In these three days series of talks, our main goal is to discuss the proof of the
Effective Bogolomov Conjecture over function fields of characteristic zero [C1]. Here is the
outline of what we plan to cover:
(1) Brief history [T] [B] of results in arithmetic geometry that concerns the finiteness of
algebraic points of small heights, with more emphasis on Bogomolov Conjectures over
several fields and their generalizations.
(2) Bogomolov Conjecture over global fields and Zhang’s work [Z1] [Z2].
(3) Effective Bogomolov Conjecture over function fields and history of previously known
results [G] [F] [M1] [M2] [M3] [M4] [Y1] [Y2] [P].
(4) Zhang’s Conjectures on polarized metrized graph invariants [Z2].
(5) Metrized graphs and their tau constant [BR] [BF] [CR] [C1] [C2] [C3] [C4] [C5].
(6) Polarized metrized graphs and their invariants [C1] [Z2] [F].
(7) Completing the proof of Effective Bogomolov Conjecture over function fields of characteristic
zero [C1].
(8) A new proof of the slope inequality for Faltings heights on moduli space of curves [Z2]
[C1] [M2] [M4].
(9) Computations of metrized graph and polarized metrized graph invariants [C4] [C5].
(10) Further related work [J1] [J2] [J3] [Y3] [GS] [BN].
References:
[B] M. Baker, Talk given at MSRI on January 19, 2006.
http://www.math.gatech.edu/ mbaker/pdf/MSRI1.pdf
[BF] M. Baker and X. Faber, Metrized graphs, Laplacian operators, and electrical networks, Quantum
graphs and their applications, 15–33, Contemp. Math., 415, Amer. Math. Soc., Providence, RI, (2006).
[BN] M. Baker and S. Norine, RiemannRoch and AbelJacobi Theory on a Finite Graph, Advances in
Mathematics, 215 (2007), 766–788.
[BR] M. Baker and R. Rumely, Harmonic analysis on metrized graphs, Canadian J. Math: 59, no. 2, 225–
275, (2007).
Earlier version http://arxiv.org/abs/math.NT/0407427
[CR] T. Chinburg and R. Rumely, The capacity pairing, J. Reine Angew. Math. 434, 1–44, (1993).
[C1] Z. Cinkir, Zhang’s Conjecture and The Effective Bogomolov Conjecture over Function Fields, Invent.
Math., Volume 183, Number 3, (2011) 517–562.
[C2] Z. Cinkir, The tau constant of a metrized graph and its behavior under graph operations, The Electronic
Journal of Combinatorics, Volume 18 (1) (2011) P81.
[C3] Z. Cinkir, The tau constant and the edge connectivity of a metrized graph, submitted, c.f. at
http://arxiv.org/abs/0901.1481v2
[C4] Z. Cinkir, The tau constant and the discrete Laplacian matrix of a metrized graph, European Journal
of Combinatorics, Volume 32, Issue 4, (2011), 639–655.
[C5] Z. Cinkir, The Tau Constant of Metrized Graphs, Thesis at the University of Georgia, 2007.
[F] X. W. C. Faber, The geometric bogomolov conjecture for curves of small genus, Experiment. Math.,
18(3):347–367, (2009).
Earlier version http://arxiv.org/abs/0803.0855v2
[G] W. Gubler, The Bogomolov conjecture for totally degenerate abelian varieties, Invent. Math.,
169(2):337–400, (2007).
[GS] B.H. Gross, and C. Schoen, The modified diagonal cycle on the triple product of a pointed curve, Ann.
Inst. Fourier (Grenoble) 45, no. 3, 649–679, (1995).
[J1] R. D. Jong, Second variation of Zhang’s lambdainvariant on the moduli space of curves, c.f.
http://arxiv.org/abs/1002.1618
[J2] R. D. Jong, Admissible constants for genus 2 curves, c.f. http://arxiv.org/abs/0905.1017
[J3] R. D. Jong, Symmetric roots and admissible pairing, c.f. http://arxiv.org/abs/0906.2112
[M1] A. Moriwaki, A sharp slope inequality for general stable fibrations of curves, J. reine angew. Math.
480, 177–195. MR 97m:14029, (1996).
Earlier version http://arxiv.org/abs/alggeom/9601003
[M2] A. Moriwaki, Bogomolov conjecture over function fields for stable curves with only irreducible fibers,
Comp. Math. 105, 125–140. CMP 97:10, (1997).
Earlier version http://arxiv.org/abs/alggeom/9505003
[M3] A. Moriwaki, Bogomolov conjecture for curves of genus 2 over function fields, J. Math. Kyoto Univ.,
36, 687–695. CMP 97:11, (1996).
Earlier version http://arxiv.org/abs/alggeom/9509008
[M4] A. Moriwaki, Relative Bogomolov’s inequality and the cone of positive divisors on the moduli space of
stable curves, J. Amer. Math. Soc., 11, 569–600, (1998).
[P] A.N. Parˇsin, Algebraic curves over function fields, I. Izv. Akad. Nauk SSSR Ser. Mat. 32:1191–1219,
(1968).
[T] P. Tzermias, The ManinMumford Conjecture: A Brief Survey Bull. London Math. Soc. (2000) 32(6):
641652.
http://math.arizona.edu/ swc/notes/files/99Tzermias.pdf
[Y1] K. Yamaki, Effective calculation of the geometric height and the Bogomolov conjecture for hyperelliptic
curves over function fields, J. Math. Kyoto Univ., 482:401–443, (2008).
Earlier version http://arxiv.org/abs/math/9903066
[Y2] K. Yamaki, Geometric Bogomolov’s conjecture for curves of genus 3 over function fields, J. Math.
Kyoto Univ., 421, 57–81, (2002).
[Y3] K. Yamaki, Graph invariants and the positivity of the height of the GrossSchoen cyle for some curves,
Manuscripta Math., 131, 149–177, (2010).
[Z1] S. Zhang, Admissible pairing on a curve, Invent. Math. 112, 171–193, (1993).
[Z2] S. Zhang, Gross–Schoen cycles and dualising sheaves, Invent. Math., 179(1), 1–73, 2010.
Earlier version: http://www.math.columbia.edu/szhang/papers/Preprints.htm
Accommodation (including breakfast, lunch) will be provided by Feza Gürsey Institute Student Hostel for participants from outside İstanbul if desired.
Travel funds are not available for participants.
All participants, including those from Istanbul, are strongly encouraged to fill in the following application form. Filling in the application form is critical: It will help us at
TÜBİTAK  Feza Gürsey Institute
to keep better records and provide the best service for all participants, including meal arrangements and alike.
Number of participants is limited to 30 people.
Deadline: August 15, 2011
To Apply: http://www.gursey.gov.tr/apps/appfrmgen.php?id=effbog1109
Web site: http://www.gursey.gov.tr/new/effbog1109/
Organizers:
Kürşat Aker (TÜBİTAK  Feza Gürsey Enstitüsü).
Contact: aker ][ gursey.gov.tr
Proof Of The Effective Bogolomov Conjecture Over Function Fields Of Characteristic Zero 

