# THEMATIC PROGRAMS

August 20, 2019

## 2008 (Fall) Thematic Program on Arithmetic Geometry, Hyperbolic Geometry and Related Topics

### GRADUATE COURSES FALL 2008, September 8 - December 5

 Taking the Course for Credit Thematic Year Homepage

*NOTE ROOM CHANGE: All courses will meet at the Fields Institute Stewart Library in the 3rd Floor unless notified otherwise.
Each meeting will run for 90 minutes.

1) I
ntroduction to Arakelov Geometry (Wednesday & Thursday @ 10 a.m)
Instructor: Henri Gillet, University of Illinois at Chicago

2) Course on Nevanlinna theory and Diophantine approximation
(Tuesday & Wednesday @ 1:00 p.m.)
Instructor: Min Ru, University of Houston

3) Course on Jet Spaces
(Mini Course In Complex Geometry)( Tuesday @ 10:00 a.m. , Thursday @ 1:00 p.m.)
Instructor: Pit-Mann Wong, University of Notre Dame

4) MAT1191HF Transcendental Methods in Algebraic Geometry
Y-T Siu
Monday and Fridays @ 10:30 a.m. -12 noon
Fields 230
Oct. 20, 27, Nov. 10, 17 [BA2155]
Sept 19, Oct 24, Nov. 14, 21 [BA2175]

Applications of L^2 \partial-estimates and multiplier ideal sheaf tech niques to problems in algebraic geometry such as the effective Nullstellensatz, the Fujita conjecture on effective global generation and very ampleness of line bundles, the effective Matsusaka big theorem, the deformational invariance of plurigenera, the finite generation of the canonical ring, and the abundance conjecture. Will also discuss hyperbolicity problems and the application of algebraic-geometric techniques to partial differential equations through multiplier ideal sheaves, especially the global regularity of the complex Neumann problem, the existence of Hermitian-Einstein and Kähler-Einstein metrics, and the global nondeformability of irreducible compact Hermitian symmetric manifolds.

### Introduction to Arakelov Geometry Instructor: Henri Gillet

Meeting Times: 90 minutes
Wednesday & Thursday 10:00 am -11:30 a.m.

The course will begin, following a review of the "non-arithmetic" theory, with a study of Arakelov's intersection theory on arithmetic surfaces as developed in Faltings. We will then develop arithmetic intersection theory for varieties of arbitrary dimesion. The main results include the relationship between arithmetic intersection theory, heights and height pairings, the arithmetic Bezout theorem, the arithmetic Hilbert-Samuel formula, and the arithmetic Riemann-Roch theorem. The course will continue with the K-theory of Hermitian vector bundles for general arithmetic varieties and the characteristic classes for this bundles, the determinant of cohomology and Quillen metrics, and the arithmetic Grothendieck-Riemann-Roch theorem. Some applications to problems in number theory will be discussed.

Syllabus:

• Review of classical intersection theory
• Intersection theory on regular schemes
• Arakelov's Intersection theory on arithmetic surfaces
• Deligne cohomology and Green currents
• Arithmetic Chow groups and arithmetic intersection theory
• Deformation to the normal bundle in arithmetic intersection theory
• Bott-Chern forms and Chern Classes of Hermitian vector bundles
• Heights and the arithmetic Bezout theorem
• Heights of Grassmannians and other special varieties
• Determinant of Cohomology and analytic torsion
• Arithmetic Riemann Roch and the arithmetic Hilbert-Samuel formula
• Recent Results and Open questions

Bost, J.-B.; Gillet, H.; Soulé, C. --Heights of projective varieties and positive Green forms. J. Amer. Math. Soc. 7 (1994), no. 4, 903--1027.

Faltings, Gerd --Calculus on arithmetic surfaces. Ann. of Math. (2) 119 (1984), no. 2, 387--424.

Gillet, Henri; Soulé, Christophe --Arithmetic intersection theory. Inst. Hautes Études Sci. Publ. Math. No. 72 (1990), 93--174 (1991).

Gillet, Henri; Soulé, Christophe --Characteristic classes for algebraic vector bundles with Hermitian metric. Ann. of Math. (2) 131 (1990), 163--238.

Lang, Serge --Introduction to Arakelov theory. Springer-Verlag, New York, 1988.

Maillot, Vincent --Un calcul de Schubert arithmétique. (French) [An arithmetic Schubert calculus] Duke Math. J. 80 (1995), no. 1, 195--221.

Soulé, Christophe --Hermitian vector bundles on arithmetic varieties. Algebraic geometry---Santa Cruz 1995, 383--419, Proc. Sympos. Pure Math., 62, Part 1, Amer. Math. Soc., Providence, RI, 1997.

Soulé, C. --Lectures on Arakelov geometry. With the collaboration of D. Abramovich, J.-F. Burnol and J. Kramer. Cambridge Studies in Advanced Mathematics, 33. Cambridge University Press, Cambridge, 1992.

Tamvakis, Harry --Height formulas for homogeneous varieties. Michigan Math. J. 48 (2000), 593--610.

### Course on Nevanlinna theory and Diophantine approximation Instructor: Min Ru, University of Houston

Meeting Times: 90 minutes
Tuesday & Wednesday 1:00 pm

Diophantine approximation is a tool to study rational points on algebraic varieties defined over a number field. On the other hand, Nevanlinna theory studies holomorphic curves in complex algebraic varieties, especially it studies how well a holomorphic curve intersects divisors in a complex algebraic variety. It has been observed by Osgood, Vojta and others that there is a striking correspondence between statements in Nevanlinna theory and in Diophantine approximation. The mini-course will cover: Roth’s theorem and Schmidt’s subspace theorem; Diophantine equations and approximation; the theory of global and local heights; Faltings’ theorem on abelian varieties; the classical theory of Nevanlinna on meromorphic functions; The Ahlfors-Cartan theory of holomorphic curves; holomorphic curves in Abelian varieties; the complex hyperbolicities and the general case of Lang’s conjecture.

Syllabus.

• Nevanlinna theory for meromorphic functions, holomorphic curves in compact Riemann surfaces.
• The theorem of Thue-Siegel-Roth and the theorem of Faltings.
• Nevanlinna’s theory for holomorphic curves, H. Cartan’s method and Ahlfors’ method.
• The theory of heights.
• Schmidt’s subspace theorem.
• The application to the study of Diophantine equations.
• Holomorphic curves in Abelian varieties and the theorem of Faltings.
• Complex hyperbolic manifolds and Lang’s conjecture.
• A survey of recent developments.

Advanced topics: These will be covered in the seminars.

Text:
Min Ru, Nevanlinna theory and its relation to Diophantine approximation. World Scientific Publishing Co., Inc., River Edge, NJ, 2001. xiv+323 pp. ISBN: 981-02-4402-9.

Here is the list of reference (with * indicates that they are most related and recommended):

1. Cowen, M. and Griffith, Ph.(1976) Holomorphic curves and metrics of nonnegative curvature'', J. Analyse Math. 29, 93--153.

2. Demailly, J.P.(1995) Algebraic criteria for Kobayashi hyperbolic varieties and jet differentials'', {\it Proc. Symp. Pur. Math. Amer. Math. Soc.} 62, Part 2, 285--360.

3.* Green, M.(1975) Some Picard theorems for holomorphic maps to algebraic varieties,'' Amer. J. Math. 97, 43--75.

4. Green, M. and Griffiths,P.(1980) Two applications of algebraic geometry to entire holomorphic mappings'', The Chern Symposium 1979, Proc. Internat. Sympos.,Berkeley, 1979, Springer-Verlag.

5. Griffiths,P. (1974) "Entire holomorphic mappings in one and several complex variables", The fifth set of Hermann Weyl Lectures, given at the Institute for Advanced Study, Princeton, N. J., October and November 1974. Annals of Mathematics Studies, No. 85. Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1976. x+99 pp.

6.* Hayman, W.(1964) Meromorphic Functions, Oxford University Press.

7.* Hindry, M. and Silverman, J.(2000) Diophantine Geometry: an Introduction, Graduate Texts in Mathematics 201, Springer-Verlag.

8.* Kobayashi, S.(1998) Hyperbolic Complex Spaces, Springer-Verlag.

9.* Lang, S.(1983) Fundamentals of Diophantine Geometry, Springer-Verlag.

10.* Lang, S. (1987) Introduction to Complex Hyperbolic Spaces, Springer-Verlag, New
York-Berlin-Heidelberg.

11.* Fujimoto, H.(1993) {\it Value Distribution Theory on the Gauss Map of Minimal
Surfaes in ${\bf R}^m$
}, Aspect of Mathematics, E21, Vieweg.

12. Lang, S. and Cherry, W.(1990) Topic in Nevanlinna theory, Lecture Notes in Math. 1433, Springer-Verlag.

13.* Noguchi, J. and Ochiai, T.(1990) Geometric Function Theory in Several Complex
Variables
, Transl. Math. Mon. 80, Amer. Math. Soc., Providence, R.I..

14. Schmidt, W.M.(1991) Diophantine approximations and diophantine equations, Lecture Notes in Math. 1467. Springer-Verlag.

15. Schmidt, W.M.(1980) Diophantine approximation, Lecture Notes in Math. 785, Springer-Verlag.

16.* Shabat, B.V.(1985) Distribution of values of holomorphic mappings,Translations of Mathematical Monographs Vol. 61, American MathematicalSociety.

17.* Siu, Y.T.(1995) Hyperbolicity Problems in Function Theory'', in Five Decades as a Mathematician and Educator - on the 80th birthday of Professor Yung-Chow Wong, ed. Kai-Yuen Chan and Ming-Chit Liu, World Scientific: Singapore, New Jersey, London, Hong Kong, pp.409--514.

18. Stoll, W.(1983) The Ahlfors-Weyl theory of meromorphic maps on parabolic manifolds, Lecture Notes in Mathematics 981, 101--219.

19. Stoll, W.(1985) Value distribution theory for meromorphic maps, Aspects of Math., E7.

20.* Vojta, P.(1987) Diophantine approximations and value distribution theory, Lecture Notes in Math. 1239, Springer-Verlag.

21. Vojta, P.(1999) Nevanlinna theory and Diophantine approximation'', Several complex variables (Berkeley, CA, 1995--1996), 535--564, Math. Sci. Res. Inst. Publ., 37, Cambridge Univ. Press, Cambridge.

22.* Wong, P.M.(1989) On the second main theorem of Nevanlinna theory'', Amer. J. Math. 549--583.

23.. Diophantine approximation and abelian varieties. Introductory lectures. Papers from the conference held in Soesterberg, April 12--16, 1992. Edited by B. Edixhoven and J.-H. Evertse. Lecture Notes in Mathematics, 1566. Springer-Verlag, Berlin, 1993. xiv+127 pp. ISBN: 3-540-57528-

### Course on Jet Spaces Instructor: Pit-Mann Wong, University of Notre Dame

Meeting Times: 90 minutes
Tuesday 10:00 am & Thursday 1:00 pm

A Mini Course In Complex Geometry

This is an outline of a mini-course in complex geometry. There are six chapters planned. The content in the rst four chapters are mostly very well known. There will be no time for complete proofs only outline and motivations will be given. Explicit references of the theorems will be given. I shall also try to provide examples to explain the ideas and clarify the concepts. A reasonable amount of details will be given in the last two chapters.

1. A brief introduction to Hermitian and Kähler Geometry.
A quick outline of the basic notions of Hermitian connection and curvature of vector bundles. The concepts of holomorphic bisectional curvature, Ricci curvature and holomorphic sectional curvature. Schwarz Lemma. Chern classes of vector bundles.

2. A brief introduction to Complex Analysis and Algebraic geometry.
Brief outline of the concepts of coherent sheaves and sheaf cohomologies. Characterizations of Stein manifolds via vanishing theorem, via strictly plurisubharmonic exhaustions and the Stein embedding theorem. The concepts of ample, big and nef bundles. Kodaira vanishing theorem and embedding theorem. Riemann-Roch theorem for coherent sheaves.

3. A brief introduction to Complex Finsler Geometry and Intrinsic Metrics.
Intrinsic metrics, mainly the Kobayashi and the Caratheodory metric, will be introduced. Positive currents and Lelong numbers. A Finsler characterization of ample bundles and big bundles will be given.

4. A brief introduction to Nevanlinna Theory.
A brief account of Nevanlinna Theory. Jensen Formula. First Main Theorem. Crofton Formula. Second Main theorem. Integrated form of Schwarz Lemma.

5. Holomorphic Jet Bundles.
The basic theory of jet bundles will be introduced with reasonable amount of details and examples.

6. Applications to Hyperbolic Geometry.
The results in the respective chapters will be applied to resolve problems in complex hyperbolic geometry.

### Taking the Institute's Courses for Credit

As graduate students at any of the Institute's University Partners, you may discuss the possibility of obtaining a credit for one or more courses in this lecture series with your home university graduate officer and the course instructor. Assigned reading and related projects may be arranged for the benefit of students requiring these courses for credit.

### Financial Assistance

As part of the Affiliation agreement with some Canadian Universities, graduate students are eligible to apply for financial assistance to attend graduate courses. Application for support now closed. Two types of support were available:

• Students outside the greater Toronto area may apply for travel support. Please submit a proposed budget outlining expected costs if public transit is involved, otherwise a mileage rate is used to reimburse travel costs. We recommend that groups coming from one university travel together, or arrange for car pooling (or car rental if applicable).

• Students outside the commuting distance of Toronto may submit an application for a term fellowship. Support is offered up to \$1000 per month.

For more details on the thematic year, see Program Page or contact thematic(PUT_AT_SIGN_HERE)fields.utoronto.ca