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The 4th CAGWS

COMPLEX ALGEBRAIC GEOMETRY WINTER SCHOOL

Nesin Mathematics Village, Şirince, İzmir
January 20-February 2, 2020



First Week

 

TIME              SPEAKER                  TITLE
Jan 20-24
9-10:30
Yoshinori Hashimoto   Introduction to Geometric Invariant Theory  
Jan 20-25
10:30-12
Tommaso Pacini Conformal Invariants and Quasi-conformal Maps
Jan 20-26
12-1:15
Mustafa Kalafat
Representation Theory of Complex Lie Algebras A
Lunch
Jan 20-26
3-4:15
Mutsuo Oka
Introduction on Mixed Hypersurface Singularity
Jan 20-26
4:30-6
Mohan Bhupal Riemann Surfaces and Fuchsian Groups
Jan 20-26
6-7:15
Buket Can Bahadır Several Complex Variables

Second Week

 

TIME              SPEAKER                  TITLE
Jan 27-Feb 1
9:10-10:45
Shin-ichi Matsumura Geometry of Holomorphic Sectional Curvature
and Rational Curves
Jan 27-Feb 2
11-12:50
Gregory Sankaran Moduli of Abelian Varieties
Lunch
Jan 27-Feb 2
2:30-4
Taro Sano Deformations of Fano and Calabi-Yau Varieties
Jan 27-Feb 2
4-5:30
Mustafa Kalafat
Representation Theory of Complex Lie Algebras B
Jan 27-Feb 2
5:30-7
Kürşat Aker* Higgs Bundles

Scientific Commitee

 

İzzet Coşkun University of Illinois at Chicago, USA
Ljudmila Kamenova Stony Brook University, USA
Lei Ni University of California at San Diego, USA
Misha Verbitsky IMPA, Brasil







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Information

The 4th CAGWS will be held at the Nesin Mathematical Village in Şirince, İzmir.
There will be about 15 mini-courses of introductory nature, related to the Geometry-Topology research subjects.
Venue-Classroom: 2nd floor of the Library building.
In the middle of the week there is an excursion to the Ancient City of Ephesus .

Wifi info: Passwords are TermoS1! at the Library networks and zeytinlik for istasyon at the registration lobby.



Application

Graduate students, recent Ph.D.s and under-represented minorities are especially encouraged to the summer school.
Partial financial support is available. Daily expenses including bed, breakfast, lunch, dinner is around 20€.

Please fill out the application form and send it to

Aslı Can Korkmaz: aslicankorkmaz@nesinvakfi.org, berkanuze@gmail.com

to attend to the winter school.

Airport: İzmir Adnan Menderes Airport - ADB is the closest one.
Visas: Check whether you need a visa beforehand.


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Abstracts



Introduction on Mixed Hypersurface Singularity

In this lecture series, I introduce the basic of mixed hypersurface singularity. A mixed polynomial is a polynomial which contains z_1,...,z_n and also their conjugates z_1,...,z_n. Hypersurace defined by {f=0} involves much wider class of geometry. We introduce the notion of Newton boundary, non-degeneracy and local Milnor fibration.

Textbook or/and course webpage:

1. M.Oka, Non-degenerate mixed functions, Kodai J. Math. 33 (1), 1-62,2010

2. J. Milnor, Singular points of complex hypersurfaces, Annals of Math. Studies 61, Princeton , N.J., 1968

Prerequisites:

Linear Algebra, holomorphic function of one variable (not a must but preferable)

Introduction to Geometric Invariant Theory

Suppose that a reductive algebraic group G acts on a projective variety X. Geometric Invariant Theory (GIT), initiated by Mumford, is a method of taking a quotient X/G in the category of varieties. It has many important applications in the theory of moduli and is also related to problems in differential geometry. This minicourse aims to be a rapid introduction to this topic; it will be mostly about the construction of the GIT quotient and the Hilbert—Mumford criterion, but related differential-geometric topics will also be mentioned.

Textbook or/and course webpage:

1. Dolgachev, I. Lectures on invariant theory. London Mathematical Society Lecture Note Series, 296. Cambridge University Press, Cambridge, 2003. xvi+220 pp. ISBN: 0-521-52548-9

2. Mukai, S. An introduction to invariants and moduli. Translated from the 1998 and 2000 Japanese editions by W. M. Oxbury. Cambridge Studies in Advanced Mathematics, 81. Cambridge University Press, Cambridge, 2003. xx+503 pp. ISBN: 0-521-80906-1

3. Huybrechts, D.; Lehn, M. The geometry of moduli spaces of sheaves. Second edition. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 2010. xviii+325 pp. ISBN: 978-0-521-13420-0

4. Chriss, N.; Ginzburg, V. Representation theory and complex geometry. Reprint of the 1997 edition. Modern Birkhäuser Classics. Birkhäuser Boston, Inc., Boston, MA, 2010. x+495 pp. ISBN: 978-0-8176-4937-1

5. Thomas, R. P. Notes on GIT and symplectic reduction for bundles and varieties. Surveys in differential geometry. Vol. X, 221--273, Surv. Differ. Geom., 10, Int. Press, Somerville, MA, 2006. https://arxiv.org/abs/math/0512411

6. Donaldson, S. K.; Kronheimer, P. B. The geometry of four-manifolds. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1990. x+440 pp. ISBN: 0-19-853553-8

7. Mumford, D.; Fogarty, J.; Kirwan, F. Geometric invariant theory. Third edition. Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], 34. Springer-Verlag, Berlin, 1994. xiv+292 pp. ISBN: 3-540-56963-4

Prerequisites: Basic complex algebraic geometry

On the Geometry of Holomorphic Sectional Curvature and Rational Curves

In this lecture, we discuss the notation of holomorphic sectional curvatures after we review hermitian metrics, Chern curvatures, and their properties of holomorphic vector bundle over complex manifolds. In particular, I explain a relation between positivity of holomorphic sectional curvatures and the geometry of rational curves (that is, 1-dim projective space embedded in manifolds). The goal of this talk is to give a structure theorem for rationally connected fibrations of projective manifolds with non-negative holomorphic sectional curvature.

Textbook or/and course webpage:

1. Foundations of Differential Geometry, written by Shoshichi Kobayashi and Katsumi Nomizu, Wiley Classics Library.

2. RC-positivity, rational connectedness and Yau's conjecture, written by Xiaokui Yang, Camb. J. Math. 6 (2018), no. 2, 183–212

3. On projective manifolds with semi-positive holomorphic sectional curvature,

4. On morphisms of compact Kähler manifolds with semi-positive holomorphic sectional curvature written Shin-ichi Matsumura, available at arXiv.

Prerequisites: Complex manifolds , (holomorphic) vector bundles(not a must but preferable)

Level: Graduate and Advanced undergraduate

Language: EN

Deformations of Fano and Calabi-Yau Varieties

In this lecture series we present an algebraic approach to deformations of Fano and Calabi-Yau varieties. Fano varieties and Calabi-Yau varieties are important objects in the classification of algebraic varieties. In the classification of vareties, it is fundamental to study their deformations. Starting from basic notions, I'll explain how to see that they have unobstructed deformations. I'll also explain the generalization to log CY varieties and normal crossing CY varieties.

Daily description is as follows.

1: Basics on complex manifolds (analytic spaces) and sheaves
2: Preliminaries on Deformation theory
3: Deformations of CY varieties (Bogomolov-Tian-Todorov theorem)
4: Deformations of log CY and normal crossing CY varieties

Textbook and References:

1. Greuel, G.-M.; Lossen, C.; Shustin, E. Introduction to singularities and deformations. Springer Monographs in Mathematics. Springer, Berlin, 2007. xii+471 pp.

2. Huybrechts, Complex geometry, An introduction. Universitext. Springer-Verlag, Berlin, 2005. xii+309 pp.

3.Namikawa, Yoshinori Calabi-Yau threefolds and deformation theory [translation of Sūgaku 48 (1996), no. 4, 337–357; MR1614448]. Suguku Expositions. Sugaku Expositions 15 (2002), no. 1, 1–29.

4. Sernesi, Edoardo Deformations of algebraic schemes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 334. Springer-Verlag, Berlin, 2006. xii+339 pp.

Prerequisites: Basic complex algebraic geometry (not a must but preferable)

Level: Graduate Advanced undergraduate

Representation Theory of Complex Lie Algebras

In this lecture series we give an introduction to the representation theory of some complex Lie algebras. We review Lie groups, Lie algebras, Cartan subalgebra, Root systems. Then, starting from concrete examples we will work on sl(2,C), sl(3,C), sl(4,C) and finally sl(n,C) and their representations. We also go into theri geometric interpretations. Also spend time on sp(2n,C) and so(m,C) if time permits.

Textbook or/and course webpage:

1. Fulton, William; Harris, Joe - Representation theory. A first course.
Graduate Texts in Mathematics, 129. Readings in Mathematics. Springer-Verlag, New York, 1991. xvi+551 pp. ISBN: 0-387-97527-6
Lectures 11,12,13,14 and 22.

2. Anthony W. Knapp. Lie groups beyond an introduction, volume 140 of Progress in Mathematics. Birkhauser Boston, Inc., Boston, MA, second edition, 2002.

Prerequisites:

Linear Algebra. Lie groups and Lie algebras.

Moduli of Abelian Varieties



Conformal Invariants and Quasi-conformal Maps

The theory of one complex variable and its geometric counterpart, the theory of Riemann surfaces, has classically been one of the main driving forces in the development of mathematics. It still offers a “treasure chest” of ideas which is worth rummaging through, both for a better understanding of current problems and as a source of inspiration for new research.

Our goal is to introduce the classical 1-dimensional conformal invariant known as extremal length and to explain its role in applications, with special focus on the notion of quasi-conformal maps underlying Teichmueller theory. We will then introduce a recent higher-dimensional analogue, extremal volume.

Daily schedule:

Lectures 1,2: Conformal invariants and extremal length.

Lecture 3: Quasi-conformal maps: the geometric definition.

Lecture 4: Quasi-conformal maps: the analytic definition via the Beltrami equation.

Lecture 5: Extremal volume.

Textbook, references and/or course webpage:

1. L. V. Ahlfors: Conformal invariants. Topics in Geometric Function Theory.

2. L. V. Ahlfors: Lectures on quasiconformal mappings.

3. J. H. Hubbard: Teichmueller Theory and applications to geometry, topology and dynamics, Vol. 1.

4. T. Pacini, Extremal length in higher dimensions and complex systolic inequalities (available on www.arxiv.org)

Prerequisites: Elementary theory of one complex variable

Level: Advanced undergraduate

Introduction to compact Riemann surfaces

In this mini-course, I will give a brief introduction to the theory of compact Riemann surfaces. After giving basic definitions and discussing topological aspects, we will use the uniformization theorem to give a bijective correspondence between the isomorphism classes of compact Riemann surfaces and the isomorphism classes of algebraic curves. We will then go on to discuss the notion of Fuchsian groups. Daily description is as follows:

RS 1: Examples of Riemann surfaces.

RS 2: Topological aspects.

RS 3: Ramified coverings and the Riemann-Hurwitz formula.

RS4: Uniformization theorem.

RS 4: Fuchsian groups.

Textbook or/and course webpage:

1. E. Girondo and G. Gonzalez-Diez, Introduction to compact Riemann surfaces and dessin d’enfants, LMS Students texts 79, Cambridge University Press, Cambridge, 2012.

2. R. Miranda, Algebraic curves and Riemann surfaces, Graduate Studies in Mathematics Vol. 5, AMS, Providence, Rhode Island, 1995.

3. H. M. Farkas and I. Kra, Riemann surfaces, Graduate Texts in Mathematics Vol. 71, Springer-Verlag, New York, second edition, 1992.

Prerequisites: Complex analysis, point set topology

Level: Graduate, advanced undergraduate

Several Complex Variables

Several Complex Variables is a rather sizely branch of mathematics with links and ties to several other branches such as algebra, differential geometry, partial differential equations, algebraic geometry, and Banach algebras. For this reason, developments in SCV has provided valuable tools for these branches. Unfortunately, for the same reason, the prerequsite list for a basic SCV class has become a rather long list. The aim of this lecture series is not to cover this extensive subject in details, but to give a quick tour, and a basic understanding of its main concepts. Daily description is as follows.

SCV 1: Review of One Variables and Some Definitions.

SCV 2: Holomorphic functions in several variables.

SCV 3: Convexity, pseudoconvexity and plurisubharmonicity.

SCV 4: CR Functions.

SCV 5: The dbar Problem.

SCV 6: Stein Manifolds and Cousin Problems.

Textbook and References:

1. L. Hormander - An Introduction to Complex Analysis in Several Variables, Volume 7 3rd Edition, North-Holland Mathematical Library, 1990.

2. Steven G. Krantz - Function Theory of Several Complex Variables, AMS Chelsea Publishing, 2001.

3. Jiri Lebl, Tasty Bits of Several Complex Variables, Open source: https://www.jirka.org/scv/

4. Steven G. Krantz - What is Several Complex Variables?, The American Mathematical Monthly, Vol. 94, No. 3 (Mar., 1987), pp. 236-256

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Contact: aslicankorkmaz@nesinvakfi.org, berkanuze@gmail.com



Activities are supported by Nesin Mathematical Village and Turkish Mathematical Society