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

Seminar


TIME	SPEAKER	TITLE
Jan 25 12:00	Ahmed Yekta Ökten	Bergman Kernel and Biholomorphic Mappings (Graduate Seminar)

Second Week


TIME	SPEAKER	TITLE
Jan 27-Feb 1 9:30-11	Shin-ichi Matsumura	Geometry of Holomorphic Sectional Curvature and Rational Curves
Jan 27-Feb 2 11:30-1	Gregory Sankaran	Moduli of Abelian Varieties
		Lunch
Jan 27-Feb 2 3:30-5:15	Taro Sano	Deformations of Fano and Calabi-Yau Varieties
Jan 27-Feb 2 5:30-7	Mustafa Kalafat	Representation Theory of Complex Lie Algebras B

Seminar


TIME	SPEAKER	TITLE
Jan 29 5:30	İlayda Barış	Introduction to the Toric Varieties (Graduate Lectures)

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

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 (Kutuphanegenel) networks, zeytinlik for istasyon and kurugolet for kisbahcesi 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.
For pick up, faculty is advised to get out of the terminal building from the domestic gate and wait there.
Also let us know before hand, your whatsapp number and/or Google-Hangouts address in case of a problem.
Small amounts of Turkish Lira may be needed in the village
which you can withdraw from ATM machines at the airport or tell the driver along the way.
In case you are coming to the Village independently, from the Airport you may take the regional trains to Selçuk,
possibly with a break at Tepeköy. Then from Selçuk take a taxi which costs about 10€.
Taxi Drivers: +90 232 892 3125 (Gar Taxi), +90 553 243 0835 (Volkan), +90 532 603 1518 (Ata).

Visas: Check whether you need a visa beforehand.

Simply Easy Learning

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
We cover the basic theory of complex abelian varieties and polatisation, and construct their moduli spaces and compactifications. Especially we aim to give an outline of toroidal and Satake compactifications, and to explain the main results about the geometry of the moduli spaces. We hope to explain the very recent progress on the geometric type of A6, which was the last case about which nothing was known

Textbook, References or/and course webpage:

1. Swinnerton-Dyer, Abelian varieties.

2. Griffiths & Harris, Principles of Algebraic Geometry.

3. Lange & Birkenhake, Complex Abelian Varieties.

Prerequisites: Complex analysis, basic algebraic geometry

Level: Graduate

Introduction to Toric Varieties
Aim of the course is to give a brief introduction to toric varieties which have strong connections with polyhedral geometry, combinatorics, commutative algebra, and topology. Because of its concreteness, one can consider the study of toric varieties as a good encounter with the techniques of modern algebraic geometry for the first time. We will try to establish some bridges between algebraic geometry and combinatorics via toric varieties. If time permits, we will study further topics related to toric varieties such as divisors, sheaves, and line bundles on toric varieties; resolution of toric singularities; and some applications in combinatorics and physics.

Daily description is as follows.

1) Preliminaries and Introduction to Affine Toric Varieties
2) Associated Combinatorial Objects; Cones, Polytopes, Fans, etc.
3) Projective Toric Varieties
4) Properties of Toric Varieties and More on Torus Actions
5) Geometry of Toric Varieties
6) Applicatons

Textbook or/and course webpage:

1. W. Fulton, Introduction to Toric Varieties, Princeton University Press, 1993.
2. D. Cox, J. Little, H. Schenck, Toric Varieties, American Mathematical Society, 2011
3. J.P. Brasselet, Geometry of Toric Varieties, 1995

Level: Graduate Advanced undergraduate

Language: TR, EN

Prerequisites:

Basics of algebraic geometry

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

Ökten : Bergman Kernel for a bounded domain D in C^n is a reproducing kernel which has interesting properties that come so naturally. In this talk I will construct Bergman Kernel in a functional analytical way and show that it gives the projection of L^2(D) onto A^2(D). I will investigate the behaviour of the Bergman Kernel under biholomorphic mappings and show that the number of its zeroes is a biholomorphic invariant. I will mention other constructions related to the Bergman Kernel such as the Bergman Representative coordinates and the Bergman metric which is a powerful tool for investigating geometry of bounded domains. At the end of this talk, I will give several results concerning the Bergman Metric and biholomorphism classes of bounded domains. The talk is expository.

Reference:

1. Greene, Robert E.; Kim, Kang-Tae; Krantz, Steven G. The geometry of complex domains. Progress in Mathematics, 291. Birkhäuser Boston, Ltd., Boston, MA, 2011.

2. Krantz, Steven G. Geometric analysis of the Bergman kernel and metric. Graduate Texts in Mathematics, 268. Springer, New York, 2013. xiv+292 pp. ISBN: 978-1-4614-7923-9; 978-1-4614-7924-6