The 4th CAGWS
COMPLEX ALGEBRAIC GEOMETRY WINTER SCHOOL
Nesin Mathematics Village, Şirince, İzmir
January 20February 2, 2020
(Invitation process still continues.
Let us know
if you have a researchminicourse proposal!)
First Week
Second Week
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

RegisterTR
Poster
Participants
Arrival
Information
The 4th CAGWS will be held at the
Nesin Mathematical Village
in Şirince, İzmir.
There will be about 15 minicourses of introductory nature, related to the GeometryTopology research subjects.
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 underrepresented 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.
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, nondegeneracy and local Milnor fibration.
Textbook or/and course webpage:
1. M.Oka, Nondegenerate mixed functions, Kodai J. Math. 33 (1), 162,2010
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)
Singular Algebraic Curves and Varieties
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 differentialgeometric 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: 0521525489
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: 0521809061
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: 9780521134200
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: 9780817649371
5. Thomas, R. P. Notes on GIT and symplectic reduction for bundles and varieties. Surveys in differential geometry. Vol. X, 221273, 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 fourmanifolds. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1990. x+440 pp. ISBN: 0198535538
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. SpringerVerlag, Berlin, 1994. xiv+292 pp. ISBN: 3540569634
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 holomorohic vector bundle over complex manifolds.
In particular, I explain a realtion between positivity of holomorphic
sectional curvatures and the geometry of rationtal curves (that is,
1dim projective space embedded in manifolds). The goal of this talk
is to give a structure theorem for rationally connected fibrations of
projective manifolds with nonnegative holomorphic sectional
curvature.
Textbook or/and course webpage:
1. Foundations of Differential Geometry,
written by Shoshichi Kobayashi and Katsumi Nomizu, Wiley Classics Library.
2. RCpositivity, rational connectedness and Yau's conjecture,
written by Xiaokui Yang, Camb. J. Math. 6 (2018), no. 2, 183–212
3. On projective manifolds with semipositive holomorphic sectional curvature,
4. On morphisms of compact Kähler manifolds with semipositive
holomorphic sectional curvature
written Shinichi 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 CalabiYau Varieties
In this lecture series we present an algebraic approach to
deformations of Fano and CalabiYau varieties. Fano varieties and
CalabiYau 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 (BogomolovTianTodorov 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.
SpringerVerlag, Berlin, 2005. xii+309 pp.
3.Namikawa, Yoshinori CalabiYau 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. SpringerVerlag, 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. SpringerVerlag, New York, 1991. xvi+551 pp. ISBN: 0387975276
2. Anthony W. Knapp. Lie groups beyond an introduction,
volume 140 of Progress in Mathematics. Birkhauser Boston, Inc., Boston, MA, second edition, 2002.
Prerequisites:
Linerar Algebra. Lie groups and Lie algebras.
Moduli of Abelian Varieties
Conformal Invariants and Quasiconformal Maps
Several Complex Variables
In this lecture series, our aim is to introduce Several Complex Variables theory which is used in many applications in Geometry. The course gives an introduction to the most central aspects and methods in the theory of holomorphic functions of several complex variables, and applications of these to approximation and mapping problems.The contents are as follows.
Holomorphic functions and mappings. Hartog's extension phenomenon, domains of holomorphy and holomorphic convexity. Plurisubharmonic functions. The dbar equation and the Levi problem. Oka's approximation theorem, Runge pairs and the Cousin problems. Polynomial convexity and applications.
Textbook or/and course webpage:
1. S. Krantz. A Crash Course in Several Complex Variables
2. Raghavan Narasimhan, "Several complex variables", Chicago lectures in mathematics.
Level: Graduate, advanced undergraduate
Prerequisites:
Complex Analysis
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Contact: aslicankorkmaz@nesinvakfi.org, berkanuze@gmail.com
Activities are supported by Nesin Mathematical Village and Turkish Mathematical Society
