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General


Description

Complex manifolds are central objects in many areas of mathematics: differential geometry, algebraic geometry, several complex variables, mathematical physics, topology, global analysis etc. Their geometry is much richer than that of real manifolds which leads to fascinating phenomena and the need for new techniques.
The present course will give a brief introduction to basic notions and methods in complex differential geometry and complex algebraic geometry. The aim is to present beautiful and powerful classical results, such as the Hodge theorem, as well as to develop enough language and techniques to make the material of current interest accessible.

Semester

Spring 2017 (Monday, January 23 to Friday, March 31)

Timetable

  • Tue 12:05 - 12:55

Prerequisites

Familiarity with basic notions of topological and differentiable manifolds, especially tensors and differential forms.
Knowledge of such Riemannian concepts as the Levi-Civita connection and curvature will be helpful, but not essential.

Syllabus

1. Complex and almost complex manifolds 2. Holomorphic forms and vector fields 3. Complex and holomorphic vector bundles 4. Hermitian bundles, metric connections, curvature 5. Chern classes 6. Hermitian and Kähler metrics 7. Dolbeaut theory and the Hodge theorem 8. Curvature of Kähler manifolds; holomorphic sectional and Ricci curvature

Lecturer


Derek Harland
Email D.G.Harland@leeds.ac.uk
Phone 0113 3435152
vcard
Photo of Derek Harland


Students


Photo of Timothy Burchell
Timothy Burchell
(Surrey)
Photo of Jacob Cable
Jacob Cable
(Manchester)
Photo of Tyrone Cutler
Tyrone Cutler
(Southampton)
Photo of Joe Driscoll
Joe Driscoll
(Leeds)
Photo of Paul Druce
Paul Druce
(Nottingham)
Photo of Maxime Fairon
Maxime Fairon
(Leeds)
Photo of Dominic Foord
Dominic Foord
(Liverpool)
Photo of Ai Guan
Ai Guan
(Lancaster)
Photo of Thomas Honey
Thomas Honey
(Manchester)
Photo of Joe Oliver
Joe Oliver
(Leeds)
Photo of Edward Pearce
Edward Pearce
(Sheffield)
Photo of Norbert Pintye
Norbert Pintye
(Loughborough)
Photo of Gregory Roberts
Gregory Roberts
(Liverpool)
Photo of Michael Roberts
Michael Roberts
(Liverpool)
Photo of Will Rushworth
Will Rushworth
(Durham)
Photo of Albert Wood
Albert Wood
(*External)


Bibliography


Foundations of Differential GeometryKobayashi and Nomizu
Lectures on Kähler geometryMoroianu and Society
Hodge theory and complex algebraic geometry. IVoisin
Principles of Algebraic GeometryGriffiths and Harris
From Holomorphic Functions to Complex ManifoldsFritzsche and Grauert
Complex Geometry: An IntroductionHuybrechts
Lectures on Kähler manifoldsBallmann
Differential analysis on complex manifoldsWells


Note:

Clicking on the link for a book will take you to the relevant Google Book Search page. You may be able to preview the book there. On the right hand side you will see links to places where you can buy the book. There is also link marked 'Find this book in a library'. This sometimes works well, but not always. (You will need to enter your location, but it will be saved after you do that for the first time.)

Assessment



This course will be assessed by an exam consisting of four questions, all of which are compulsory. The time period for the assessment will be the usual two-week period, from 24th April to 7th May.

MAGIC044 Complex Geometry Exam 2017

Files:Exam paper
Released: Monday 24 April 2017 (63.4 days ago)
Deadline: Sunday 7 May 2017 (49.7 days ago)
Instructions:

Solutions must be submitted before 6pm on Friday 7th May. Answer all three questions. All questions carry equal weight. A pass will be awarded to any script judged to answer more than half of the exam correctly.



Files


Files marked L are intended to be displayed on the main screen during lectures.

Week(s)File
exercise1.pdf
slides2017.pdf
solution1.pdf


Recorded Lectures


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