Algebraic Geometry and Linear Algebra

Gian Pietro Pirola
University of Pavia (Italy)
Abstract
We would like to present some methods of linear algebra that are frequently used in algebraic geometry and conversely.
The first lecture will illustrate some basic “tricks” as Hopf’s and Gieseker’s lemmas, some antisymmetric variations and the so called ker-coker lemma.
In the second and in the third lecture using some rudiment from the Hodge theory and the theory of Chern classes we give some application of the above methods to the topology of the algebraic varieties.
The preliminaries for this course are:
  • Projective space and Grassmannian.
  • Vector bundle and the operation with vector bundles.
  • Definition of Chern classes for a complex vector bundle.
  • The basic statement of the Hodge theory (decomposition theorem, dd* lemma, hodge diamond).  
References:
[1] Griffiths and Harris, Principles of Algebraic Geometry, Wiley, New York, 1978. (Chapter 0: Complex manifolds, Hodge theory; Chapter 1: Grassmannians).
[2] Bott and Tu, Differential Forms in Algebraic TopologyVolumen 82 de Graduate Texts in Mathematics, Springer New York, 1995. (§20 Chern Classes of complex vector bundles).
[3] Wikipedia could also be useful for a beginner.