Mathematical Institute

University of Oxford

Andrew Wiles Building

Radcliffe Observatory Quarter (550)

Woodstock Road

Oxford, OX2 6GG

Email : tiago.jardimdafonseca **at** maths.ox.ac.uk

Office : N2.04

Tuesdays 13:30 - 15:30. First lecture: 30th of April.

**No lecture on the 7th of May.** Replacement on Friday **14/06/19 from 14:00 to 16:00**.

This is a *basic* course in algebraic geometry, and can be regarded as a natural continuation of a first course on the theory of schemes, such as Oxford's C2.6 Introduction to Schemes.

The main goal of the first part of these lectures is to introduce some differential techniques in the context of algebraic geometry. Therefore, we will talk about differential forms, tangent spaces, smoothness, etc. These are very concrete (and purely algebraic!) notions which enable us to better understand the geometry of schemes. We will illustrate the general theory in the simplest possible case, that of algebraic curves. For instance, the important concept of genus and the classical Riemann-Roch theorem will be discussed in detail.

In a second moment, we will do some integral calculus in the guise of *algebraic de Rham cohomology*. This is a cohomology theory for algebraic varieties which is similar in spirit to the usual de Rham cohomology for smooth manifolds. By considering families of algebraic varieties, we will discover the *Gauss-Manin connection*, which can be thought as a modern viewpoint on the classical *Picard-Fuchs equations*. These differential equations of algebro-geometric origin lie at the heart of many recent developments and open questions pertaining to number theory and algebraic geometry (and physics!). Depending on time constraints, we will end the lectures by discussing some general theorems concerning these equations, such as the regularity of the Gauss-Manin connection and the local monodromy theorem.

**Prerequisites:** a basic understanding of scheme theory and sheaf cohomology (e.g. Q. Liu's chapters 2, 3 and 5).

**Warm-up reading:** the beautiful expository paper Euler and algebraic geometry by Burt Totaro explains some of the historical roots, and discuss some current open questions, related to the subject of these lectures.

Please, beware that the following lecture notes are not polished and might contain mistakes.

- Kähler differentials
- Smoothness 1
- Smoothness 2
- Tangent bundle and curves
- Connections and local systems
- De Rham cohomology 1
- De Rham cohomology 2; Gauss-Manin connection
- The regularity and the local monodromy theorems

Here is a short list of exercises (regularly updated). If you want more, the exercises concerning smoothness and differentials in Liu's book are also good.

- R. Hatshorne,
*Algebraic Geometry*, GTM Springer-Verlag (1977). - Q. Liu,
*Algebraic Geometry and Arithmetic Curves*, Oxford University press (2006).

- J. Bertin, J.-P. Demailly, L. Illusie, C. Peters,
*Introduction to Hodge Theory*, SMF/AMS Texts and Monographs 8 (2002). - P. Deligne,
*Equations différentielles à points singuliers réguliers*, LNM 163 (1970). - A. Grothendieck,
*On the de Rham cohomology of algebraic varieties*. Publ. Math. de l'I.H.E.S., tome 29 (1966). - A. Grothendieck et al.,
*Revêtements étales et groupe fondamental (SGA 1)*. Augmenté de deux exposés de Mme M. Raynaud. EDP Sciences (2003). - N. M. Katz, T. Oda,
*On the differentiation of de Rham cohomology classes with respect to parameters*. J. Math. Kyoto Univ. 8-2, 199-213 (1968). - N. M. Katz,
*The Regularity Theorem in Algebraic Geometry*Acts, Congrès intern. math. (1970). - J.-P. Serre,
*Géométrie Algébrique et Géométrie Analytique*Annales de l'Institut Fourier 6 (1956).