# Geometric invariant theory for geometers

I am trying to learn "Geometric invariant theory" like it was introduced by Mumford. But I do not have a strong background in algebraic geometry since I work in geometric topology and geometry.

So my question is if there is any nice reference where people explain geometric invariant theory from a geometric viewpoint. In particular, I am looking for a good reference where the analogies between algebraic geometry and differential geometry are pointed out.

• Algebraic geometry is not geometry? – Angelo Aug 2 '12 at 14:57
• @ Angelo: As I tried to make precise I mean differential geometry or geometric topology by the notion geometry. – berl13 Aug 2 '12 at 15:09
• Maybe Richard Thomas' "Notes on GIT and symplectic reduction for bundles and varieties" [ arxiv.org/abs/math.AG/0512411 ] ? – user9154 Aug 2 '12 at 15:14
• Chapter 1 of this thesis arxiv.org/abs/math/0611002 is also very useful. – YangMills Aug 2 '12 at 17:36
• I think these kinds of questions are a bit silly; imagine asking about a reference for elliptic operators that doesn't use any theory of Sobolev spaces! At the very least, I think the OP should clarify more carefully what is his background in (presumably complex) differential geometry, and precisely how far along he is in learning algebraic geometry. – Jason Starr Aug 2 '12 at 18:29

If you just want to get a feeling for invariant theory, here are some books that aren't necessarily comprehensive but nevertheless are enlightening at a more leisurely pace as compared to GIT, which would be useful for someone who isn't as familiar with algebraic groups and algebraic geometry:

• Santos and Rittatore - Actions and Invariants of Algebraic Groups: Minimal prerequisites. A very gentle introduction to some aspects of invariant theory, including some motivation via Hilbert's 14th problem. This book also contains most of the required theory of linear algebraic groups.

• Dolgachev - Lectures on Invariant Theory: This takes a more geometric viewpoint and might be something you are interested in. This only requires some basic knowledge of algebraic geometry.

• Schmitt - Geometric Invariant Theory and Decorated Principal Bundles: this might also be interesting if you are interested in the geometric applications and the related geometry, though I haven't looked into this book very much, but Part 1 does contain a fairly leisurely-looking introduction to GIT

There is also Popov's and Vinberg's treatise "Invariant Theory" in the Ecyclopedia of Mathematical Sciences Volume 55 (Springer) which contains a good summary of the classical results in characteristic zero.

I would recommend a look at chapter 8 of the third edition of Geometric invariant theory by Mumford, Forgarty and Kirwan. It describes a connection between GIT and Hamiltonian group actions in symplectic geometry.

(edit) You may also like Moment maps and geometric invariant theory by Chris Woodward.

• Hi Eugene, nice to see you here. – alvarezpaiva Aug 3 '12 at 8:58
• The longer version being Kirwan's thesis, published as "Cohomology of Quotients in Symplectic and Algebraic Geometry". – Allen Knutson Aug 5 '12 at 9:59

For a "more classical" point of view:

"An introduction to Invariants and Moduli " by S. Mukai.

Fogarty J. "Invariant theory" (Benjamin, 1969)

For a introduction to Mumford's:

P. E. Newstead, "Introduction to Moduli Problems and Orbit Spaces"

Anyway you have to learn (before or after) a Gothendieck-categorical background:

Fondements de la géométrie algébrique (Grothendick). The Hilbert schema chapter is very important (need the Hartshorne "Algebraic Geometry" as base)

Or in more gentle way: Fundamental Algebraic Geometry. Grothendieck's FGA Explained - Fantechi B., Göttsche, L., Illusie L.

• Typo correction: "Mukay" --> "Mukai". – Jason Starr Aug 3 '12 at 12:10
• Typo correction: "Moduly" --> "Moduli". (Sorry could not resist. :) ) – user9072 Aug 3 '12 at 14:25

Read the survey in my article and go over the references therein. It is written with exactly similar intentions you have asked for.