MATH 7320, Modern Algebraic Geometry: Invariant theory (Fall 2017)
Meeting times: MW, 2-3.30pm. The first meeting will be on Sept 6.
Room: 509 Lake.
If you are a Northeastern student planning to take this class, please consider doing this for credit and
Instructor: Prof. Ivan Losev
Office: 519 Lake
Office hours: by appointment.
Content: Here is a list of topics I hope to cover (in roughly this order, some of the topics will
occupy several lectures):
0) A brief intro to Invariant theory: origins and goals.
I) Invariant theory of finite groups: finiteness properties, Noether theorem
(a bound on degrees of generators), Chevalley-Shephard-Todd theorem (on invariants of
complex reflection groups).
II) Birational invariants: separation of generic orbits by birational invariants.
III) Invariants under reductive group actions: a brief review of reductive groups,
Hilbert's theorems on quotients and quotient morphisms. Algebro-geometric properties
IV) Closed orbits in orbit closures: the Hilbert-Mumford theorem, Kempf's theorem of optimal
V) Connection to moment maps for compact group actions (the Kempf-Ness theorem)
VI) Computation of invariants: Classical invariant theory.
VII) Invariants under more general algebraic group actions, U-invariants.
VIII) GIT quotients: construction and properties. Linearization of line bundles.
Applications to moduli spaces in Algebraic geometry.
IX) Local structure of the actions (Luna's etale slice theorem and applications).
X) Actions with good invariant theoretic properties and Vinberg's \theta-groups.
Homework 1 (corrected verison (typo in problem 3 fixed, thanks to Borya for noticing) due Oct 4, either in class or by e-mail by the end of the day; your official score
for every homework is the minimum of 10 and the actual score). Hints. Solutions to problems
1 to 3 and a solution to the extra-credit problem.
Homework 2, due Oct 18. Hints. Solutions.
Homework 3, due Nov 1 (typo in Problem 1 fixed, thanks to Mohamed for noticing). Hints.
Homework 4, due Nov 20, hints. Solutions.
Homework 5, due Dec 7, hints.
References: Our main reference is the great survey [PV]. Occasionally, we will also use other
sources that will be listed here later.
[B] N. Bourbaki, Lie groups and Lie algebras.
Chapters 1-3, Chapters 4-6,
Chapters 7-9, Springer.
[CG] N. Chriss, V. Ginzburg. Representation theory and Complex geometry.
[C] I. Coskun. Course notes.
[CM] D. Collingwood, W. McGovern.
Nilpotent Orbits In Semisimple Lie Algebra: An Introduction, CRC Press, 1993.
[D] I. Dolgachev, Lectures on Invariant
[Dr] J.-M. Drezet, Luna's slice theorem and applications.
[E] D. Eisenbud,
Commutative algebra: with a view toward Algebraic geometry, GTM 150, Springer, 1995.
[K] H. Kraft,
Geometrische Methoden in der Invariantentheorie, 1984. I'm afraid, there's no English translation,
though there is a Russian translation.
[L] I. Losev, Kempf-Ness theorem and Invariant theory.
[MFK] D. Mumford, J. Fogarty, F. Kirwan,
Geometric Invariant theory, Springer 1994.
[OV] A. Onishchik, E. Vinberg, Lie groups and
[PV] V. Popov and E. Vinberg,
Invariant theory in Algebraic geometry 4, Encyclopaedia of Mathematical Sciences, vol. 55, Springer Verlag.
[S] T. Springer,
Invariant Theory, Springer 1977.
Prerequisites: A basic knowledge of algebraic geometry (algebraic varieties, including affine and projective ones, morphisms and their special
classes: affine, finite, projective, etc., properties of algebraic varieties such as smoothness, normality etc.) and the related
commutative algebra. Complex semisimple Lie groups,
Lie algebras and their finite dimensional representations.
A very official syllabus can be found
here. It contains an actual grading policy and grade cut-offs. In particular, there
will be five homeworks.
1) Sept 6, Wednesday: An intro to Invariant theory. This is going to be a mix between a historial introduction,
a motivation and an overview. References include [PV, Section 0], [E, 1.3-1.7], [K, introduction].
2) Sept 11, Monday: Invariants of finite groups, I (finiteness properties and quotients).
References include [S, Section 3]. Here's an alternative (hopefully, more clear) write-up of the proof
of Proposition 5 (the images of the disjoint G-invariant closed subsets under the quotient morphism
3) Sept 13, Wednesday: Invariants of finite groups II (complex reflection groups and the Chevalley-Shephard-Todd theorem).
References include [B, Chapter 5, Section 5].
4) Sept 18, Monday: Actions of algebraic groups and birational invariants. References include [OV, Section 3.1], [PV, Chapters 1 and 2].
write-up of the proof of Rosenlicht's theorem.
5) Sept 20, Wednesday: Structure theory of algebraic groups I (solvable groups). Reference [OV, Section 3.2].
Corrected proof of Theorem 4 (Zariski generic in Step 1 became Weil generic, thanks for
Sveta for noticing the mistake).
6) Sept 25, Monday: Jordan decompositions, Lie algebras of algebraic groups, reductive algebraic groups.
Reference [OV, Sections 3.2, 3.3, 4.1].
7) Sept 27, Wednesday: Classification and representations of reductive algebraic groups.
Reference [OV, Sections 4.1-4.3, 5.2]. Proof of the classification of simple groups.
8) Oct 2, Monday: Invariants strike back! We discuss categorical quotients for reductive group
actions on affine varieties and the normality properties. References include [PV 3.4, 3.9, 4.3-4.5].
9) Oct 4, Wednesday: Adjoint action: semisimple and nilpotent elements, sl_2-triples [CM, Sections 1-3].
HW1 DUE!!! A write-up of some proofs omitted in the lecture.
10) Oct 12, Thursday, 4.45-6.15, Dodge 070 (note unusual date, time and location, Dodge is building 43
on this map). We cover the Chevalley restriction
theorem and discuss orbit closures and invariants for representations of tori. References include [PV, Sections 3.8, 5.3, 5.4].
11) Oct 16, Monday: The Hilbert-Mumford and Kempf-Ness theorems. References include [PV, Sections 5.3, 6.12],
[K, 2.4]. Correction to the Calculus Lemma in Section 2.1 (thanks for Daniil for catching that),
the proof of the KN theorem is the same.
12) Oct 18, Wednesday: We show that closed orbits in vector spaces have reductive stabilizers.
And we discuss characteristics of nilpotent elements. References include [L], [PV, 5.5]. Homework 2 is due!
13)Oct 23, Monday: We prove Luna's criterium for an orbit to be closed and start discussing
the Classical invariant theory. References include [L],[PV, 9.2-9.4].
14)Oct 25, Wednesday: We finish studying the Classical invariant theory by covering the fundamental theorem
for SL_n. The we wrap up and start a new topic by discussing homogeneous spaces. References include [PV, Section 9], [OV, 3.1].
15)Oct 30, Monday: We talk more about homogeneous spaces and start to study U-invariants.
References include [K, III.3].
16)Nov 1, Wednesday: We finish our discussion of U-invariants and apply them to prove that the
determinantal schemes are normal. References include [K, III.3]. Homework 3 is due!
17)Nov 6, Monday: We discuss GIT quotients and semistable points. References include
[PV, Section 4.6], [D, Section 9].
18) Nov 8, Wednesday: We discuss Hilbert scheme and start a discussion of G-equivariant
line bundles. References include [CG, Section 5.1].
19) Nov 13, Monday: We discuss equivariant structures on line bundles. And also talk about the moduli
space of cubic curves in P^2. References include [CG, Section 5.1], [PV, Section 0.14].
20) Nov 15, Wednesday: We discuss Mumford's work on moduli spaces for higher genus curves.
References include [MFK], [C]. If time permits, we have a preliminary discussion of slices -- in the case of compact
group actions on C^\infty-manifolds.
21) Nov 20, Monday: We start our discussion of Luna's etale slice theorem. References include
[Dr] and [PV, 6.1-6.3]. HW4 is due!
22) Nov 27, Monday: We prove the Fun Lemma from the previous lecture (note that it wasn't stated correctly
in the previous lecture, this will be fixed). Time permitting we'll explain some applications of the etale
slice theorem. References are the same as in the previous lecture.
Nov 29 lecture is cancelled.
23) Dec 4, Monday: We discuss applications and an example of the slice theorem. Then we start our discussion
of \theta-groups. References include [PV, 6.7, 8.5], [OV, 4.4].
24) Dec 6, Wednesday: We continue studying \theta-groups. References include [PV, 6.7, 8.5], [OV, 4.4].
HW 5 is due on Dec 7.