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Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.

12 votes

Using algebraic geometry to understand class field theory

A good reference is Neukirch's Algebraic Number Theory. It takes the analogy between algebraic number theory and algebraic geometry seriously, and it includes coverage of class field theory. But you'd …
Michael Joyce's user avatar
3 votes

Reading list for Equivariant Cohomology

Some additional resources, which are more on the algebraic side than the symplectic side: 1) Bill Fulton's Eilenberg lectures on Equivariant Cohomology in Algebraic Geometry, available at David Ander …
Michael Joyce's user avatar
7 votes
Accepted

Uniqueness of the wonderful compactification of a semi-simple group

I assume you mean the variety $G$ considered as a $G \times G$ variety via the action $(g,h) \cdot x = gxh^{-1}$, which is the standard interpretation in the literature. The variety $G$ is spherical a …
LSpice's user avatar
  • 12.9k
0 votes

if V(f) is irreducible, then how to show that the polynomial f itself is irreducible?

As pointed out, the statement is false as written. To get a true statement, you must assume also that $f$ is square-free (or equivalently that the scheme $V(f)$ is reduced). Then, if you assume that …
Michael Joyce's user avatar
8 votes

Algebraic Geometry for non-mathematician

You might be better off studying an analytic approach to algebraic geometry first. The classical reference for this approach to algebraic geometry is Griffith and Harris' Principles of Algebraic Geom …
Michael Joyce's user avatar
2 votes
0 answers
126 views

Seeking a generalization of group embedding of symmetric varieties

I am looking for generalizations of the following construction. Let $G$ be a connected, reductive group and let $\theta : G \rightarrow G$ be an involution. Let $H = G^{\theta}$ be the subgroup of $ …
5 votes

Cohomology Ring of the Flag Manifolds, Cartan Subalgebras, and Weyl Groups

(1) Assuming you are referring to the coefficient field for your cohomology theory, then yes, the result immediately extends to $\mathbb{R}$ and $\mathbb{C}$ coefficients. (2) Two sources are Fulton' …
Michael Joyce's user avatar
4 votes
Accepted

Intersection theory for $G$-varieties - an action on the chow ring?

If you are interested in intersection theory of varieties with $G$-actions, then you want to study equivariant intersection theory. This theory exploits the $G$-action in a way that leads to deeper i …
Michael Joyce's user avatar
7 votes

References request on the algebraic geometry of projective homogeneous spaces

Michel Brion's Lectures on the Geometry of Flag Varieties answers all of your questions in the special case $G = SL_n$ and $P = B$ (the Borel subgroup of upper triangular matrices). See Section 1.4. …
Michael Joyce's user avatar
6 votes
Accepted

The pseudoeffective cone does not contain lines

First, whether a class is pseudoeffective or not depends only on its numerical equivalence class. (The pseudoeffective cone is the closure of the cone of big classes, and $D$ is big if and only if $n …
Michael Joyce's user avatar
5 votes
0 answers
226 views

Is there a brief name for the symmetric space $SL_{2n} / Sp_{2n}$?

Let $V$ be a complex vector space of even dimension. Then the homogeneous space $SL_{2n}(V) / SP_{2n}(V)$ is known to parametrize the space of non-degenerate skew-symmetric bilinear forms on $V$. (1 …
2 votes
Accepted

Chains in $K\backslash G/B$ lying over a closed $K$-orbit

Malheureusement, this is not true, not even for the weak order. This can be seen for example when $G = GL(4)$ and $K = GL(2) \times GL(2)$. Then $K \backslash G / B$ is parameterized by involutions …
Allen Knutson's user avatar
13 votes

Learning Tropical geometry

Several years ago, I participated in a learning seminar in tropical algebraic geometry and collected several helpful survey articles. (This was before Maclagan and Sturmfels' book was written, which …
Michael Joyce's user avatar
3 votes
Accepted

Dimension of spaces of invariants/tableaux functions

The numbers you refer to are known as Kostka numbers. They are discussed in standard references like Fulton's Young Tableaux and Stanley's Enumerative Combinatorics. The weights of a tableaux are of …
Michael Joyce's user avatar
4 votes

Configuration space of flags

(This should be a comment but I don't have enough reputation to leave one.) The quotient of $U$ by $PGL(2)$ is $\mathbb{P}^1 \backslash \{ 0, 1, \infty \}$. To get something compact, you need to all …
Michael Joyce's user avatar

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