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Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.
8
votes
2
answers
2k
views
Global Affine Flag Variety and Affine Flag Variety
There is a construction of a global affine flag variety over $\mathbb{A}^1$ (or another curve) $Fl_{\mathbb{A}_1}$ such that each fiber above $\epsilon \neq 0$ is isomorphic to a direct product of the …
7
votes
1
answer
710
views
Bialynicki-Birula Decomposition and moment polytopes/graphs
Let $X$ be a possibly singular projective scheme which admits a torus $T$ action and has finitely many $T$ fixed points and one-dimensional $T-$orbits. There are many such schemes in the Grassmannian/ …
6
votes
1
answer
498
views
Lattice model for Affine Grassmannians of non type A
There is a Lattice model for affine Grassmannians of type A, due to Lusztig. It describes affine Grassmannians of type A as the moduli space of certain subspaces in an infinite-dimensional $\mathbb{C} …
6
votes
0
answers
448
views
Cohomology of Bott-Samelson varieties?
How is the cohomology of Bott-Samelson varieties (desingularizations of Schubert Varieties ) usually calculated? Let's fix the Lie group to be $GL_n(\mathbb{C})$ or $SL_n(\mathbb{C})$ here.
Is there …
5
votes
1
answer
547
views
Intersections of $B$ and $B^-$ orbits in the flag variety $G/B$
Let $G = SL_n(\mathbb{C})$, $B$ be a Borel subgroup, and $B^-$ be the opposite Borel.
Both the $B$ and $B^-$ orbits on the flag variety $G/B$ are indexed by the Weyl group $W$. Let $S_{w_1}$ and $S^ …
5
votes
0
answers
170
views
Intersections of the B-orbits and the orbits of some other Borel subgroups in the flag varie...
This is a follow-up of this previous question below:
Intersections of $B$ and $B^-$ orbits in the flag variety $G/B$
Let $G = SL_n(\mathbb{C})$, $B$ be the standard Borel subgroup, and consider some …
4
votes
0
answers
189
views
Fibers of torus equivariant moment maps
Given a closed (possibly singular) projective variety $V$ with a symplectic structure and a torus action, there is a moment map
$\mu: V \rightarrow Lie(T)^*$. Note that the dimension of $T$ could be …
4
votes
1
answer
821
views
Equi-dimensionality of special fibers in a flat family
Given a flat map $f: X \rightarrow Y$ such that $X$ is a projective variety and $Y$ is a smooth curve. Each generic fiber is isomorphic to an irreducible projective variety $A$ of dimension $d$.
The …
4
votes
5
answers
2k
views
Elliptic curves and algebraic stacks
I am a student almost without background on algebraic geometry (but I do know basic graduate algebra and topology). Now I am trying to understand something about algebraic stacks.
I want to start wit …
3
votes
Accepted
Global Affine Flag Variety and Affine Flag Variety
Now let me attempt to give an answer myself.
There are very concrete descriptions of the fibers $Fl_{\epsilon}$ in $Fl_{\mathbb{A}^1}$ for each $\epsilon \in \mathbb{A}^1$.
$Fl_{\epsilon} \cong LG …
3
votes
2
answers
686
views
Closure relations between Bruhat cells on the flag variety
Given a Lie group $G$ over $\mathbb{C}$ and a Borel subgroup $B$. There is this famous Bruhat decomposition of the flag variety $G/B$.
How do we prove the closure relations between the cells, which …
3
votes
1
answer
236
views
Singularity of torus fixed points from combinatorial data
May I ask what are the relations between the geometry and combinatorics near a torus fixed point? Any references?
In particular, let $S$ be a scheme that is torus invariant with finitely many zero and …
2
votes
2
answers
749
views
Du Val Singularities and Dynkin diagrams references
May I ask whether there are good references for computing blowups of the Du Val Singularities? Also, how are these singularities related to the Dynkin diagrams?
2
votes
1
answer
485
views
Fibers of the Bott-Samelson Resolution of Schubert Varieties
Is there an explicit (perhaps visual) description of the fibers of the Bott-Samelson Resolutions of Schubert Varieties? Let's fix $G$ to be $GL_n(\mathbb{C})$.
Also, how would the answer to the quest …
2
votes
0
answers
355
views
$G$-equivariant coherent sheaves on Bott$-$Samelson resolutions
Let $G$ be a Lie group and $B$ a Borel subgroup. $G/B$ is the corresponding flag variety.
Let $w$ be an element of the Weyl group $W$ with a reduced expression
$w = s_1 \cdots s_n$. Let $X_w$ be th …