Reference for Hodge decomposition for flag variety
$begingroup$
Any references for proof of the following facts:
The cohomology of the (complex) flag variety is always in $(p, p)$-type of Hodge Decomposition.
The natural map $G/T → G_mathbb{C}/B$ is a diffeomorphism, where $G$ is compact connected real Lie group and $T$ is its maximal torus. $G_mathbb{C}$ denote the complexification of $G$ and choose a Borel subgroup $B$ containing the complexification of $T$.
algebraic-geometry reference-request homology-cohomology complex-geometry hodge-theory
asked Nov 29 '18 at 17:29
userabcuserabc
19210
$endgroup$
add a comment |
$begingroup$
Any references for proof of the following facts:
The cohomology of the (complex) flag variety is always in $(p, p)$-type of Hodge Decomposition.
The natural map $G/T → G_mathbb{C}/B$ is a diffeomorphism, where $G$ is compact connected real Lie group and $T$ is its maximal torus. $G_mathbb{C}$ denote the complexification of $G$ and choose a Borel subgroup $B$ containing the complexification of $T$.
algebraic-geometry reference-request homology-cohomology complex-geometry hodge-theory
asked Nov 29 '18 at 17:29
userabcuserabc
19210
$endgroup$
3
$begingroup$
1 : The cohomology of flag variety is generated by the Bialynicki-Birula cells which are algebraic so of type $(p,p)$ : a reference is the book "Complex geometry and representation theory" by Chriss and Ginzburg, chapter 3.
$endgroup$
– Nicolas Hemelsoet
Nov 29 '18 at 19:57
3
$begingroup$
2 : G being compact form of $G_{mathbb{C}}$ is a maximal compact subgroup of $G_{mathbb{C}}$. Now using Iwasaw Decomposition for $G_{mathbb{C}}$ we see that $G_{mathbb{C}} = G.B$ and hence a surjective map $G rightarrow G_{mathbb{C}}/B$. One more observation is that $G cap B = T$, since $G$ has only semisimple elements. Then we are done. Details can be looked up in the chapter Iwasawa decomposition in Lie Groups by D.Bump.
$endgroup$
– random123
Nov 30 '18 at 5:44
add a comment |
$begingroup$
Any references for proof of the following facts:
The cohomology of the (complex) flag variety is always in $(p, p)$-type of Hodge Decomposition.
The natural map $G/T → G_mathbb{C}/B$ is a diffeomorphism, where $G$ is compact connected real Lie group and $T$ is its maximal torus. $G_mathbb{C}$ denote the complexification of $G$ and choose a Borel subgroup $B$ containing the complexification of $T$.
algebraic-geometry reference-request homology-cohomology complex-geometry hodge-theory
asked Nov 29 '18 at 17:29
userabcuserabc
19210
$endgroup$
Any references for proof of the following facts:
The cohomology of the (complex) flag variety is always in $(p, p)$-type of Hodge Decomposition.
The natural map $G/T → G_mathbb{C}/B$ is a diffeomorphism, where $G$ is compact connected real Lie group and $T$ is its maximal torus. $G_mathbb{C}$ denote the complexification of $G$ and choose a Borel subgroup $B$ containing the complexification of $T$.
algebraic-geometry reference-request homology-cohomology complex-geometry hodge-theory
algebraic-geometry reference-request homology-cohomology complex-geometry hodge-theory
asked Nov 29 '18 at 17:29
userabcuserabc
19210
asked Nov 29 '18 at 17:29
userabcuserabc
19210
asked Nov 29 '18 at 17:29
userabcuserabc
19210
asked Nov 29 '18 at 17:29
userabcuserabc
19210
asked Nov 29 '18 at 17:29
userabcuserabc
19210
19210
3
$begingroup$
1 : The cohomology of flag variety is generated by the Bialynicki-Birula cells which are algebraic so of type $(p,p)$ : a reference is the book "Complex geometry and representation theory" by Chriss and Ginzburg, chapter 3.
$endgroup$
– Nicolas Hemelsoet
Nov 29 '18 at 19:57
3
$begingroup$
2 : G being compact form of $G_{mathbb{C}}$ is a maximal compact subgroup of $G_{mathbb{C}}$. Now using Iwasaw Decomposition for $G_{mathbb{C}}$ we see that $G_{mathbb{C}} = G.B$ and hence a surjective map $G rightarrow G_{mathbb{C}}/B$. One more observation is that $G cap B = T$, since $G$ has only semisimple elements. Then we are done. Details can be looked up in the chapter Iwasawa decomposition in Lie Groups by D.Bump.
$endgroup$
– random123
Nov 30 '18 at 5:44
add a comment |
3
$begingroup$
1 : The cohomology of flag variety is generated by the Bialynicki-Birula cells which are algebraic so of type $(p,p)$ : a reference is the book "Complex geometry and representation theory" by Chriss and Ginzburg, chapter 3.
$endgroup$
– Nicolas Hemelsoet
Nov 29 '18 at 19:57
3
$begingroup$
2 : G being compact form of $G_{mathbb{C}}$ is a maximal compact subgroup of $G_{mathbb{C}}$. Now using Iwasaw Decomposition for $G_{mathbb{C}}$ we see that $G_{mathbb{C}} = G.B$ and hence a surjective map $G rightarrow G_{mathbb{C}}/B$. One more observation is that $G cap B = T$, since $G$ has only semisimple elements. Then we are done. Details can be looked up in the chapter Iwasawa decomposition in Lie Groups by D.Bump.
$endgroup$
– random123
Nov 30 '18 at 5:44
3
3
$begingroup$
1 : The cohomology of flag variety is generated by the Bialynicki-Birula cells which are algebraic so of type $(p,p)$ : a reference is the book "Complex geometry and representation theory" by Chriss and Ginzburg, chapter 3.
$endgroup$
– Nicolas Hemelsoet
Nov 29 '18 at 19:57
$begingroup$
1 : The cohomology of flag variety is generated by the Bialynicki-Birula cells which are algebraic so of type $(p,p)$ : a reference is the book "Complex geometry and representation theory" by Chriss and Ginzburg, chapter 3.
$endgroup$
– Nicolas Hemelsoet
Nov 29 '18 at 19:57
3
3
$begingroup$
2 : G being compact form of $G_{mathbb{C}}$ is a maximal compact subgroup of $G_{mathbb{C}}$. Now using Iwasaw Decomposition for $G_{mathbb{C}}$ we see that $G_{mathbb{C}} = G.B$ and hence a surjective map $G rightarrow G_{mathbb{C}}/B$. One more observation is that $G cap B = T$, since $G$ has only semisimple elements. Then we are done. Details can be looked up in the chapter Iwasawa decomposition in Lie Groups by D.Bump.
$endgroup$
– random123
Nov 30 '18 at 5:44
$begingroup$
2 : G being compact form of $G_{mathbb{C}}$ is a maximal compact subgroup of $G_{mathbb{C}}$. Now using Iwasaw Decomposition for $G_{mathbb{C}}$ we see that $G_{mathbb{C}} = G.B$ and hence a surjective map $G rightarrow G_{mathbb{C}}/B$. One more observation is that $G cap B = T$, since $G$ has only semisimple elements. Then we are done. Details can be looked up in the chapter Iwasawa decomposition in Lie Groups by D.Bump.
$endgroup$
– random123
Nov 30 '18 at 5:44
add a comment |
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3
$begingroup$
1 : The cohomology of flag variety is generated by the Bialynicki-Birula cells which are algebraic so of type $(p,p)$ : a reference is the book "Complex geometry and representation theory" by Chriss and Ginzburg, chapter 3.
$endgroup$
– Nicolas Hemelsoet
Nov 29 '18 at 19:57
3
$begingroup$
2 : G being compact form of $G_{mathbb{C}}$ is a maximal compact subgroup of $G_{mathbb{C}}$. Now using Iwasaw Decomposition for $G_{mathbb{C}}$ we see that $G_{mathbb{C}} = G.B$ and hence a surjective map $G rightarrow G_{mathbb{C}}/B$. One more observation is that $G cap B = T$, since $G$ has only semisimple elements. Then we are done. Details can be looked up in the chapter Iwasawa decomposition in Lie Groups by D.Bump.
$endgroup$
– random123
Nov 30 '18 at 5:44