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There is a proper map $f: X \rightarrow Y$ of projective varieties. The Decomposition Theorem of Beilinson–Bernstein–Deligne-Gabber states that $$Rf∗IC_X  \cong \oplus_a IC_{\bar{Y_a}}(L_a)[shifts] …

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 …

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 …

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 …

We have a flat family of projective varieties with a torus $T$ action, over $\mathbb{A}^1$. How do the moment map images of the fibers change when we pass from the generic fiber to the special fiber …

Given a possibly singular, connected, symplectic algebraic variety with a torus action, every fiber of the moment map admits a torus action. Is each fiber of this moment map connected? Any examples or …

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 …

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 …

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/ …