0
votes
1answer
164 views

Are period domains ever contractible

Which simply-connected period domains are contractible? Examples. Siegel upper-half space? Poincare upper-half plane? Universal cover of a Shimura variety? Are these contractible?
1
vote
0answers
180 views

Good covers on complex algebraic varieties with normal crossings singularities

Let $X$ be a topological space. A good cover on $X$ is an open cover such that all finite non-empty intersections are contractible. It is a theorem of Hironaka that (complex) algebraic sets admit ...
7
votes
2answers
356 views

The topology of open semi-algebraic sets (appl.: totally positive matrices)

Let $P_1,\ldots,P_r$ be polynomials over ${\mathbb R}^N$. I am interested in the homotopy class of the semi-algebraic set defined by $$P_j(x_1,\ldots,x_N)>0,\qquad j=1,\ldots,r.$$ Is there a ...
2
votes
1answer
233 views

Sequences of groups, exact not just in etale but also in the Zariski topology

Given $A=k[[x,y]]$, with say $k=\mathbb{C}$ and $X=Spec(A)$. $B$ denotes a free $A$-algebra of rank $e^2$, actually we have $B=A[u,v]$ with $u^e=x$, $v^e=y$ and $uv=\xi_e vu$, where $\xi_e$ is an e-th ...