**The short:** Let


$X= \{(x,y,z) \in \mathbb{C}^\* \times \mathbb{C} \times \mathbb{C} \, |\, yz-x\neq 0\}$


Compute $H^\*\_c(X)$ (say with $\mathbb{C}$-coefficients).


**The long:** Unless I messed something up, the answer should be


$H\_c^3(X) = \mathbb{C}$, $H\_c^4(X) = \mathbb{C}^2$, $H\_c^5(X) = \mathbb{C}^2$, $H\_c^6(X)=\mathbb{C}$, and $0$ otherwise.


However, as will become clear I did this through an extremely convoluted argument and I am hoping someone can explain to me a simple way of doing this. 


**Some context:** this question is closely related to


[Intersection of plus/minus cells in Bialynicki-Birula decomposition](https://mathoverflow.net/questions/111682/intersection-of-plus-minus-cells-in-bialynicki-birula-decomposition) 


Namely, $X$ is the intersection of the big Bruhat cell and the big opposite Bruhat cell for $SL\_3$.


It is also closely related to 


[Are Kazhdan-Lusztig $R$-polynomials the Poincare polynomials of the corresponding affine varieties](https://mathoverflow.net/questions/83877/are-kazhdan-lusztig-r-polynomials-the-poincare-polynomials-of-the-corresponding) 


(the Hodge-Euler characteristic of $X$ is the $R$-polynomial corresponding to the identity and the longest element in type $A\_2$; this is a special case of a general fact about $R$-polynomials).


**My convoluted argument:**  Considering the alternatives $y\neq 0$ and $y= 0$, one obtains a decomposition of $X$ into


$X = (\mathbb{C}^\*)^3 \sqcup \mathbb{C}\times \mathbb{C}^\*$.


This gives rise to a long exact sequence that puts several restrictions on $H^\*(X)$ (but doesn't fully determine it, namely $H\_c^3(X), H\_c^4(X), H\_c^5(X)$ aren't fully determined). 


So far this is nice, but now the convoluted bit starts.
It is not too hard to see that $H^{\*-3}(X)$ equals $Ext^\*(\Delta\_e, \Delta\_{w\_0})$ where $\Delta\_e$ is the unique simple Verma and $\Delta\_{w\_0}$ is the unique projective Verma in the principal block of the BGG-category $\mathcal{O}$ of $\mathfrak{sl}\_3$. 


**Aside:** this a special case of a statement connecting extensions of Verma modules with cohomology of intersections of Bruhat cells and opposite Bruhat cells (and also why I am interested in the cohomology of these intersections).


Now some standard representation theoretic facts about these $Hom$ spaces combined with the decomposition above yield what I claimed the answer to be.


I would love a simpler/more geometric way of going about this computation!



A cohomology computation request.
Reladenine -- your $X$ is the complement in the affine 3-space of the union of two hypersurfaces $Y$ and $Z$, the first given by $x=0$, the second by $yz=x$. The intersection $Y\cap Z$ is the union of two intersecting affine lines. Moreover, both $Y$ and $Z$ are isomorphic to $\mathbb{C}^2$ (note that $Z$ is the graph of a function). So the Borel-Moore homology of $Y\cup Z$ is given by $H^{BM}\_i(Y\cup Z)=\mathbb{C}^2$ if $i=4,3$ and $\mathbb{C}$ if $i=2$. So, by the Alexander duality $\tilde H^j(X)\cong H^{BM}\_{6-1-j}(Y\cup Z)$ where $\tilde H$ stands for the reduced cohomology, $H^i(X)=\mathbb{C}$ if $i=0,3$ and $\mathbb{C}^2$ if $i=1,2$. Applying the Poincar\'e duality $H^i(X)\cong H^{6-i}\_c(X)$ one gets the answer you give in your posting.


