Are virtual Betti numbers of a projective algebraic variety obtained from a cell decomposition the same as the usual Betti numbers?

To be more precise, let $X$ be an algebraic variety (not necessary irreducible) over the complex numbers, we define the $i$-th Betti number to be the dimension of the $i$-th singular cohomology group of $X$ with coefficients in $\mathbb{C}$.

Assume that $X$ has a cell decommposition, i.e. there is a filtration of closed subvarieties
$ F^1\subset \cdots \subset F^n =X$
such that the complements $F^i \setminus F^{i-1}$ are disjoint union of affine spaces, which we call cells.

Then, the virtual Betti numbers (following Byalinicki-Birula) are

$ b_{2i+1}=0$, $b_{2i}=$ number of $i$-dim cells.

Here are two remarks on it.

1) If one takes Borel-Moore homology instead of singular cohomology the equality of virtual Betti numbers with Betti numbers is a direct consequence of the long exact localization sequence (for an open subset and its closed complement) and the knowledge of the BM-homology on affine spaces. This does not use the assumption of $X$ being projective. So, for spaces with cell decomposition is the BM-homology isomorphic to singular cohomology? But there must be assumptions on $X$ to get an isomorphism, else
$X=\mathbb{C}^n$ would be a counterexample.

2) In Fresse: Betti numbers of Springer fibre in type A, p.12 they state the equality of virtual Betti numbers with the dimensions of the usual sheaf cohomology with coefficients in $\mathbb{Q}$. He gives as source Kashiwara, Schapira (Sheaves on Manifolds), section 4.6 which does not seem to exist.

Of course, when you assume $X$ to be smooth (and projective), or having an operation of a torus with only finitely many fixed points, there are much more methods available.