Let $X$ be a smooth complex algebraic variety endowed with a $\mathbb{C}^*$ action. We assume also to have an antiholomorphic involution $\sigma$ over $X$ such that it anticommutes with the action above i.e $$\sigma(t \cdot x)=\bar{t}\cdot \sigma(x) .$$

Let us also assume that the $\mathbb{C}^*$-action respects the properties in order to get a well behaved Byalinicki-Birula decomposition.

For every $x\in X$ there exists $\lim_{t \to 0} t \cdot x$ and the fixed point set is a projective variety. (I think $X$ is called semiprojective in this case).

In this case, we know that the Poincare polynomial of $X$ can be expressed as $$P(X,t)=\sum_{i \in I}P(F_i,t)t^{d_i} $$ where $$X^{\mathbb{G}_m}=\bigsqcup_{i \in I} F_i $$ and $F_i$ are connected and $d_i$ are some positive integers associated to the action.

The fixed point set $X^{\sigma}$ of the antiholomorphic involution is a smooth real manifold. The hypothesis tells us that $\sigma(X^{\mathbb{G}_m})=X^{\mathbb{G}_m}$. Is it still true somehow that $$P(X^{\sigma},t)=\sum_{i \in I}P(F_i^{\sigma},t)t^{h_i} $$ or not ?

Is this known in the literature? I'm totally new to real algebraic geometry.

EDIT: The answer below indicates this is not true in general. However, I'd be interested in the following more specific situation. $X$ should be given the structure of an hyperkahler manifold with complex structures $I,J,K$ such that the complex algebraic variety we are looking at is the one induced by $I$.

The involution $\sigma$ should then be antiholomorphic with respect to structure $I,J$ and holomorphic with respect to $K$. The $\mathbb{C}^*$ action should be algebraic with respect to the structure induced by $I$ while in general it is not clear what happens with respect to the other structures.

The setting to think of is that of Non Abelian Hodge theory: it is known that we have an hyperkahler manifold $M$ such that with respect to $I$ it is the moduli space of stable higgs bundle $(\mathcal{E},\phi)$ of fixed rank $n$ and degree $d$. There we have the action $$t(\mathcal{E},\phi)=(\mathcal{E},t\phi) .$$ With respect to the structure $J$ is the de Rham moduli space of connections and with respect to $K$ is the associated (twisted)character variety.

As suggested below, it is likely that hyperkahler structure should help because of parity of dimension of the cells involved somehow, but I wasn't able to prove this neither to find some references.

Morsenot Hodge. $\endgroup$