Let $X$ be any complex projective (not necessarily smooth) variety. Let $P(X)$ be the virtual Poincare polynomial of $X$ defined by Deligne's mixed Hodge theory, that is,
$$P(X)=\sum_{i,j} (-1)^{i+j} \dim_{\mathbb{Q}}(\text{gr}^j_W H^i_{\text{cpt}}(X;\mathbb{Q}) ) t^j,$$
where $\text{gr}^j_W H^i_{\text{cpt}}(X;\mathbb{Q})$ is the $j$-th weight-graded piece of the $i$-th rational
cohomology group of $X$ with compact supports.
Let $c_j=\sum_{i} (-1)^{i+j} \dim_{\mathbb{Q}}(\text{gr}^j_W H^i_{\text{cpt}}(X;\mathbb{Q}) )$, i.e., the coefficient of $t^j$. My first question is
(1) Which sequences $(c_0, c_1, ...)$ can arise as sequences of coefficients of virtual Poincare polynomials? If $X$ is non-compact, any sequence might occur, but I'm interested in proper varieties.
This may be very hard. So an easier question is
(2) Is there an example of a projective variety $X$ with $c_1=c_3=c_5=\cdots=0$ and $P(X)|_{t=1}$<0? Smooth projective varieties cannot satisfy both conditions. It seems to me that if such a variety exists, then its desingularization has a badly-behaved exceptional locus.
Thank you for any comments.
