Is there example of hypersurface $X \subset \mathbb{P}^n$ satisfying 


1. $X$ is of degree 2. (I mean, the Poincare dual PD(X) is 2u, where u is a generator of
$H^2(\mathbb{P}^n, \mathbb{Z})$.
2. Some odd betti number is non-zero.


Thank you for any comment. 



Hypersurface of complex projective space
As noticed by Sasha in his comment, the answer is **no**. 


The following proof also shows that this result cannot be generalized for higher values of the degree.


For any smooth complex hypersurface of degree $d$, say $X\_d \subset \mathbb{P}^{n+1}$, by standard arguments involving Lefschetz theorem we have $$H^k(X\_d)= H^k(\mathbb{P}^n) \quad \mathrm{for} \; k \neq n.$$


In particular, all the odd Betti numbers are zero, except possibly the middle Betti number 
when $n$ is odd. On the other hand, the Euler-Poincare characteristic of $X\_d$ is equal to $$\chi(X\_d)= \langle c\_n(T\_{X\_d}), [X\_d] \rangle =\frac{1}{d}[(1-d)^{n+2}-1]+n+2,$$
so for $n$ odd and $d=2$ the middle cohomology group must be zero too. Notice that for $n$ odd and $d >2$ one always has a non-zero middle Betti number. For instence, if $X \subset \mathbb{P}^4$ is a smooth cubic hypersurface, then $b\_3(X)=10$. 


A good reference for these results is Dimca's book *Singularities and topology of hypersurfaces*, Chapter 5, which also considers the case of hypersurfaces in $\mathbb{P}^{n+1}$ with isolated singularities. 


