TITLE: Fanos with $\chi_{top} = 0$
QUESTION [8 upvotes]: Question 1:Do there exist smooth projective Fano varieties over $\mathbb{C}$ with topological Euler characteristic $0$? 
Question 2:If so what is the lowest dimension in which such examples occur?

REPLY [10 votes]: Every odd $n$-dimensional smooth (2,2)-complete intersection has Euler characteristic 0. I'll leave this website as evidence (http://pbelmans.ncag.info/cohomology-tables/), although this result should also follow from a careful Chern class calculation of $c_n(T_X)$.
When $n\ge 3$ these are Fano varieties. By Remy's argument $n=3$ is the smallest dimension where this can happen.
The reason for this, I believe, is that the intermediate Jacobian of these Fano threefolds is the Jacobian of a hyperrelliptic curve of genus (n+1)/2 (this should be contained in Miles Reid's thesis http://homepages.warwick.ac.uk/~masda/3folds/qu.pdf). Thus the middle cohomology has dimension n+1. The sum of the dimensions of the even cohomology groups is also n+1, which easily follows from the Lefschetz theorems.