**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?



Fanos with $\chi_{top} = 0$
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.


