If $k$ is an algebraically closed field of characteristic $0$, then the map
$$X \mapsto N(X) := \#(\text{connected components of }X)$$
defined for smooth projective varieites extends to a ring homomorphism $K_0(Var/k) \to \mathbf{Z}$. So if both $X$ and $Y$ are smooth projective, then $[X] + [Y] = 0$ implies that $N(X) = N(Y) = 0$, hence $X$ and $Y$ are both empty.