TITLE: What does it mean that $[X]+[Y]=0$ in the Grothendieck ring of varieties?
QUESTION [11 upvotes]: This is a really basic question. If I have two non-isomorphic varieties $X$ and $Y$, is it possible that $[X]+[Y]=0$ in the Grothendieck ring?
If so, what does this mean geometrically? Obviously $[\emptyset]-[X]-[Y]$ is not one of the standard relations modded into the ring, so $[X]+[Y]$ has to be some non-trivial combination of such relations. I'm having trouble seeing how this could happen though. An example would be especially nice.

REPLY [15 votes]: The element $[X]+[Y]$ of $K_0({\rm Var}_k)$ is the class of the disjoint union $X\cup Y$. So your question amounts to understand whether the class of a variety, say $X$, can vanish in $K_0({\rm Var}_k)$.
As is implicit in the questions, and in the other answers, the ring $K_0({\rm Var}_k)$ is hard to understand, and is essentially only understood thanks to a few geometrically motivated ``motivic measures'':

If $k$ is finite, the counting measure $N\colon K_0({\rm Var}_k)\to\mathbf Z$;
The Euler characteristic using any cohomology theory with compact supports you can imagine;
If $k=\mathbf C$, the Hodge-Deligne polynomial, valued in $\mathbf Z[u,v]$;
Using Deligne's theory of weights on étale cohomology, one can define a Poincaré polynomial, valued in $\mathbf Z[t]$;
If $k$ is a field of characteristic $0$, Larsen and Lunts have defined an exotic — and useful — motivic measure with valued in the free abelian group of stable birational class of varieties.

In any case, any of these motivic measures shows that the class of a non-empty variety does not vanish. They can also be useful to show that the classes of two varieties are not equal.