I stumbled upon this isolated singularity of a Calabi-Yau fourfold:
\begin{equation}
x_1x_2+x_3x_4+x_5^2=0
\end{equation}
as a hypersurface in $\mathbb{C}^5$.
Clearly, I can resolve this by a simple blow-up. This does not seem to be a crepant resolution however.
So my question is:
Is there some simple criterion which allows me to tell if there exists a crepant resolution ? In case the answer is yes, how would one go about constructing one ?
In the corresponding threefold case there even exists a small resolution. A second question is hence: Are there results on the existence of small resolutions of isolated fourfold singularities ?
Best Answer
That particular singularity only can have a crepant resolution if it has a small resolution.
Terminal singularities
If a variety has terminal singularities (like yours does), the then every further blow-up also has discrepancies $> 0$. (In other words, whether or not the variety has terminal singularities can be read off from a single log resolution). In particular, every discrete valuation has discrepancy $ > 0$. It follows that:
Fact: If a variety has terminal singularities, then it only has a (commutative) crepant resolution if it has a small resolution
In your example above, the variety clearly has terminal singularities, which can be read off from the blowup you mentioned.
Small resolutions
Ok, so now you want to check whether your variety has a small resolution. The first thing you can do is look at THIS ANSWER by Sándor Kovács. The upshot is there is only hope for a small resolution if your singularity is not factorial.
After staring at your singularity for 4 minutes, I don't see why its not a UFD (hopefully someone, ie Hailong Dao, can tell us easily -- i have a recollection that higher dimensional isolated hypersurface singularities are factorial).
EDIT: Olivier Benoist points out in a comment below that this singularity is factorial due to Samuel's conjecture (proven by Grothendieck).
However, if you see any given singularity that is not factorial, here's what I would try:
Find a Weil divisor that is not Cartier (the existence of such is equivalent to it being not factorial). Blowup the ideal of that Weil divisor. (Maybe do this more than once). The reason I suggest this is because:
Fact: The only way to get a small resolution is to blowup ideals whose vanishing locus has codimension one.
Here's a quick proof, Blowup an ideal $J$ on a (Noetherian!) variety who vanishing locus $W = V(J)$ has codimension $\geq 2$. The blowup turns that ideal $J$ into a Cartier divisor (it principalizes the ideal). In particular, there is now a divisor lying over $W$. Thus this isn't a small resolution.
Of course, blowing up Weil divisors that are not Cartier is how you get the small resolutions in the $xy-uv = 0$ example.