# (semi-)Small resolutions of Peterson varieties

Peterson varieties (in type A) can be described as the subvarieties of the full flag variety

$$\{(F_{i})\;|\; F_{i}\subset \mathbb{C}^{n}, \; \dim F_{i} =i,\; N(F_{i})\subset F_{i+1}\}$$

where $N$ is a regular nilpotent endomorphism of $\mathbb{C}^{n}$. Peterson introduced these types of varieties in the 90s in his work on quantum cohomology of flag varieties and Kostant showed they are singular and gave partial descriptions of their singularities; this work has been recently extended by Insko and Yong here.

Question: do there exist (semi)small resolutions of these varieties?

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I would love to be contradicted, but my educated guess is that this is not known. –  Alexander Woo May 12 '13 at 6:21