A Categorical Quantum Toroidal Action on Hilbert Schemes

Yu Zhao

We categorify the commutation of Nakajima's Heisenberg operators $P_{\pm 1}$
and their infinitely many counterparts in the quantum toroidal algebra
$U_{q_1,q_2}(\ddot{gl_1})$ acting on the Grothendieck groups of Hilbert
schemes. By combining our result with arxiv:1804.03645 , one obtains a
geometric categorical $U_{q_1,q_2}(\ddot{gl_1})$ action on the derived category
of Hilbert schemes.
In order to construct the categorification, we construct two triple moduli
spaces and study their singularity structure from the viewpoint of the minimal
model program. We prove that one moduli space is a canonical singularity and
the other one is semi-divisorial log terminal. It leads to the natural
transformations in derived categories.