Derived categories of Quot schemes on smooth curves and tautological bundles

We define a categorical action of the shifted quantum loop group of $\mathfrak{sl}_2$ on the derived categories of Quot schemes of finite length quotient sheaves on a smooth projective curve. As an application, we obtain a semi-orthogonal decomposition of the derived categories of Quot schemes, of representation theoretic origin. We use this decomposition to calculate the cohomology of interesting tautological vector bundles over the Quot scheme.

💡 Research Summary

In this paper Marian and Negut develop a new bridge between representation theory and the geometry of Quot schemes on smooth projective curves. The authors start with a locally free sheaf (V) of rank (r) on a smooth complex projective curve (C) and consider the Grothendieck Quot scheme (\operatorname{Quot}_d(V)) parametrising zero‑dimensional quotients of length (d). They construct three families of Fourier–Mukai functors between the derived categories (D^b(\operatorname{Quot}_d)) for varying (d):

• (e_i : D^b(\operatorname{Quot}d) \to D^b(\operatorname{Quot}{d+1}\times C)) and
• (f_i : D^b(\operatorname{Quot}_{d+1}) \to D^b(\operatorname{Quot}_d\times C))

which are defined using the nested Quot scheme (\operatorname{Quot}{d,d+1}) and the tautological line bundle (\mathcal L) on it, together with the natural projections (p{\pm}). The third family, (m_j), is given by tensoring with the exterior powers (\wedge^j\mathcal E) of the universal kernel (\mathcal E) on (\operatorname{Quot}_d\times C).

The first major result (Theorem 1) is a precise list of natural transformations among compositions of these functors. For example, for (i\le j) there is a morphism (e_j\circ e_i\to e_i\circ e_j) whose cone admits a filtration whose graded pieces are pull‑backs of functors (e_k\circ e_{i+j-k}). Analogous relations hold for (f)’s, for the mixed compositions (f_j\circ e_i), and for the interaction of (m_j) with (e_i) and (f_i). These relations match exactly the defining relations of the shifted quantum loop algebra (U_q(L\mathfrak{sl}2)). Passing to K‑theory, the authors recover the algebraic action of this quantum group on (\bigoplus{d\ge0}K(\operatorname{Quot}_d)) that they previously constructed in