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Derived categories of Quot schemes on smooth curves and tautological bundles

Alina Marian, Andrei Neguţ

Abstract

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.

Derived categories of Quot schemes on smooth curves and tautological bundles

Abstract

We define a categorical action of the shifted quantum loop group of 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.

Paper Structure

This paper contains 46 sections, 29 theorems, 379 equations.

Table of Contents

  1. Introduction
  2. Quot schemes over curves
  3. The functors ${e}_i,{f}_i$ and ${m}_i$
  4. The categorical relations
  5. The operators $e_i,f_i,m_i$ and shifted quantum $L\mathfrak{sl}_2$
  6. Semi-orthogonal decomposition
  7. Cohomology of tautological bundles
  8. Acknowledgements
  9. Fundamentals
  10. Derived categories of projective bundles
  11. Virtual projective bundles
  12. $K$-theory
  13. Nested Quot schemes
  14. Morphisms between nested Quot schemes
  15. Functors and commutation relations
  16. ...and 31 more sections

Key Result

Theorem 1

(a) (Proposition prop:quadratic relation e) For any $i \leq j \in \mathbb{Z}$, there is a natural transformation of functors $\mathsf{D}_{\mathop{\mathrm{\mathsf{Quot}}}\nolimits} \rightarrow \mathsf{D}_{\mathop{\mathrm{\mathsf{Quot}}}\nolimits \times \textcolor{red}{C} \times \textcolor{blue}{C}}$, whose cone has a filtration with associated graded (b) (Proposition prop:e and f) For all $i,j \i

Theorems & Definitions (57)

  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Remark 1
  • Theorem 3
  • Lemma 1
  • Lemma 2
  • Definition 1
  • Lemma 3
  • Lemma 4
  • ...and 47 more