Lecture series for virtual invariants of Quot schemes of surfaces

Woonam Lim

Abstract : Let S be a smooth projective surface. Grothendieck's Quot scheme on S admits a 2-term perfect obstruction theory when it parametrizes torsion quotients. This allows us to define various virtual invariants of Quot schemes, including homological (K-theoretic) descendent invariants and virtual Segre/Verlinde numbers. The study of these invariants is partially motivated by the parallel theory of moduli of sheaves. A special feature of the Quot scheme theory is the conjectural rationality of the homological (K-theoretic) descendent series. We explain how this can be proven for all surfaces with p_g>0 using the multiplicative structural formula involving Seiberg-Witten invariants. The same method applies to the study of the virtual Segre/Verlinde series. We also explain the virtual Segre/Verlinde correspondence and a special symmetry for punctual Quot schemes which is reminiscent of the numerical strange duality.

First lecture : Based on the deformation theory of Quot schemes, we give a list of examples where Quot schemes admit perfect obstruction theory. To motivate the study of virtual geometry of Quot schemes, we briefly discuss applications of Quot schemes to the study of moduli of stable bundles of curves.

Second lecture : We study virtual invariants of Quot schemes of surfaces via virtual localization. Residual computations of each fixed loci can be done by using the theory of nested Hilbert schemes of points and curves. This naturally relates the theory of Quot schemes to the Seiberg-Witten invariants. Under the simplifying assumptions on the Seiberg-Witten aspect of the problem, we prove the multiplicative structural formula of the generating series.

Third lecture : We study the multiplicative universal series in the structural formula. This uses the simplifications of the virtual invariants of Quot schemes of K3 surfaces and their blow ups. Rather surprisingly, this computation determines all the universal series in the formula up to certain change of variables. We explain how this can be used to prove the rationality of the descendent series and virtual Segre/Verlinde correspondence.

References for virtual invariants of Quot schemes of surfaces:

  1. [Oprea, Pandharipande] Quot schemes of curves and surfaces: virtual classes, integrals, Euler characteristics.
  2. [Johnson, Oprea, Pandharipande] Rationality of descendent series for Hilbert and Quot schemes of surfaces.
  3. [Lim] Virtual chi_y-genera of Quot schemes on surfaces.
  4. [Arbesfeld, Johnson, Lim, Oprea, Pandharipande] The virtual K-theory of Quot schemes of surfaces.

List of other useful references (non-exhaustive):

# Quot schemes of curves and their applications.

  1. [Marian, Oprea] On the intersection theory on the moduli space of rank two bundles.
  2. [Marian, Oprea] Virtual intersections on the Quot scheme and Vafa-Intriligator formulas.
  3. [Marian, Oprea] The level-rank duality for nonabelian theta functions.
  4. [Bertram, Daskalopoulos, Wentworth] Gromov Invariants for Holomorphic Maps from Riemann Surfaces to Grassmannians.

# Virtual classes for Quot schemes of 3-fold.

  1. [Beentjes, Ricolfi] Virtual counts on Quot schemes and the higher rank local DT/PT correspondence.
  2. [Ricolfi] Virtual classes and virtual motives of Quot schemes on threefolds.

# (nested) Hilbert scheme of points and curves.

  1. [Ellingsrud, Gottsche, Lehn] On the Cobordism Class of the Hilbert Scheme of a Surface.
  2. [Duerr, Kabanov, Okonek] Poincare invariants.
  3. [Gholampour, Sheshmani, Yau] Nested Hilbert schemes on surfaces: Virtual fundamental class.
  4. [Gholampour, Thomas] Degeneracy loci, virtual cycles and nested Hilbert schemes II.

# Virtual invariants of moduli of sheaves on surfaces.

  1. [Gottsche, Kool] Sheaves on surfaces and virtual invariants.
  2. [Gottsche, Kool] Virtual refinements of the Vafa-Witten formula.
  3. [Gottsche, Kool] Virtual Segre and Verlinde numbers of projective surfaces.
  4. [Tanaka, Thomas] Vafa-Witten invariants for projective surfaces I: stable case.
  5. [Laarakker] Monopole contributions to refined Vafa-Witten invariants.