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Virtual invariants of Quot schemes of surfaces

  • Author(s): Lim, Woonam
  • Advisor(s): Oprea, Dragos
  • et al.
Abstract

Quot schemes are fundamental objects in the moduli theory of algebraic geometry. Quot schemes of surfaces admit natural perfect obstruction theories if we consider 1-dimensional quotients of trivial vector bundles. We study various virtual invariants of such Quot schemes using the structure of Seiberg-Witten invariants and Hilbert schemes of points.

The main result expresses the virtual Quot scheme invariants universally in terms of Seiberg-Witten invariants and certain cohomological data of a surface. When a curve class is of Seiberg-Witten length N, we prove the multiplicative structural formula for the generating series of equivariant virtual Quot scheme invariants in terms of the universal series and Seiberg-Witten invariants. Furthermore, the universal series are completely determined up to the change of variables. As an application, we prove the rationality of the homological and K-theoretic descendent series for any curve classes of Seiberg-Witten length N. Explicit formulas are available in several cases for specializations to the generating series of virtual Euler characteristics and virtual $\chi_{-y}$-genera.

For K3 surfaces, the usual virtual Quot scheme invariants vanish. We thus define and study the reduced invariants of Quot schemes. Rather surprisingly, we show that the reduced $\chi_{-y}$-genera of Quot schemes and Pair spaces are equal when N=1. This implies that the reduced $\chi_{-y}$-genera of Quot schemes are given by the Kawai-Yoshioka formula.

We study the virtual Segre and Verlinde series of Quot schemes as a variation of the non-virtual Segre and Verlinde series of Hilbert schemes of points. Analogously to the conjectural algebraicity of the Segre and Verlinde series of Hilbert schemes of points, we prove that the virtual Segre and Verlinde series of Quot schemes are algebraic for any curve classes of Seiberg-Witten length N. Furthermore, we prove the virtual Segre and Verlinde correspondence relating the universal series. This, in particular, matches the virtual Segre and Verlinde series for punctual Quot schemes up to a sign.

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