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The virtual \(K\)-theory of Quot schemes of surfaces. (English) Zbl 1476.14010

The authors study virtual invariants on the Quot scheme of 1-dimensional quotients over surfaces. To be more precise, consider the moduli space \(\mathrm{Quot}_X(\mathbb{C}^N, \beta, n)\) which parametrizes short exact sequences \[ 0\to S\to \mathbb{C}\otimes \mathcal{O}_X\to Q\to 0, \] where \(\mathrm{rk} Q=0\), \(c_1(Q)=\beta\) and \(\chi(Q)=n\). The Quot scheme \(\mathrm{Quot}_X(\mathbb{C}^N, \beta, n)\) carries a canonical 2-term perfect obstruction theory and a virtual fundamental class of virtual dimension \(\mathrm{vdim}=Nn+\beta^2\). Previous work of (part of) the authors dealt with studying the virtual invariants obtained by integrating various classes against the virtual fundamental class, such as virtual Euler characteristics, virtual \(\chi_y\)-genera and descendant invariants. In particular, the authors speculated on the rationality of such invariants and their relation with Seiberg-Witten theory.
The present paper deals with \(K\)-theoretic invariants. The perfect obstruction theory induces a virtual structure sheaf \(\mathcal{O}^{\mathrm{vir}}\) on \(\mathrm{Quot}_X(\mathbb{C}^N, \beta, n)\), by means of which one defines for any class \(V\in K^0(\mathrm{Quot}_X(\mathbb{C}^N, \beta, n))\) the invariants \[ \chi^{\mathrm{vir}}(\mathrm{Quot}_X(\mathbb{C}^N, \beta, n), V)=\chi(\mathrm{Quot}_X(\mathbb{C}^N, \beta, n), V\otimes \mathcal{O}^{\mathrm{vir}}). \] For a class \(\alpha\in K^0(X)\), define the tautological class \[ \alpha^{[n]}:=R\pi_{1*}(\mathcal{Q}\otimes \pi_2^*\alpha)\in K^0(\mathrm{Quot}_X(\mathbb{C}^N, \beta, n)), \] where \(\mathcal{Q}\) is the universal quotient and \(\pi_1, \pi_2\) are the natural projections. The authors main conjecture is that the series \[ Z_{X,N,\beta}^K(\alpha_1, \dots, \alpha_l|k_1, \dots, k_l)=\sum_{n\in \mathbb{Z}}q^n\chi^{\mathrm{vir}}(\Lambda^{k_1}\alpha_1^{[n]}\otimes\dots \otimes \Lambda^{k_l}\alpha_l^{[n]}) \] is the Laurent expression of a rational function in \(q\). Analogous conjectures include the twist by \(\Lambda_y\Omega^{\mathrm{vir}}\) and some virtual and non-virtual invariants where the Quot scheme is replaced by the Hilbert scheme of points \(X^{[n]} \) and the moduli space of stable pairs \(P_n(X, \beta)\). These conjectures have been in checked in this paper in various geometries.
Secondly, the Segre and Verlinde series are defined as follows, using the usual smooth structure on the Hilbert scheme of points \(X^{[n]}\) \[ S_\alpha^{\mathrm{Hilb}}=\sum_{n=0}^\infty q^n\int_{X^{[n]}} s(\alpha^{[n]}), \]
\[ V_\alpha^{\mathrm{Hilb}}=\sum_{n=0}^\infty q^n\chi(X^{[n]}, \det\alpha^{[n]}). \] These two series are shown to match (under suitable identifications) in several geometries, and shown not to match in some other cases.
Finally, consider a nonsingular surface \(X\) which admits a nonsingular canonical curve \(C\subset X\). One can analogously define virtual invariants on the Quot scheme on the curve \(\mathrm{Quot}_X(\mathbb{C}^N, n)\). The authors find an 8-fold equivalence among (virtual) Segre integrals and (virtual/twisted) Verlinde numbers, defined either on the surface \(X\) or the canonical curve \(C\). These equivalence comes by combining the Segre-Verlinde correspondence, various symmetry results and the cosection localization principle. Nevertheless, a complete geometric understanding of these correspondences has to be found.
This paper is rich of examples and complete computations of the invariants studied.

MSC:

14C05 Parametrization (Chow and Hilbert schemes)
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
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