The chi-y genera of relative Hilbert schemes for linear systems on
  Abelian and K3 surfaces

There is no author summary for this article yet. Authors can add summaries to their articles on ScienceOpen to make them more accessible to a non-specialist audience.

Abstract

For an ample line bundle on an Abelian or K3 surface, minimal with respect to the polarization, the relative Hilbert scheme of points on the complete linear system is known to be smooth. We give an explicit expression in quasi-Jacobi forms for the chi-y genus of the restriction of the Hilbert scheme to a general linear subsystem. This generalizes a result of Yoshioka and Kawai for the complete linear system on the K3 surface, a result of Maulik, Pandharipande, and Thomas on the Euler characteristics of linear subsystems on the K3 surface, and a conjecture of the authors.

Related collections

Most cited references 5

Record: found
Abstract: found
Article: found

Is Open Access

Curve counting via stable pairs in the derived category

, (2008)

For a nonsingular projective 3-fold \(X\), we define integer invariants virtually enumerating pairs \((C,D)\) where \(C\subset X\) is an embedded curve and \(D\subset C\) is a divisor. A virtual class is constructed on the associated moduli space by viewing a pair as an object in the derived category of \(X\). The resulting invariants are conjecturally equivalent, after universal transformations, to both the Gromov-Witten and DT theories of \(X\). For Calabi-Yau 3-folds, the latter equivalence should be viewed as a wall-crossing formula in the derived category. Several calculations of the new invariants are carried out. In the Fano case, the local contributions of nonsingular embedded curves are found. In the local toric Calabi-Yau case, a completely new form of the topological vertex is described. The virtual enumeration of pairs is closely related to the geometry underlying the BPS state counts of Gopakumar and Vafa. We prove that our integrality predictions for Gromov-Witten invariants agree with the BPS integrality. Conversely, the BPS geometry imposes strong conditions on the enumeration of pairs.