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For a K3 surface S, we study motivic invariants of stable pairs moduli spaces associated to 3-fold thickenings of S. We conjecture suitable deformation and divisibility invariances for the Betti realization. Our conjectures, together with earlier calculations of Kawai-Yoshioka, imply a full determination of the theory in terms of the Hodge numbers of the Hilbert schemes of points of S. The work may be viewed as the third in a sequence of formulas starting with Yau-Zaslow and Katz-Klemm-Vafa (each recovering the former). Numerical data suggest the motivic invariants are linked to the Mathieu M_24 moonshine phenomena. The KKV formula and the Pairs/Noether-Lefschetz correspondence together determine the BPS counts of K3-fibered Calabi-Yau 3-folds in fiber classes in terms of modular forms. We propose a framework for a refined P/NL correspondence for the motivic invariants of K3-fibered CY 3-folds. For the STU model, a complete conjecture is presented.
We define the BPS invariants of Gopakumar-Vafa in the case of irreducible curve classes on Calabi-Yau 3-folds. The main tools are the theory of stable pairs in the derived category and Behrends constructible function approach to the virtual class. We
The moduli space of stable pairs on a local surface $X=K_S$ is in general non-compact. The action of $mathbb{C}^*$ on the fibres of $X$ induces an action on the moduli space and the stable pair invariants of $X$ are defined by the virtual localizatio
We generalize the multiple cover formula of Y. Toda (proved by Maulik-Thomas) for counting invariants for semistable coherent sheaves on local K3 surfaces to semistable twisted sheaves over twisted local K3 surfaces. As applications we calculate the
We study the virtual geometry of the moduli spaces of curves and sheaves on K3 surfaces in primitive classes. Equivalences relating the reduced Gromov-Witten invariants of K3 surfaces to characteristic numbers of stable pairs moduli spaces are proven
We prove the KKV conjecture expressing Gromov-Witten invariants of K3 surfaces in terms of modular forms. Our results apply in every genus and for every curve class. The proof uses the Gromov-Witten/Pairs correspondence for K3-fibered hypersurfaces o