We conjecture an equivalence between the Gromov-Witten theory of 3-folds and the holomorphic Chern-Simons theory of Donaldson and Thomas. For Calabi-Yau 3-folds, the equivalence is defined by the change of variables e iu = −q, where u is the genus parameter of Gromov-Witten theory and q is the Euler characteristic parameter of Donaldson-Thomas theory. The conjecture is proven for local Calabi-Yau toric surfaces.

relative geometry. We derive a formula for the equivariant vertex measure in the degree 0 case and prove Conjecture 1 ′ of [14] in the toric case. A degree 0 relative formula is also proven. AcknowledgmentsWe thank J. Li for explaining his definition of relative Donaldson-Thomas theory to us. An outline of his ideas is presented in Section 3.2.1. We thank

Let V be a nonsingular, complex, projective variety containing a nonsingular divisor W . The absolute Gromov-Witten theory of V is defined by integrating descendent classes over the moduli space of stable maps to V . The relative Gromov-Witten theory of the pair (V, W ) is defined by descendent integration over the space of stable relative maps to V with prescribed tangency data along W .We present here a systematic study of relative Gromov-Witten theory via universal relations. We find the relative theory does not provide new invariants: the relative theory is completely determined by the absolute theory. The relation between the relative and absolute theories is guided by a strong analogy to classical topology.Our results open new directions in the subject. For example, we present a complete mathematical determination of the Gromov-Witten theory (in all genera) of the Calabi-Yau quintic hypersurface in P 4 . Leray-HirschLet X be a nonsingular, complex, projective variety equipped with a line bundle L. Let Y be the projective bundle P(L ⊕ O X ), and let π be the projection map, π : Y → X.

We construct new compactifications with good properties of moduli spaces of maps from nonsingular marked curves to a large class of GIT quotients. This generalizes from a unified perspective many particular examples considered earlier in the literature.

We prove the equivariant Gromov-Witten theory of a nonsingular toric 3-fold X with primary insertions is equivalent to the equivariant Donaldson-Thomas theory of X. As a corollary, the topological vertex calculations by Agangic, Klemm, Mariño, and Vafa of the GromovWitten theory of local Calabi-Yau toric 3-folds are proven to be correct in the full 3-leg setting.

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 proved. As a consequence, we prove the Katz–Klemm–Vafa conjecture evaluating λg integrals (in all genera) in terms of explicit modular forms. Indeed, all K3 invariants in primitive classes are shown to be governed by modular forms. The method of proof is by degeneration to elliptically fibred rational surfaces. New formulas relating reduced virtual classes on K3 surfaces to standard virtual classes after degeneration are needed for both maps and sheaves. We also prove a Gromov–Witten/Pairs correspondence for toric 3‐folds. Our approach uses a result of Kiem and Li to produce reduced classes. In Appendix A, we answer a number of questions about the relationship between the Kiem–Li approach, traditional virtual cycles, and symmetric obstruction theories. The interplay between the boundary geometry of the moduli spaces of curves, K3 surfaces, and modular forms is explored in Appendix B by Pixton.

Given a K3 surface X over a field of characteristic p, Artin conjectured that if X is supersingular (meaning infinite height) then its Picard rank is 22. Along with work of Nygaard-Ogus, this conjecture implies the Tate conjecture for K3 surfaces over finite fields with p \geq 5. We prove Artin's conjecture under the additional assumption that X has a polarization of degree 2d with p > 2d+4. Assuming semistable reduction for surfaces in characteristic p, we can improve the main result to K3 surfaces which admit a polarization of degree prime-to-p when p \geq 5. The argument uses Borcherds' construction of automorphic forms on O(2,n) to construct ample divisors on the moduli space. We also establish finite-characteristic versions of the positivity of the Hodge bundle and the Kulikov-Pinkham-Persson classification of K3 degenerations. In the appendix by A. Snowden, a compatibility statement is proven between Clifford constructions and integral p-adic comparison functors.Comment: Some minor edits made; German error fixed; comments still welcom

Let G denote a complex, semisimple, simply-connected group and B its associated flag variety. We identify the equivariant quantum differential equation for the cotangent bundle T * B with the affine Knizhnik-Zamolodchikov connection of Cherednik and Matsuo. This recovers Kim's description of quantum cohomology of B as a limiting case. A parallel result is proven for resolutions of the Slodowy slices. Extension to arbitrary symplectic resolutions is discussed.