Universality in enumerative geometry and Vafa-Witten theory -Universaliteit in aftellende meetkunde en Vafa-Wittentheorie

Abstract

Chapter One of this thesis is devoted to a generalisation of the famous Göttsche Conjecture. For a relative effective divisor C on a smooth projective family of surfaces q : S -> B, we consider the locus in B over which the fibres of C are d-nodal curves. We prove a conjecture by Kleiman and Piene on the universality of an enumerating cycle on this locus. We propose a bivariant class motivated by the BPS calculus of Pandharipande and Thomas, and show that it can be expressed universally as a polynomial in certain Chern classes associated to the pair (S,C). Under an ampleness assumption, we show that our class is the rational equivalence class of a natural effective cycle with support equal to the closure of the locus of d-nodal curves. Finally, we apply our method to calculate node polynomials for plane curves intersecting general lines in three dimensional projective space. We verify our results using 19th century geometry of Schubert. In Chapter Two we study the monopole contribution to (refined) Vafa-Witten invariants, recently defined by Thomas et. al. as invariants of moduli spaces of compactly supported sheaves on a local surface. Here we work on under the assumption that stability and semistability of such sheaves agree (the stable case). We apply the results of Gholampour and Thomas to prove a universality result for the generating series of vertical contributions, i.e. contributions of equivariant Higgs pairs with one dimensional weight spaces. For prime rank, these account for the entire monopole contribution by a theorem of Thomas. We use toric computations to determine part of the generating series, and find agreement with the conjectures of Göttsche and Kool for rank two and three. In Chapter Three we show that vertical contributions to Vafa-Witten invariants in the semistable case are well defined for surfaces that admit a holomorphic two form, partially proving conjectures of Tanaka and Thomas. Moreover, we show that such contributions are computed by the same tautological integrals as in the stable case. Using the work of Kiem and Li, we show that stability of universal families of vertical Joyce- Song pairs is controlled by a cosections of the obstruction sheaves of such families. Combining these results, we extend the universality theorem of Chapter Two to the semistable case.