| The Gromov-Witten invariants are virtual intersection numbers of Kontsevich's moduli stack Mg,n (X) of stable maps from marked nodal curves to a smooth projective variety X. The information contained in these invariants can be used (under certain hypotheses) to give the cohomology ring H*(X, Q ) the structure of an algebra over the homology wheeled PROP of the stack M=⨆g ,nMg ,n of all stable marked curves.;In this dissertation, we define analogous invariants, constructed as the indices of certain "admissible" K-theory classes on a completion of the moduli stack of all principal Cx -bundles on stable marked curves. We interpret these invariant as the Gromov-Witten invariants of pt / Cx . More precisely, one may expect that these invariants can be used to endow the twisted equivariant K-theory KhCx&parl0; Cx&parr0; with the structure of an algebra over the K-homology wheeled PROP K*( M ).;Our completion, which we denote M&d5;g,n (pt / Cx ), is defined using ideas of Gieseker; we allow projective lines carrying degree 1 bundles to appear at the nodes of stable curves. It carries evaluation maps evi with target pt / Cx and a forgetful morphism F : M&d5;g,n (pt / Cx ) → Mg,n . Taking the index of a K-theory class on this stack - which amounts to constructing the pushforward of a K-theory class along the forgetful morphism F - is non-trivial, as the fibers of F generally have infinitely many connected components, each of which is typically non-separated and of infinite type.;The main theorem of this dissertation asserts the coherence of the right-derived pushforward RF*alpha of a complex representing an admissible class. (As Mg,n is smooth and projective, this implies that the index of the class [alpha] is well-defined.) This theorem has two main ingredients, which may be of independent interest: We prove vanishing and coherence theorems for the local cohomology of the fibers of F which allows us to reduce the question of coherence on the infinite-type fibers to the question of coherence on finite type substacks of these fibers. And we prove that these finite type substacks are proper, providing an alternate answer to an old question about the construction of compactifications of the Jacobian of a nodal curve. |
