Jim Bryan

The enumerative geometry of 3 surfaces and modular forms

We prove the conjectures of Yau-Zaslow and Gottsche concerning the number curves on K3 surfaces. Specifically, let X be a K3 surface and C be a holomorphic curve in X representing a primitive homology class. We count the number of curves of geometric genus g with n nodes passing through g generic points in X in the linear system |C| for any g and n satisfying C^2=2g+2n-2. When g=0, this coincides with the enumerative problem studied by Yau and Zaslow who obtained a conjectural generating function for the numbers. Recently, Gottsche has generalized their conjecture to arbitrary g in terms of quasi-modular forms. We prove these formulas using Gromov-Witten invariants for families, a degeneration argument, and an obstruction bundle computation. Our methods also apply to P^2 blown up at 9 points where we show that the ordinary Gromov-Witten invariants of genus g constrained to g points are also given in terms of quasi-modular forms.

The local Gromov-Witten theory of curves

The local Gromov-Witten theory of curves is solved by localization and degeneration methods. Localization is used for the exact evalu- ation of basic integrals in the local Gromov-Witten theory of P 1 . A TQFT formalism is defined via degeneration to capture higher genus curves. Together, the results provide a compete and effective solution. The local Gromov-Witten theory of curves is equivalent to the lo- cal Donaldson-Thomas theory of curves, the quantum cohomology of the Hilbert scheme points of C 2 , and the orbifold quantum cohomol- ogy the symmetric product of C 2 . The results of the paper provide the local Gromov-Witten calculations required for the proofs of these equivalences.

Generating functions for colored 3D Young diagrams and the Donaldson-Thomas invariants of orbifolds

We derive two multivariate generating functions for three-dimensional (3D) Young diagrams (also called plane partitions). The variables correspond to a coloring of the boxes according to a finite Abelian subgroup G of SO (3). These generating functions turn out to be orbifold Donaldson-Thomas partition functions for the orbifold [C 3/G]. We need only the vertex operator methods of Okounkov, Reshetikhin, and Vafa for the easy case G = Z n; to handle the considerably more difficult case G = Z 2 × Z 2, we also use a refinement of the authors recent q-enumeration of pyramid partitions. In the appendix, we relate the diagram generating functions to the Donaldson-Thomas partition functions of the orbifold [C 3/G]. We find a relationship between the Donaldson-Thomas partition functions of the orbifold and its G-Hilbert scheme resolution. We formulate a crepant resolution conjecture for the Donaldson-Thomas theory of local orbifolds satisfying the hard Lefschetz condition.

The orbifold quantum cohomology of ℂ²/ℤ₃ and Hurwitz-Hodge integrals

Let Z_3 act on C^2 by non-trivial opposite characters. Let X =[C^2/Z_3] be the orbifold quotient, and let Y be the unique crepant resolution. We show the equivariant genus 0 Gromov-Witten potentials of X and Y are equal after a change of variables -- verifying the Crepant Resolution Conjecture for the pair (X,Y). Our computations involve Hodge integrals on trigonal Hurwitz spaces which are of independent interest. In a self contained Appendix, we derive closed formulas for these Hurwitz-Hodge integrals.

BPS states of curves in Calabi-Yau 3-folds

The Gopakumar-Vafa conjecture is defined and studied for the local geometry of a curve in a Calabi-Yau 3-fold. The integrality predicted in Gromov-Witten theory by the Gopakumar-Vafa BPS count is verified in a natural series of cases in this local geometry. The method involves Gromov-Witten computations, Mobius inversion, and a combinatorial analysis of the numbers of etale covers of a curve.

GENERATING FUNCTIONS FOR THE NUMBER OF CURVES ON ABELIAN SURFACES

Let X be an Abelian surface and C a holomorphic curve in X representing a primitive homology class. The space of genus g curves in the class of C is g dimensional. We count the number of such curves that pass through g generic points and we also count the number of curves in the fixed linear system |C| passing through g-2 generic points. These two numbers, (defined appropriately) only depend on n and g where 2n=C^2+2-2g and not on the particular X or C (n is the number of nodes when a curve is nodal and reduced). Gottsche conjectured that certain quasi-modular forms are the generating functions for the number of curves in a fixed linear system. Our theorem proves his formulas and shows that (a different) modular form also arises in the problem of counting curves without fixing a linear system. We use techniques that were developed in our earlier paper for similar questions on K3 surfaces. The techniques include Gromov-Witten invariants for families and a degeneration to an elliptic fibration. One new feature of the Abelian surface case is the presence of non-trivial Pic^0(X). We show that for any surface S the cycle in the moduli space of stable maps defined by requiring that the image of the map lies in a fixed linear system is homologous to the cycle defined by requiring the image of the map meets b_1 generic loops in S representing the generators of the first integral homology group (mod torsion).

Curves in Calabi-Yau threefolds and Topological Quantum Field Theory

We continue our study of the local Gromov-Witten invariants of curves in Calabi-Yau threefolds. We define relative invariants for the local theory which give rise to a 1+1-dimensional TQFT taking values in the ring Q[[t]]. The associated Frobenius algebra over Q[[t]] is semisimple. Consequently, we obtain a structure result for the local invariants. As an easy consequence of our structure formula, we recover the closed formulas for the local invariants in case either the target genus or the degree equals 1.

Surface bundles over surfaces of small genus

We construct examples of non-isotrivial algebraic families of smooth complex projective curves over a curve of genus 2. This solves a problem from Kirbys list of problems in low-dimensional topology. Namely, we show that2 is the smallest possible base genus that can occur in a 4-manifold of non-zero signature which is an oriented fiber bundle over a Riemann surface. A refined version of the problem asks for the minimal base genus for fixed signature and fiber genus. Our constructions also provide new (asymptotic) upper bounds for these numbers.

Instantons on S4 and CP2, rank stabilization, and Bott periodicity

We study the large n limit of the moduli spaces of Gn-instantons on S4andCP2 where Gn is SU(n),Sp(n/2),orSO(n). We show that in the direct limit topology, the moduli space is homotopic to a classifying space. For example, the moduli space of Sp(∞)orSO(∞) instantons on CP2 has the homotopy type of BU(k) where k is the charge of the instantons. We use our results along with Taubes’ result concerning the k→∞ limit to obtain a novel proof of the homotopy equivalences in the eight-fold Bott periodicity spectrum. We work with the algebro-geometric realization of the instanton spaces as moduli spaces of framed holomorphic bundles on CP2andCP2 blown-up at a point. We give explicit constructions for these moduli spaces (see Table 1).

Motivic classes of commuting varieties via power structures

We prove a formula, originally due to Feit and Fine, for the class of the commuting variety in the Grothendieck group of varieties. Our method, which uses a power structure on the Grothendieck group of stacks, allows us to prove several refinements and generalizations of the Feit-Fine formula. Our main application is to motivic Donaldson-Thomas theory.