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In this paper we determine all Kobayashi-hyperbolic 2–dimensional complex manifolds for which the group of holomorphic automorphisms has dimension 3. This work concludes a recent series of papers by the author on the classification of hyperbolic –dimensional manifolds, with automorphism group of dimension at least , where .
Witten’s top Chern class is a particular cohomology class on the moduli space of Riemann surfaces endowed with –spin structures. It plays a key role in Witten’s conjecture relating to the intersection theory on these moduli spaces.
Our first goal is to compute the integral of Witten’s class over the so-called double ramification cycles in genus 1. We obtain a simple closed formula for these integrals.
This allows us, using the methods of the first author [Int. Math. Res. Not. 38 (2003) 2051-2094], to find an algorithm for computing the intersection numbers of the Witten class with powers of the –classes over any moduli space of –spin structures, in short, all numbers involved in Witten’s conjecture.
In this paper we define instanton Floer homology groups for a pair consisting of a compact oriented –manifold with boundary and a Lagrangian submanifold of the moduli space of flat –connections over the boundary. We carry out the construction for a general class of irreducible, monotone boundary conditions. The main examples of such Lagrangian submanifolds are induced from a disjoint union of handle bodies such that the union of the –manifold and the handle bodies is an integral homology –sphere. The motivation for introducing these invariants arises from our program for a proof of the Atiyah–Floer conjecture for Heegaard splittings. We expect that our Floer homology groups are isomorphic to the usual Floer homology groups of the closed –manifold in our main example and thus can be used as a starting point for an adiabatic limit argument.
Using the combinatorial approach to knot Floer homology, we define an invariant for Legendrian knots (or links) in the three-sphere, with values in knot Floer homology. This invariant can also be used to construct an invariant of transverse knots.
If is a compact set, a topological contraction is a self-embedding such that the intersection of the successive images , , consists of one point. In dimension 3, we prove that there are smooth topological contractions of the handlebodies of genus whose image is essential.
Let be an ring spectrum. We use the description from our preprint [math.AT/0612165] of the cyclic bar and cobar construction to give a direct definition of topological Hochschild homology and cohomology of using the Stasheff associahedra and another family of polyhedra called cyclohedra. This construction builds the maps making up the structure into , and allows us to study how varies over the moduli space of structures on .
As an example, we study how topological Hochschild cohomology of Morava –theory varies over the moduli space of structures and show that in the generic case, when a certain matrix describing the noncommutativity of the multiplication is invertible, topological Hochschild cohomology of –periodic Morava –theory is the corresponding Morava –theory. If the structure is “more commutative”, topological Hochschild cohomology of Morava –theory is some extension of Morava –theory.
In this paper we develop a new theory of infinitesimal harmonic deformations for compact hyperbolic –manifolds with “tubular boundary”. In particular, this applies to complements of tubes of radius at least around the singular set of hyperbolic cone manifolds, removing the previous restrictions on cone angles.
We then apply this to obtain a new quantitative version of Thurston’s hyperbolic Dehn surgery theorem, showing that all generalized Dehn surgery coefficients outside a disc of “uniform” size yield hyperbolic structures. Here the size of a surgery coefficient is measured using the Euclidean metric on a horospherical cross section to a cusp in the complete hyperbolic metric, rescaled to have area 1. We also obtain good estimates on the change in geometry (eg volumes and core geodesic lengths) during hyperbolic Dehn filling.
This new harmonic deformation theory has also been used by Bromberg and his coworkers in their proofs of the Bers Density Conjecture for Kleinian groups.
Lagrangian cobordisms are three-dimensional compact oriented cobordisms between once-punctured surfaces, subject to some homological conditions. We extend the Le–Murakami–Ohtsuki invariant of homology three-spheres to a functor from the category of Lagrangian cobordisms to a certain category of Jacobi diagrams. We prove some properties of this functorial LMO invariant, including its universality among rational finite-type invariants of Lagrangian cobordisms. Finally, we apply the LMO functor to the study of homology cylinders from the point of view of their finite-type invariants.
Given a quiver algebra with relations defined by a superpotential, this paper defines a set of invariants of counting framed cyclic –modules, analogous to rank– Donaldson–Thomas invariants of Calabi–Yau threefolds. For the special case when is the non-commutative crepant resolution of the threefold ordinary double point, it is proved using torus localization that the invariants count certain pyramid-shaped partition-like configurations, or equivalently infinite dimer configurations in the square dimer model with a fixed boundary condition. The resulting partition function admits an infinite product expansion, which factorizes into the rank– Donaldson–Thomas partition functions of the commutative crepant resolution of the singularity and its flop. The different partition functions are speculatively interpreted as counting stable objects in the derived category of –modules under different stability conditions; their relationship should then be an instance of wall crossing in the space of stability conditions on this triangulated category.
We give an explicit formula for the difference between the standard and reduced genus-one Gromov–Witten invariants. Combined with previous work on geometric properties of the latter, this paper makes it possible to compute the standard genus-one GW-invariants of complete intersections. In particular, we obtain a closed formula for the genus-one GW-invariants of a Calabi–Yau projective hypersurface and verify a recent mirror symmetry prediction for a sextic fourfold as a special case.
An important step in the calculation of the triply graded link homology of Khovanov and Rozansky is the determination of the Hochschild homology of Soergel bimodules for . We present a geometric model for this Hochschild homology for any simple group , as –equivariant intersection cohomology of –orbit closures in . We show that, in type A, these orbit closures are equivariantly formal for the conjugation –action. We use this fact to show that, in the case where the corresponding orbit closure is smooth, this Hochschild homology is an exterior algebra over a polynomial ring on generators whose degree is explicitly determined by the geometry of the orbit closure, and to describe its Hilbert series, proving a conjecture of Jacob Rasmussen.