Obstruction Bundle Research Articles

Tangent curves to degenerating hypersurfaces

Abstract We study the behaviour of rational curves tangent to a hypersurface under degenerations of the hypersurface. Working within the framework of logarithmic Gromov–Witten theory, we extend the degeneration formula to the logarithmically singular setting, producing a virtual class on the space of maps to the degenerate fibre. We then employ logarithmic deformation theory to express this class as an obstruction bundle integral over the moduli space of ordinary stable maps. This produces new refinements of the logarithmic Gromov–Witten invariants, encoding the degeneration behaviour of tangent curves. In the example of a smooth plane cubic degenerating to the toric boundary we employ localisation and tropical techniques to compute these refinements. Finally, we leverage these calculations to describe how embedded curves tangent to a smooth cubic degenerate as the cubic does; the results obtained are of a classical nature, but the proofs make essential use of logarithmic Gromov–Witten theory.

S1$S^1$‐equivariant contact homology for hypertight contact forms

In a previous paper, we showed that the original definition of cylindrical contact homology, with rational coefficients, is valid on a closed three-manifold with a dynamically convex contact form. However, we did not show that this cylindrical contact homology is an invariant of the contact structure. In the present paper, we define ‘nonequivariant contact homology’ and ‘ S 1 $S^1$ -equivariant contact homology’, both with integer coefficients, for a contact form on a closed manifold in any dimension with no contractible Reeb orbits. We prove that these contact homologies depend only on the contact structure. Our construction uses Morse–Bott theory and is related to the positive S 1 $S^1$ -equivariant symplectic homology of Bourgeois-Oancea. However, instead of working with Hamiltonian Floer homology, we work directly in contact geometry, using families of almost complex structures. When cylindrical contact homology can also be defined, it agrees with the tensor product of the S 1 $S^1$ -equivariant contact homology with Q ${\mathbb {Q}}$ . We also present examples showing that the S 1 $S^1$ -equivariant contact homology contains interesting torsion information. In a subsequent paper, we will use obstruction bundle gluing to extend the above story to closed three-manifolds with dynamically convex contact forms, which in particular will prove that their cylindrical contact homology has a lift to integer coefficients which depends only on the contact structure.

An obstruction bundle relating Gromov-Witten invariants of curves and Kähler surfaces

In a previous paper the authors defined symplectic ``Local Gromov-Witten invariants'' associated to spin curves and showed that the GW invariants of a K\"ahler surface $X$ with $p_g>0$ are a sum of such local GW invariants. This paper describes how the local GW invariants arise from an obstruction bundle (in the sense of Taubes) over the space of stable maps into curves. Together with the results of our earlier paper, this reduces the calculation of the GW invariants of elliptic and general-type complex surfaces to computations in the GW theory of curves with additional classes: the Euler classes of the (real) obstruction bundles.

Global Stringy Orbifold Cohomology, K-Theory and de Rham Theory

There are two approaches to constructing stringy multiplications for global quotients. The first one is given by first pulling back and then pushing forward. The second one is given by first pushing forward and then pulling back. The first approach has been used to define a global stringy extension of the functors K0 and Ktop by Jarvis–Kaufmann–Kimura, A* by Abramovich–Graber–Vistoli, and H* by Chen–Ruan and Fantechi–Gottsche. The second approach has been applied by the author in the case of cyclic twisted sector and in particular for singularities with symmetries and for symmetric products. The second type of construction has also been discussed in the de Rham setting for Abelian quotients by Chen–Hu. We give a rigorous formulation of de Rham theory for any global quotient from both points of view. We also show that the pull–push formalism has a solution by the push–pull equations in the setting case of cyclic twisted sectors. In the general, not necessarily cyclic case, we introduce ring extensions and treat all the stringy extension of the functors mentioned above also from the second point of view. A first extension provides formal sections and a second extension fractional Euler classes. The formal sections allow us to give a pull–push solution while fractional Euler classes give a trivialization of the co-cycles of the pull–push formalism. The main tool is the formula for the obstruction bundle of Jarvis–Kaufmann–Kimura. This trivialization can be interpreted as defining the physics notion of twist fields. We end with an outlook on applications to singularities with symmetries aka. orbifold Landau–Ginzburg models.

Quantum cohomology of [ℂN/μr

AbstractWe give a construction of the moduli space of stable maps to the classifying stack Bμr of a cyclic group by a sequence of rth root constructions on $\overline {M}_{0, n}$. We prove a closed formula for the total Chern class of μr-eigenspaces of the Hodge bundle, and thus of the obstruction bundle of the genus-zero Gromov–Witten theory of stacks of the form [ℂN/μr]. We deduce linear recursions for genus-zero Gromov–Witten invariants.

The age grading and the Chen-Ruan cup product

We prove that the obstruction bundle used to define the cup-product in Chen-Ruan cohomology is determined by the so-called `age grading' or `degree-shifting numbers'. Indeed, the obstruction bundle can be directly computed using the age grading. We obtain a Kunneth Theorem for Chen-Ruan cohomology as a direct consequence of an elementary property of the age grading, and explain how several other results - including associativity of the cup-product - can be proved in a similar way.

The full orbifold K-theory of abelian symplectic quotients

AbstractIn their 2007 paper, Jarvis, Kaufmann, and Kimura defined the full orbifoldK-theory of an orbifold , analogous to the Chen-Ruan orbifold cohomology of in that it uses the obstruction bundle as a quantum correction to the multiplicative structure. We give an explicit algorithm for the computation of this orbifold invariant in the case when arises as an abelian symplectic quotient. To this end, we introduce the inertial K-theory associated to a T -action on a stably complex manifold M, where T is a compact abelian Lie group. Our methods are integral K-theoretic analogues of those used in the orbifold cohomology case by Goldin, Holm, and Knutson in 2005. We rely on the K-theoretic Kirwan surjectivity methods developed by Harada and Landweber. As a worked class of examples, we compute the full orbifold K-theory of weighted projective spaces that occur as a symplectic quotient of a complex affine space by a circle. Our computations hold over the integers, and in the particular case of these weighted projective spaces, we show that the associated invariant is torsion-free.

Obstruction bundles over moduli spaces with boundary and the action filtration in symplectic field theory

Branched covers of orbit cylinders are the basic examples of holomorphic curves studied in symplectic field theory. Since all curves with Fredholm index one can never be regular for any choice of cylindrical almost complex structure, we generalize the obstruction bundle technique of Taubes for determining multiple cover contributions from Gromov–Witten theory to the case of moduli spaces with boundary. Our result proves that the differential in rational symplectic field theory and contact homology is strictly decreasing with respect to the natural action filtration.

The Chen-Ruan cohomology of almost contact orbifolds

Comparing to the Chen-Ruan cohomology theory for the almost complex orbifolds, we study the orbifold cohomology theory for almost contact orbifolds. We define the Chen-Ruan cohomology group of any almost contact orbifold. Using the methods for almost complex orbifolds, we define the obstruction bundle for any 3-multisector of the almost contact orbifolds and the Chen-Ruan cup product for the Chen-Ruan cohomology. We also prove that under this cup product the direct sum of all dimensional orbifold cohomology groups constitutes a cohomological ring. Finally we calculate two examples.

Stringy product on twisted orbifold K-theory for abelian quotients

In this paper we present a model to calculate the stringy product on twisted orbifold K-theory of Adem-Ruan-Zhang for abelian complex orbifolds. In the first part we consider the non-twisted case on an orbifold presented as the quotient of a manifold acted by a compact abelian Lie group. We give an explicit description of the obstruction bundle, we explain the relation with the product defined by Jarvis-Kaufmann-Kimura and, via a Chern character map, with the Chen-Ruan cohomology, we explicitly calculate the stringy product for a weighted projective orbifold. In the second part we consider orbifolds presented as the quotient of a manifold acted by a finite abelian group and twistings coming from the group cohomology. We show a decomposition formula for twisted orbifold K-theory that is suited to calculate the stringy product and we use this formula to calculate two examples when the group is (ℤ/2) 3 .

Gluing pseudoholomorphic curves along branched covered cylinders II

This paper and its prequel (“Part I”) prove a generalization of the usual gluing theorem for two index 1 pseudoholomorphic curves U+ and U− in the symplectization of a contact 3-manifold. We assume that for each embedded Reeb orbit γ, the total multiplicity of the negative ends of U+ at covers of γ agrees with the total multiplicity of the positive ends of U− at covers of γ. However, unlike in the usual gluing story, here the individual multiplicities are allowed to differ. In this situation, one can often glue U+ and U− to an index 2 curve by inserting genus zero branched covers of R-invariant cylinders between them. This paper shows that the signed count of such gluings equals a signed count of zeroes of a certain section of an obstruction bundle over the moduli space of branched covers of the cylinder. Part I obtained a combinatorial formula for the latter count and, assuming the result of the present paper, deduced that the differential ∂ in embedded contact homology satisfies ∂2 = 0. The present paper completes all of the analysis that was needed in Part I. The gluing technique explained here is in principle applicable to more gluing problems. We also prove some lemmas concerning the generic behavior of pseudoholomorphic curves in symplectizations, which may be of independent interest.

The Chen–Ruan Cohomology of Weighted Projective Spaces

AbstractIn this paper we study the Chen–Ruan cohomology ring of weighted projective spaces. Given a weighted projective space we determine all of its twisted sectors and the corresponding degree shifting numbers. The main result of this paper is that the obstruction bundle over any 3-multisector is a direct sum of line bundles which we use to compute the orbifold cup product. Finally we compute the Chen–Ruan cohomology ring of weighted projective space

Notes on Certain (0,2) Correlation Functions

In this paper we shall describe some correlation function computations in perturbative heterotic strings that, for example, in certain circumstances can lend themselves to a heterotic generalization of quantum cohomology calculations. Ordinary quantum chiral rings reflect worldsheet instanton corrections to correlation functions involving products of Dolbeault cohomology groups on the target space. The heterotic generalization described here involves computing worldsheet instanton corrections to correlation functions defined by products of elements of sheaf cohomology groups. One must not only compactify moduli spaces of rational curves, but also extend a sheaf (determined by the gauge bundle) over the compactification, and linear sigma models provide natural mechanisms for doing both. Euler classes of obstruction bundles generalize to this language in an interesting way.

The Chen-Ruan Cohomology Ring of Mirror Quintic

We compute the Chen-Ruan orbifold cohomology ring of the Batyrev mirror of a smooth quintic hypersurface in ℙ4. We identify the obstruction bundle for this example by using the Riemann bilinear relations for periods. We expect that our method can be used to compute the Chen-Ruan ring for Calabi-Yau hypersurfaces in projective simplicial toric varieties when every obstruction bundle is a direct sum of orbifold line bundles.

Large {$N$} duality for compact Calabi-Yau threefolds

We study geometric transitions for topological strings on compact Calabi-Yau hypersurfaces in toric varieties. Large N duality predicts an equivalence between topological open and closed string theories connected by an extremal transition. We develop new open string enumerative techniques and perform a high precision genus zero test of this conjecture for a certain class of toric extremal transitions. Our approach is based on a) an open string version of Gromov-Witten theory with convex obstruction bundle and b) an extension of Chern-Simons theory treating the framing as a formal variable.

On the Cappell–Lee–Miller gluing theorem

We formulate a more conceptual interpretation of the Cappell-Lee-Miller glueing/splitting theorem in terms of asymptotic maps and asymptotic exact sequences. Additionally, we show this gluing result is equivalent to a MayerVietoris-type long exact sequence. We also present applications to eigenvalue estimates, approximation of obstruction bundles and glueing of determinant line bundles arising frequently in gauge theory. All these results are true in a slightly more general context than in [6]. We work with operators which differ from translation invariant ones by exponentially decaying terms.

Equivariant Seiberg-Witten Floer homology

In this paper we construct, for all compact oriented three- manifolds, a U(1)-equivariant version of Seiberg-Witten Floer homology, which is invariant under the choice of metric and perturbation. We give a detailed analysis of the boundary structure of the monopole moduli spaces, compactified to smooth manifolds with corners. The proof of the independence of metric and perturbation is then obtained via an analysis of all the relevant obstruction bundles and sections, and the corresponding gluing theorems. The paper also contains a discussion of the chamber structure for the Seiberg-Witten invariant for rational homology 3-spheres, and proofs of the wall crossing formula, obtained by studying the exact sequences relating the equivariant and the non-equivariant Floer homologies and by a local model at the reducible monopole.

The enumerative geometry of $K3$ 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.

Obstruction bundles, semiregularity, and Seiberg–Witten invariants

We compare the deformation theory and the analytic structure of the Seiberg-Witten moduli spaces of a K\ahler surface to the corresponding components of the Hilbert scheme, and show that they are isomorphic. Next we show how to compute the invariant in case the moduli space is smooth but not of the expected dimension, and apply this study to elliptic surfaces. Finally we discuss ruled surfaces, both products and more general ruled surfaces. For product ruled surfaces we relate the infinitesimal structure of the moduli spaces to Brill-Noether theory and compute the invariant in special cases. For more general ruled surfaces, we relate the geometry of the Hilbert scheme to properties of stable bundles and give more general computations.

Quantitation of variations in regional left ventricular function in isolated disease of the left anterior descending coronary artery.

The pattern of contraction of the endocardial wall of the left ventricle in the right anterior oblique cineangiogram was studied by using a frame by frame radial technique and a fixed centroid without correction for rotation and translation motion during the cardiac cycle. Spatial defects of contraction were quantitated by measuring the shortening fraction of each radius and temporospatial defects by using a time-contraction integral. Twelve normal subjects were used as a basis for comparison. Thirty-two patients with isolated disease of the left anterior descending (LAD) coronary artery were divided into seven arbitrary clinicopathological subsets. Five subsets showed significant quantitative differences in contraction from the normal subjects but there was no significant difference between the subsets. They had a typical defect of contraction in the distal two thirds of the anterior wall, the apex and distal quarter of the inferior wall of the left ventricle. The subsets included: (1) patients who had undergone an anterior myocardial infarction and who had total occlusion of the LAD artery and a large anterior infarction on ECG; (2) patients with a previous classical myocardial infarction but with only 95% residual narrowing of the LAD; (3) patients with an anterior infarction and total occlusion of the LAD with return of the R waves in the anterior precordial leads; (4) patients with anterior infarction, LAD obstruction and left bundle branch block and (5) patients with anterior infarction but with early successful reperfusion with intracoronary streptokinase. Two other subsets, (1) patients with total obstruction of the LAD without a clinical myocardial infarction or (2) subtotal occlusion of the LAD without infarction, had mild left ventricular dysfunction at rest and did not differ significantly from normal.