Vortices on surfaces with cylindrical ends
We consider Riemann surfaces obtained from nodal curves with infinite cylinders in the place of nodal and marked points, and study the space of finite energy vortices defined on these surfaces. To compactify the space of vortices, we need to consider stable vortices – these incorporate breaking of cylinders and sphere bubbling in the fibers. In this paper, we prove that the space of gauge equivalence classes of stable vortices representing a fixed equivariant homology class is compact and Hausdorff under the Gromov topology. We also show that this space is homeomorphic to the moduli space of quasimaps defined by Ciocan-Fontanine, Kim and Maulik in .
The moduli space of stable quasimaps described by Ciocan-Fontanine, Kim and Maulik in  is a compactification of the space of maps from non-singular curves to targets that are Geometric Invariant Theory (GIT) quotients. The domain curves have genus and marked points and the complex structure is allowed to vary. The points in the boundary of the quasimap moduli space exhibit two phenomena: first, maps may acquire a finite number of base points, and second, the domain curve may degenerate to a nodal curve. Base points are required to be away from marked points and nodal points. This paper provides a symplectic version of stable quasimaps.
The quasimap compactification is different from Kontsevich’s stable map compactification, which works for general targets. The boundary points of the Kontsevich moduli space are maps whose domains are nodal curves. In contrast, quasimaps with base points are no longer honest maps from a nodal curve to the target. We explain this point. Let be a connected complex reductive group and be a polarized affine -variety. The GIT quotient is an open substack of the quotient stack . A quasimap is a map from a curve to the stack . In the complement of the set of base points in , the map has image in . Maps from a curve to the quotient stack correspond to -maps from a principal -bundle on to . The GIT quotient is the stack-theoretic quotient of the semistable locus . Base points are thus the points on that map to .
Suppose is a maximal compact subgroup of . In symplectic geometry, the maps are analogous to -vortices from to . When is as above, the action of on is Hamiltonian and has a moment map . By choosing an -invariant metric on , we assume there is an identification . A vortex consists of a connection on a principal -bundle and a holomorphic section with respect to that satisfies the equation
This equation requires a choice of area form on . To make sure that base points are away from special points on , we ‘blow up’ the area form at the special points. Punctured neighborhoods of these points will be isometric to semi-infinite cylinders. The blowing up of the area form near the special points ensures that the limit of , as we approach the special points, lies in and hence is in the semistable locus . We define a smooth family of metrics, called the neck-stretching metrics on stable nodal genus , -pointed curves such that the metric blows up at special points in the above-mentioned way, so any such curve now corresponds to a Riemann surface with cylindrical ends. The space of vortices representing a given equivariant homology class in defined on stable nodal curves with neck-stretching metric is not compact. To compactify it, we allow breaking of cylinders (as in Floer theory) and sphere bubbles in -fibers (as in the work of Ott ). The resulting objects are called stable vortices. The space of stable vortices in a fixed class of modulo -gauge transformations is compact and Hausdorff under the Gromov topology.
Suppose is the coarse moduli space of stable nodal curves of genus with marked points. We assume , and for stability . If a vortex has finite energy and bounded image in , a removal of singularity result applies at the cylindrical ends, which means that is well-defined over a closed complex curve. Then, represents a class in . Let be the space of vortices on stable genus , -pointed curves equipped with the neck-stretching metric, such that modulo the group of (unitary) gauge transformations. Removal of singularity at the cylindrical ends ensures that the evaluation maps
are well-defined for marked points . By the definition of , there is a forgetful map
In the compactification of , the domain may not be stable, is defined as the stabilization of the domain, achieved by contracting unstable components. Our first result is:
Suppose is a -Hamiltonian symplectic manifold that is equivariantly convex at (see Definition 2.5), has a proper moment map and has free action of on . The compactification of , called , is a compact Hausdorff space under the Gromov topology. The forgetful map and the evaluation maps are well-defined and continuous on .
Compactifications of the space of symplectic vortices have been constructed by ,  and . Ziltener () constructs a compactification of the space of vortices on the complex plane . In this case, besides sphere bubbling in the fibers, there is bubbling at infinity that produces sphere bubbles in the quotient and vortices on attached to these bubbles. Our situation for vortices on surfaces with cylindrical ends is simpler in comparison.
Mundet-Tian () have constructed a compactification for vortices with varying domain curve, equipped with a finite volume metric. In this case, when the domain curve degenerates to a nodal curve, the map can degenerate to a chain of gradient flow lines of the moment map (, so maps to ). In our approach, by allowing infinite volume at nodal points we avoid these structures. The infinite volume also ensures that nodal points map to the semistable locus, where the group action is free, which helps us avoid meromorphic connections present in . Further, the behavior of vortices away from base points is similar to that of holomorphic curves on GIT quotient . This phenomenon ties in with the philosophy of gauged Gromov-Witten theory where the moduli spaces of vortices with target is related to the moduli space of holomorphic curves on the quotient by wall-crossing as in Toda  and Woodward (, , ).
Let be the space of stable quasimaps whose domains are nodal -pointed curves of genus and which represent the homology class . Assuming that the -action on the semistable locus is free and that is an affine variety, the paper  proves that the moduli space of quasimaps is a Deligne-Mumford stack that is proper over the affine quotient. The next result of this paper is that is homeomorphic to the space of stable vortices that are in the equivariant homology class . We state the following theorem for the case that the affine quotient is a single point, or in other words, the GIT quotient is projective. In that case, can be realized as a Kähler Hamiltonian -manifold with a proper moment map that is equivariantly convex at infinity.
Let , , , , be as above and . Suppose the GIT quotient is projective. There is a homeomorphism
that commutes with the evaluation maps to the quotient and the forgetful map to .
The map in the above Theorem is still a homeomorphism in the case of a general affine quotient, i.e. when is not projective – this generalization is discussed in Section 7.5. The proof of the Theorem 0.2 does not require that is affine. It only requires that is compact. Therefore, we expect to be compact in more general situations, provided is aspherical. In order to remove the asphericity assumption, the definition of quasimaps will have to be broadened to include sphere bubbles in the fiber.
In the proof of Theorem 0.2, the bijection is a Hitchin-Kobayashi correspondence established in . The notion of stability required by this correspondence – of the point at infinity (marked points and nodal points in the case of ) mapping to the semistable locus – is part of the definition of quasimaps. The proof of continuity of relies on the convexity of the moment map – but there are analytic difficulties arising because of the non-compact domains. To overcome these, we crucially rely on a stronger version of removal of singularity at infinity for vortices. The original such result proved by Ziltener in his thesis () for the affine case gives only control on the decay of the connection. But in the cylindrical case, we are able to show a similar result (Proposition 2.9) giving control.
The paper is organized as follows. Section 1 constructs a smooth family of neck-stretching metrics on stable nodal curves parametrized by . Sections 2-5 describe vortices and prove Theorem 0.1, the proof appears in Section 5. Section 6 introduces quasimaps and Section 7 establishes the homeomorphism , the proof of Theorem 0.2 appears in Section 7.4.
Acknowledgements: I want to thank Chris Woodward for suggesting the idea for this paper and many discussions that helped me along the way. I was a post-doctoral fellow in Tata Institute of Fundamental Research at the time the article was written. I was also hosted by Department of Mathematics, Rutgers University for a month, whose hospitality I am grateful for. Finally, I thank the referee for carefully reading the paper and suggesting improvements.
1. Description of neck stretching metrics
In this section, we construct a family of metrics for stable nodal marked curves with special points deleted so that punctured neighborhoods in the curves are isometric to semi-infinite cylinders. The presentation is similar to Section 2 in Gonzalez-Woodward .
1.1. Stable curves, gluing
A compact complex nodal curve is obtained from a collection of smooth compact curves by identifying a collection of distinct nodal points
Points on the curves that are not nodal points are called smooth points. A nodal curve with marked points comes with a collection of distinct smooth points . A marked nodal curve is stable if it has finite automorphism group, i.e. every genus 0 component has at least 3 special points (marked or nodal point) and a genus 1 component has at least 1 special point. The genus of a nodal curve is the genus of a “smoothing” of the curve. For example, the genus of the curve in Figure 1 is one.
A family of nodal curves over a scheme is a proper flat morphism such that each fiber , is a nodal curve. One can ask if there is a space and a family such that for any family , there is a unique map such that is isomorphic to the pullback . For marked nodal curves with genus , such a family does not exist because there are curves with a non-trivial automorphism group - see p. 267 in the book . For stable curves, the automorphism group is finite. In that case, there is a coarse moduli space – this means for any family of curves , there is a unique map . There is a universal curve that maps to by forgetting the last point and contracting unstable components. For any stable curve , the fiber over the point in is isomorphic to . Given a family of curves over a pointed scheme , the finite automorphism group of the central curve acts on an open neighborhood of in (which we continue to call ). Further, the map factors through the quotient:
The action of lifts to and the quotient is isomorphic to the pullback . The moduli space has the structure of a complex orbifold. We use the definition of an orbifold in Satake , which says that locally a complex orbifold is homeomorphic to a neighborhood of the quotient of under the holomorphic action of a finite group, and the transition functions are bi-holomorphisms. This definition has problems and more sophisticated definitions have been provided, for example Lerman  says that orbifolds should be thought of as Deligne-Mumford stacks. But Satake’s definition is enough for the purposes of this paper.
The combinatorial type of a marked nodal curve is a modular graph and a genus function . The vertices are the components of , the edges in are nodes in and the infinite edges correspond to markings. An edge has two end points , so it is incident on two vertices . It is possible that these two vertices are the same – see for example Figure 1. For the markings, there is an ordering of the set by a bijection and each edge is incident on only one vertex .
A modular graph is stable if any curve corresponding to it is stable. The stabilization of a nodal curve is obtained by contracting curve components that have genus and less than 3 marked points. For a graph , the stabilization is the combinatorial type of the stabilization of any curve of type . A morphism of modular graphs corresponds to a sequence of moves, each of one of the two forms:
- Contracting a non-loop edge:
Contract an edge that is incident on vertices , , the genus of the combined vertex is the sum of the genus of and .
- Contracting a loop edge:
Delete an edge both whose end-points are incident on the same vertex and increment the genus of by .
In terms of the related nodal curves, each move smooths out a nodal singularity. Based on the combinatorial type , we can define a stratification of . If there is a morphism , then . The lowest stratum consists of points representing smooth curves, and it is an open set in . Denote by the subspace parametrizing curves of combinatorial type . Then, the boundary of consists of
Given a combinatorial type , we next discuss how a neighborhood in fits into – this is done through deformation theory. Suppose is a compact curve. Then a deformation of by a pointed scheme , , is a proper flat morphism plus an isomorphism between and the central fiber . A deformation of is universal if given any other deformation , for any sufficiently small neighborhood of in , there is a unique morphism such that is isomorphic to the fibered product . Stable curves possess universal deformations. Suppose is a universal deformation of the curve . If has a non-trivial stabilizer , then for a sufficiently small neighborhood of , the action of extends to compatible actions on and (theorem 6.5, chapter 11, ). If is small enough and -invariant, then there is an injection , and thus deformations of curves provide holomorphic orbifold charts for . Suppose is a universal deformation of the curve . Then, the space of infinitesimal universal deformations of , denoted by is the tangent space . If is a nodal curve of type , a deformation of of type is a family of curves where every fiber is a curve of type , along with an isomorphism between and the central curve . If is a universal deformation of of type , then the space of infinitesimal deformations of type , denoted by , is defined as the tangent space . For a nodal curve of type , given a universal deformation of type , we can construct a universal deformation using a gluing procedure. Suppose is the normalization of . By Proposition 3.32 in ,
Let be a nodal curve and suppose for all nodes , we are given holomorphic coordinates in small neighborhoods of the lifts of the nodes in the normalization . To every small , we can associate a curve .
is constructed as follows. The tensor is a complex number and we can define an equivalence
Define the glued curve as a quotient . If , we let the node stay in place in . Since the coordinates are holomorphic, the quotient relation respects complex structure, and so has a complex structure.∎
The gluing process in Construction 1.1 can be done in families also. Suppose, is a family of curves of type and we have a family of holomorphic co-ordinates on neighborhoods of the lifts of nodes in , where . Let be a vector bundle whose fiber over is
In a neighborhood of the zero section of , we can associate a glued curve to every point. If were a universal deformation of type of a curve , by Theorem 3.17, chapter 11 in , the gluing procedure above produces a universal deformation of the curve .
1.2. Riemann surfaces with cylindrical ends
Let be a nodal curve with marked points and nodal points . A Riemann surface with cylindrical ends 111Strictly speaking, the terminology is Riemann surface with metric that has cylindrical ends at punctures. To shorten notation, we assume cylindrical ends implies an underlying metric. associated to the curve is the punctured curve equipped with a metric satisfying the following property: for any there is a neighborhood of in the normalization of , and an isometry
such that . The metric on the right hand side is . We further require that the volume of is finite and any preserves the metric on .
In the context of cylindrical co-ordinates, will denote .
Given integers such that , the goal of this section is to show that to any stable -pointed curve of genus , we can associate a Riemann surface with cylindrical ends, and that the metric varies smoothly as we vary . In other words, if is a family of such curves, we need to put a metric on the fibers of that varies smoothly with . We call such a family of metrics neck-stretching metrics, because they stretch the ‘neck region’ in the glued surfaces described in Construction 1.1. A family of neck-stretching metrics is constructed using a family of holomorphic co-ordinates defined on on neighborhoods of marked points and lifts of nodal points. Note that in the construction 1.1, for a curve , if a node stays in place, i.e. , then the coordinates on also induce coordinates in the neighborhoods of on . A family of holomorphic coordinates defined on a family of nodal curves is said to be compatible if the co-ordinate on and agree, when is in a small neighborhood of .
There exists a family of neck-stretching metrics on stable curves parametrized by . The space of neck-stretching metrics on is contractible.
Assume is a compatible family of holomorphic coordinates defined on in neighborhoods of marked points and lifts of nodal points. We first describe a neck-stretching metric on a component of a curve, which is called and has special points . Holomorphic co-ordinates are defined on neighborhoods of . Define map as . In the neighborhood , define co-ordinates as . If for some edge , we use the co-ordinates . A cylindrical metric is given on by pulling back the Euclidean metric by . This metric can be extended to all of so that the volume of is finite.
For a combinatorial type , the family of metrics defined on the stratum can be extended to a small neighborhood of by the gluing construction for families. Given a curve of type , and , we now describe the metric on the curve . In the previous paragraph, by pulling back the Hermitian metric on via to a neighborhood , the element can be identified to a complex number. Let . The identification (1) used in the construction of can be re-written as
To construct the punctured curve with metric , we start with . For any node , parts of the semi-infinite cylinders are discarded and the remainder of the cylinders are identified via (2) to produce a finite cylinder in . The two regions being identified are isometric, so the metric on induces a metric on . If a modular graph is obtained from by contracting a single edge, the family of metrics can be extended to . If we assume that the metrics on curves of type are invariant under the action of the automorphism group of the curve, the condition would also be satisfied for the metrics on curves of type . A family of metrics on can thus be constructed inductively.
Finally we prove that the space of such metrics is contractible. For a fixed family of holomorphic co-ordinates in the neighbourhood of nodes in curves parametrized by , the space of metrics on stable curves is convex, hence contractible: Given two families of neck-stretching metrics , , for any , is also a neck-stretching metric. This is because the metrics and agree on the neighborhoods of special points. Now, consider different choices of , say and . We remark that multiplying or by a constant non-zero complex number does not affect the metric constructed above - it only modifies the identification , i.e. would now be relabelled by a multiple of . So, the family of co-ordinates can be multiplied by a family of scalars so that for any curve and special point , . Such a modification is possible because any strata of is simply connected. Now, we see that for , the interpolation is also a family of holomorphic coordinates in (possibly smaller) neighborhoods of special points (see Remark 2.1.1 in ). ∎
We make a choice of a family of neck-stretching metrics and fix it for the rest of the paper. Now, we can talk about a ‘Riemann surface with cylindrical ends corresponding to a stable nodal curve’.
2. Preliminaries: vortices on surfaces with cylindrical ends
Let be a compact connected Lie group, be a Riemann surface with metric and be a principal -bundle. Let be a Kähler manifold with a -action that preserves and . We assume the -action on is Hamiltonian, whose meaning we now define.
(Hamiltonian actions) A moment map is a -equivariant map such that , where the vector field is the infinitesimal action of on , i.e. . The action of is Hamiltonian if there exists a moment map . Since is compact, has an -invariant metric. We fix such a metric so the moment map becomes a map .
(Connections, curvature and gauge transformations) A connection is a -equivariant one-form that satisfies for all . The space of connections is an affine space modeled on where is the adjoint bundle. The form is basic, so it descends to a two-form , which is the curvature of the connection . A gauge transformation is an automorphism of – it is an equivariant bundle map . The group of gauge transformations on is denoted by .
In case is a trivial bundle , there is a trivial connection . The adjoint bundle has a trivialization . A connection is then of the form where . The formula of curvature is A gauge transformation acts on the connection as
On an associated bundle , a connection defines a covariant derivative on sections of :
On a local trivialization of the bundle , where , the covariant derivative of a section is . At a point , is the infinitesimal action of at .
Suppose is a complex -manifold whose complex structure is preserved by . A connection on determines a holomorphic structure on the associated bundle via the operator . This construction can be reversed when is the complexified group . Then, the associated bundle is a principal -bundle containing as a sub-bundle. Given a holomorphic structure on , the co-dimension one distribution corresponds to a connection on (see ).
(Gauged holomorphic maps) A gauged holomorphic map from to consists of a connection and a section of that is holomorphic with respect to . The space of gauged holomorphic maps from to is called .
(Symplectic vortices) A symplectic vortex is a gauged holomorphic map that satisfies
where is an area form on .
(Energy) The energy of a gauged holomorphic map is
For a compact base manifold , one can define Sobolev completions of the spaces , in a standard way (see Appendix B of the book ). But for a manifold with cylindrical ends , we restrict our attention to a trivial principal bundle . Let , such that . The space of connections is where is the trivial connection on . The space of gauge transformations is . Unlike in the case of a compact base manifold, these Sobolev spaces are dependent on the choice of trivialization and hence it is not possible to define these concepts for a general principal bundle. The space acts smoothly on . For integers , we also use the Sobolev spaces , which are the same as .
We assume the following in the rest of this article.
acts freely on .
Definition 2.5 (Equivariant convexity at infinity).
The -Hamiltonian manifold is equivariantly convex if there is a proper -invariant function , and a constant such that and ,
The above notion is defined by Cieliebak et al. in  and is an equivariant version of the idea of convexity in symplectic geometry introduced by Eliashberg and Gromov . If the moment map is proper and is equivariantly convex, the image of a finite energy vortex with bounded image is contained in the compact set (see Lemma 2.7 in ). An important example of equivariantly convex spaces are symplectic vector spaces with a linear group action and a proper moment map.
2.2. Asymptotic behavior on cylindrical ends
On a non-compact base space, vortices with bounded image and finite energy have good asymptotic properties (see Ziltener ). In this section we prove a stronger result (Proposition 2.9) in the special case that the base space has cylindrical ends.
Proposition 2.6 (Decay for vortices on the half cylinder).
Let be a half cylinder
with the standard metric . Let be a Hamiltonian -manifold with a proper moment map and such that acts freely on . There exists a constant that satisfies the following. Suppose is a finite energy vortex from the trivial bundle to whose image is contained in a compact subset of . Then, there exists a constant so that
This result is similar to Theorem 1.3 in Ziltener  and the proof carries over. It is an equivariant version of the isoperimetric inequality. But this result is much weaker, because in , is arbitrarily close to . The hypothesis in  place some condition on in order to make the metric on ‘admissible’, which can be dropped if we do not require an optimal value of .
The following result is an easy consequence of Proposition 2.6. It says that for a finite energy vortex on a semi-infinite cylinder, the connection asymptotically converges to a limit connection defined on . Further the map has a limit at infinity after it is twisted by a gauge transformation determined by the limit connection. It is enough to consider trivial principal bundles for this result because on a Riemann surface which is homotopic to a one-cell, any principal bundle with connected structure group is trivializable.
Proposition 2.8 (Removal of singularity for vortices at infinity).
Suppose is a finite energy vortex on a trivial -bundle on the half cylinder whose image is contained in a compact subset of . Then, there exist a point and a gauge transformation such that
Suppose, the restriction of in radial gauge to the circle is , there exist constants such that for all ,
The above result ensures that the -orbit is well-defined.
Let . Suppose is a finite energy vortex on a trivial -bundle on the half cylinder whose image is contained in a compact subset of . There is a gauge transformation , such that if , then,
for some positive constant and . Hence .
We first describe a cover of , in which all but a finite number of open sets have identical geometry. On these sets, the connection can be uniformly bounded using Uhlenbeck compactness. Fix any . Let . For any integer ,
There is an integer such that for all , , where is the constant in Uhlenbeck’s local Theorem (Theorem A.7). This bound ensures that the connection can be put in Coulomb gauge in each of these sets. By Uhlenbeck compactness, there is a gauge transformation on for all , such that is in Coulomb gauge, i.e. denoting ,
The constants , are independent of because the domains are identical to each other. We remark that since the set has corners, Uhlenbeck’s Theorem A.7 will have to be applied to closed sets with smooth boundary that are slightly bigger than . From Proposition 2.6, for some .
Next, we patch all these gauge transformations to get one on , which is a cover of . First look at one of the components of given by , we call these . On , let , where . Let be a cut-off function on that is in the neighborhood of and supported away from . Let . Define a gauge transformation on as being equal to on and equal to or outside . Define . We have to show that
For this it is enough to show that , which in turn can be shown by a similar bound on . The inequality (4) is unchanged if is multiplied by a constant factor. So we can assume there is a point on which and agree and so . We know . By a standard argument (see for example proof of Lemma 2.4 in ), we get
By a similar process, we can patch and for each and obtain a gauge transformation on all of on . Denote . We have bounds
Finally, we produce a gauge transformation on . The gauge transformation is defined by patching as above, but we cannot get bounds using the above technique. This is because for the other component of , given by and , the gauge transformations may not agree at any point as we no longer have the flexibility of modifying anything by a constant. But, we can get a similar bound if we can show that
This inequality is proved as follows. Denote by the holonomy of around the loop . By Proposition 2.8,
Now, denote by the holonomy of along the path – note that the end points are not identified in , but since we are on a trivial bundle there is a canonical identification between the fibers at the end points. So, we have
Recall is defined as follows: denote , let be given by the ODE
then . Since , we have , and this implies . Together with (6) and (7), (5) follows. The resulting gauge transformation can be extended to all of . The choice of extension is not important because we only require an asymptotic bound. ∎
Let . Suppose is a finite energy vortex on a Riemann surface with cylindrical ends whose image is contained in a compact subset of . We assume , where is compact. Further, for any , let be cylindrical coordinates on the punctured neighborhood . Then, there exists a gauge transformation and so that if on , then,
and hence, .
It is enough to assume that there is only one cylindrical end corresponding to , so we can drop from the notation. We fix a trivialization of the principal -bundle . Apply Proposition 2.9 to the restriction of to the half cylinder , and call the resultant gauge transformation . The homotopy equivalence class of the map will contain a geodesic loop , where . Now,