
TL;DR
This paper introduces a tropical analogue of the Hodge bundle on the moduli space of tropical curves, exploring its combinatorial structure and providing explicit examples and computations.
Contribution
It proposes a novel tropical Hodge bundle on $M_g^{trop}$ and analyzes its combinatorial properties, extending classical algebraic geometry concepts to tropical geometry.
Findings
Defined a tropical Hodge bundle structure
Analyzed combinatorial properties of the bundle
Provided explicit examples and computations
Abstract
The moduli space of tropical curves of genus is a generalized cone complex that parametrizes metric vertex-weighted graphs of genus . For each such graph , the associated canonical linear system has the structure of a polyhedral complex. In this article we propose a tropical analogue of the Hodge bundle on and study its basic combinatorial properties. Our construction is illustrated with explicit computations and examples.
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∎
11institutetext: Bo Lin 22institutetext: Department of Mathematics, University of California, Berkeley, Berkeley, CA 94720, 22email: [email protected] 33institutetext: Martin Ulirsch 44institutetext: Fields Institute for Research in Mathematical Sciences, University of Toronto, 222 College Street, Toronto, Ontario M5T 3J1 44email: [email protected]
Towards a tropical Hodge bundle
Bo Lin and Martin Ulirsch
Abstract
The moduli space of tropical curves of genus is a generalized cone complex that parametrizes metric vertex-weighted graphs of genus . For each such graph , the associated canonical linear system has the structure of a polyhedral complex. In this article we propose a tropical analogue of the Hodge bundle on and study its basic combinatorial properties. Our construction is illustrated with explicit computations and examples.
1 Introduction
Let and denote by the moduli space of smooth algebraic curves of genus . The Hodge bundle is a vector bundle on whose fiber over a point in is the vector space of holomorphic differentials on . One can think of the total space of as parametrizing pairs consisting of a smooth algebraic curve and a differential on . Since for every curve the canonical linear system can be identified with the projectivization \mathbb{P}\big{(}H^{0}(C,\omega_{C})\big{)}, the total space of the projectivization of parametrizes pairs consisting of a smooth algebraic curve and a canonical divisor on ; it is referred to as the projective Hodge bundle. Let be the universal curve on . We may define formally as the pushforward of the relative dualizing sheaf on over .
The Hodge bundle is of fundamental importance when describing the geometry of . For example, its Chern classes, the so-called -classes, form an important collection of elements in the tautological ring on (see Vakil_tautologicalring for an introductory survey). The Hodge bundle admits a natural stratification by prescribing certain pole and zero orders such that and the study of natural compactifications of these components has recently seen an surge from both the perspective of algebraic geometry as well as from Teichmüller theory (see e.g. BainbridgeChenGendronGrushevskyMoeller_abeliandifferentials ).
In tropical geometry, the natural analogue of is the moduli space that parametrizes isomorphism classes of stable tropical curves of genus . In Section 2 below we are going to recall the construction of this moduli space. In particular, we are going to see how this moduli space naturally admits the structure of a generalized cone complex whose cones are in a natural order-reversing one-to-one correspondence with the boundary strata of the Deligne-Mumford compactification of (see AbramovichCaporasoPayne_tropicalmoduli as well as Section 2 below for details).
We also refer the reader to GathmannKerberMarkwig_tropicalfans , GathmannMarkwig_tropicalKontsevich , Mikhalkin_ICM , and Mikhalkin_Gokova for the theory in genus (with marked points), to BrannettiMeloViviani_tropicalTorelli , CaporasoViviani_tropicalTorelli , Chan_tropicalTorelli , ChanMeloViviani_tropicalmoduli , and Viviani_tropvscompTorelli for its connections to the tropical Torelli map, as well as to AbramovichCaporasoPayne_tropicalmoduli , CavalieriHampeMarkwigRanganathan_tropicalHassett , CavalieriMarkwigRanganathan_tropicalHurwitz , and Ulirsch_tropicalHassett for connections of (and some of its variants) to non-Archimedean analytic geometry and to Chan_topologyM2n and ChanGalatiusPayne_tropicalmoduliII for an in-depth study of the topology of . We, in particular, highlight the two survey papers Caporaso_tropicalmoduli and Caporaso_tropicalmoduliII .
Let be a tropical curve. We denote by the canonical divisor on and by the group of piecewise integer linear functions on (see Section 3 below for details). In this note we propose tropical analogues of the affine and the projective Hodge bundle, and study their basic combinatorial properties.
Definition 1
As a set, the tropical Hodge bundle is given as
[TABLE]
and the projective tropical Hodge bundle is given as
[TABLE]
The associations \big{(}[\Gamma],f\big{)}\mapsto[\Gamma] and \big{(}[\Gamma],D\big{)}\mapsto[\Gamma] define natural projection maps and , which, in a slight abuse of notation, we denote both by .
In GathmannKerber_RiemannRoch ; HaaseMusikerYu_linearsystems ; MikhalkinZharkov_tropicalJacobians the authors describe a structure of a polyhedral complex on the linear system associated to a divisor on a tropical curve ; we are going to review this description in Section 3 below. We also, in particular, highlight the first authors Lin_linearsystems , where he presents algorithms for computing this polyhedral complex. Our main result is the following:
Theorem 1.1
Let .
- (i)
The tropical Hodge bundle and the projective tropical Hodge bundle carry the structure of a generalized cone complex. 2. (ii)
The dimensions of and are given by
[TABLE]
respectively. 3. (iii)
There is a proper subdivision of such that, for all in the relative interior of a cone in this subdivision, the canonical linear systems
[TABLE]
have the same combinatorial type.
We are going to refer to this subdivision of as the wall-and-chamber decomposition of . In general, the generalized cone complexes and are not equi-dimensional. So Theorem 1.1 (ii) really states that the dimension of a maximal-dimensional cone in (or ) has dimension (or respectively).
As a first example we refer the reader to Figure 1 below, which depicts the face lattice of the tropical Hodge bundle in the case .
Let us give a quick outline of the contents of this contribution. In Section 2 we recall the construction of the moduli space of stable tropical curves and in Section 3 the polyhedral structure of linear systems on tropical curves respectively. In Section 4 we prove Theorem 1.1 by describing the polyhedral structures of both and simultanously. Section 5 contains a selection of explicit (sometimes partial) calculations of the polyhedral structure of in some small genus cases. Finally, in Section 6 we describe a natural tropicalization procedure for the projective algebraic Hodge bundle via non-Archimedean analytic geometry and exhibit a natural realizability problem.
2 Moduli of tropical curves
A tropical curve is a finite metric graph (with a fixed minimal model ) together with a genus function . The genus of (or of ) is defined to be
[TABLE]
where denotes the Betti number of . In the above sum one should think of the vertex-weight terms as the contributions of infinitesimally small loops at every vertex . We say a tropical curve (or the graph ) is stable, if for every vertex we have
[TABLE]
where denotes the valence of at .
Definition 2
As a set, the moduli space of stable tropical curves of genus is given as
[TABLE]
Let us now recall from AbramovichCaporasoPayne_tropicalmoduli the description of as a generalized extended cone complex.
Proposition 1 (AbramovichCaporasoPayne_tropicalmoduli Section 4)
The moduli space carries the structure of a generalized rational polyhedral cone complex that is equi-dimensional of dimension .
First, recall that a morphism between rational polyhedral cones is said to be a face morphism, if it induces an isomorphism onto a face of . Note that we explicitly allow the class of face morphisms to include all isomorphisms. A generalized (rational polyhedral) cone complex is a topological space that arises as a colimit of a finite diagram of face morphisms (see (AbramovichCaporasoPayne_tropicalmoduli, , Section 2) and (Ulirsch_functroplogsch, , Section 3.5) for details).
In order to understand this structure on , we observe that it is given as a colimit
[TABLE]
of rational polyhedral cones taken over a category . Let us go into some more detail:
The category is defined as follows:
- •
its objects are stable vertex-weighted graphs of genus , and
- •
its morphisms are generated by weighted edge contractions for an edge of as well as by the automorphisms of all .
Here a weighted edge contraction is an edge contraction such that for every vertex in we have
[TABLE] 2. 2.
Moreover, for every graph we denote by
[TABLE]
the parameter space of all possible edge lengths on . 3. 3.
The association defines a contravariant functor from to the category of rational polyhedral cones. It associates to a weighted edge contraction the embedding of the corresponding face of and to an automorphism of the automorphism of that permutes the entries correspondingly.
We note hereby that we have a decomposition into locally closed subsets
[TABLE]
where the disjoint union is taken over the objects in , i.e. over all isomorphism classes of stable finite vertex-weighted graphs of genus .
Example 1 (Chan_tropicalTorelli Theorem 2.12)
For a -dimensional cone complex , its -vector is defined as , where is the number of -dimensional cones in . The -dimensional moduli space has cells; its -vector is given by
[TABLE]
Remark 1
Earlier approaches, such as BrannettiMeloViviani_tropicalTorelli , Caporaso_tropicalmoduli , CaporasoViviani_tropicalTorelli , Chan_tropicalTorelli , ChanMeloViviani_tropicalmoduli , and Viviani_tropvscompTorelli , used to refer to the structure of a generalized cone complex as a stacky fan. Since there is a closely related, but not equivalent, notion of the same name in the theory of toric stacks we prefer to follow the terminology of generalized cone complexes introduced in AbramovichCaporasoPayne_tropicalmoduli .
3 Linear systems on tropical curves
Let be a tropical curve. A divisor on is a finite formal -linear sum
[TABLE]
over points in , i.e. is an element in the free abelian group on the points of . The degree of a divisor is defined to be the integer . We say is effective, if for all .
A rational function on is a continuous function whose restriction to every edge is a piecewise linear integral affine function. Given a rational function on as above and a point , the order of at is defined to be the sum of the outgoing slopes of emanating from . Observe that is equal to zero for all but finitely many points . So we have a map
[TABLE]
Divisors of the form for a function form a subgroup of and are referred to as the principal divisors on . Two divisors and on are said to be equivalent (written as ), if , i.e. if there is a rational function such that . Note that the continuity of implies that .
Let us now define the main players of this game:
Definition 3
Let be a divisor of degree on a tropical curve .
Denote by the set
[TABLE]
For , the divisor is supported in points (counted with multitplicity). We may therefore define:
[TABLE] 2. 2.
The linear system associated to is the set
[TABLE]
Observe that , where the symmetric group acts on by permutation of the points . Moreover, the additive group operates on by adding a constant function and, taking the quotient under this operation, we obtain that
[TABLE]
since if and only if is a constant function on .
The spaces , , and are known to carry the structure of a polyhedral complex (see e.g. MikhalkinZharkov_tropicalJacobians or GathmannKerber_RiemannRoch ). The following proposition is a more detailed version of (GathmannKerber_RiemannRoch, , Lemma 1.9).
Proposition 2
Given a divisor on a tropical curve , the space has the structure of a polyhedral complex. Choose an orientation for each edge of , identifying it with the open interval . Then the cells of can be described by the following (discrete) data:
- (i)
a partition of into disjoint subsets and (indexed by and edges ) that tells us on which edge (or at which vertex) every is located, 2. (ii)
a total order on each , and 3. (iii)
the outgoing slope of at the starting point of
such that for every vertex the equality
[TABLE]
holds. Furthermore, this polyhedral structure descends from to and .
Proof
Set and . We claim that the points in a cell of can be parametrized by the following two types of continuous data:
- •
the value at a vertex , as well as
- •
the distance of every from .
The distances immediately determine the . In order to reconstruct (if it exists) we write for points on , where the positive integers indicate the number of that are all located at the same point . The rational function is then determined by taking the value at the origin of every edge and continuing it piecewise linearly with slope until we hit , at which point we change the slope to until we hit , where we change the slope to , and so on until we hit the vertex at the end of . So, by continuity, for every edge we obtain the linear condition
[TABLE]
on the parameters of a cell in . This, together with the inequalities determines the polyhedral structure of a cell in . Note that our parameters are still overdetermined in the sense that there may be no rational function such that and which also fulfills all of the above inequalities; in this case we obtain an empty cell.
The conditions on the cells of are all discrete and the points within one cell are all parametrized by the distances and the values subject to these discrete conditions. Therefore is a polyhedral complex that does not depend on the choice of the orientation of .
The action of on every cell is affine linear and therefore the polyhedral structure descends to . Moreover, the additive group acts on by adding a constant to all and therefore the polyhedral structure also descends to . ∎
4 Structure of the tropical Hodge bundle
Let be a tropical curve with a fixed minimal model . As explained in (AminiCaporaso_RiemannRoch, , Section 5.2), the canonical divisor on is defined to be
[TABLE]
where denotes the valence of the vertex . Observe that . The -term in the sum should hereby be thought of as contributing infinitely small loops at the vertex . In fact, given a semistable curve whose dual graph is , the canonical divisor is the multidegree of the dualizing sheaf on (see (AminiBaker_metrizedcurvecomplexes, , Remark 3.1)). We recall Definition 1 from the introduction.
Definition 4
Let . As a set, the tropical Hodge bundle is defined to be
[TABLE]
and the projective tropical Hodge bundle as
[TABLE]
The tropical Hodge bundles come with natural projection maps
[TABLE]
given by \big{(}[\Gamma],f\big{)}\mapsto[\Gamma] and \big{(}[\Gamma],D\big{)}\mapsto[\Gamma], which, in abuse of notation, we both denote by .
In order to understand the structure of the tropical Hodge bundle we consider the pullback of and to , defined as
[TABLE]
and
[TABLE]
In analogy with the space , as in Section 3 above, we also set
[TABLE]
Proposition 3
- (i)
The action of on that permutes the points induces a natural bijection
[TABLE] 2. (ii)
The action of the additive group on , given by adding constant functions to , induces a natural bijection
[TABLE]
Proof
The projections and are both invariant under the action of and . Therefore our claims follow from the respective identities on the fibers. ∎
Let us now recall Theorem 1.1 from the introduction.
Theorem 4.1
Let .
- (i)
The tropical Hodge bundles and canonically carry the structure of a generalized cone complex. 2. (ii)
The dimensions of and are given by
[TABLE]
respectively. 3. (iii)
There is a proper subdivision of such that, for all in the relative interior of a cone in this subdivision, the canonical linear systems |K_{\Gamma}|=\pi_{g}^{-1}\big{(}[\Gamma]\big{)} have the same combinatorial type.
Proof (Proof of Theorem 1.1)
Part (i): We are going to show that canonically carries the structure of a cone complex. Then, by Proposition 3 above, both and carry the structure of a generalized cone complex.
Choose an orientation for each edge of , identifying it with the closed interval . As in Proposition 2 above, we can describe the cells of by the following discrete data:
- (i)
a partition of into disjoint subsets and (indexed by vertices and edges ) that tells us on which edge (or at which vertex) each is located, 2. (ii)
a total order on each , and 3. (iii)
the integer slope of at the starting point of
such that for every vertex the equality
[TABLE]
holds, where and . The continuous parameters describing all elements in our cell are given by
- (i)
the values , 2. (ii)
the distances of from , and 3. (iii)
the lengths .
In order to find the conditions on those parameters, we again write for . Using this notation we have as conditions on the as well as by the continuity of :
[TABLE]
Eliminating the non-parameters we can combine the system of equations to
[TABLE]
Since these conditions are invariant under multiplying all parameters simultaneously by elements in , every non-empty cell in has the structure of a rational polyhedral cone.
Finally, the natural action of on , given by
[TABLE]
for is compatible with both the - and the -operation. Moreover, given a weighted edge contraction of , the natural map identifies with the subcomplex of given by the condition in the above coordinates.
Therefore we can conclude that both
[TABLE]
where the limits are taken over the category as in Section 2 above, carry the structure of a generalized cone complex.
Part (ii): We need to show that the dimension of a maximal-dimensional cone in is . By (BrannettiMeloViviani_tropicalTorelli, , Proposition 3.2.5 (i)), we have and, by (Lin_linearsystems, , Corollary 7), the dimension of the fiber of a point is at most . This shows that the dimension of is at most .
In addition we now exhibit a -dimensional cone in as follows: Consider the tropical curve as indicated in Figure 2 and note that it has vertices and edges.
Lemma 1
Let be a tropical curve with minimal model . Let be a divisor on such that the support of is contained in . Then the combinatorial structure of is independent of the length of any loop or bridge in .
A proof of Lemma 1 is provided below. Lemma 1 implies that the combinatorial structure of is independent of the edge lengths, so we can choose a generic chamber. We obtain a divisor as indicated in Figure 3.
By (HaaseMusikerYu_linearsystems, , Proposition 13), the divisor belongs to a -dimensional face in . Thus there is a -dimensional cone in .
Part (iii): We use the coordinates described in part (i) above. For every edge of with we have an equation which is parametrized by the . So suppose and set . Then (2) can be rewritten as
[TABLE]
Eliminating from this equation, subject to the condition we obtain
[TABLE]
and therefore we obtain that the images of cells in are polyhedra. Moreover, the combinatorial type of is independent under scaling all edge lengths with a factor in and thus all these polyhedra determine a subdivision of such that on each relatively open cell of this subdivision, the corresponding has the same set of cells, i.e., the combinatorial type of is constant. ∎
Proof (of Lemma 1)
Suppose is a bridge in . Let and be two tropical curves with minimal model such that the length of in and is and respectively (where ), and for , the lengths of in and are the same. It suffices to show that the sets of cells in and are exactly the same.
In , we view the bridge as the open interval . For any cell in , its data consists of an integer and a partition of nonnegative integers . Suppose a divisor is on the bridge , where . Here the rational function is unique up to a translation. So we may assume that the value of is zero on the endpoint [math] of . Then on the function is defined inductively as follows:
- •
For it is given by
[TABLE]
- •
Given for we have
[TABLE]
Now on we also view the bridge as the open interval , with the same orientation. We construct a rational function on . First, we define on the bridge inductively as follows:
- •
For we define by
[TABLE]
- •
and, given for , we set
[TABLE]
Since is a bridge in , the graph consists of two connected components. We denote them by and , where contains the endpoint [math] of and contains the endpoint of . For convenience we let
[TABLE]
and
[TABLE]
Then we define on as follows:
[TABLE]
By definition, and admit the same data on for . In addition, on the bridge , both functions admit the integer and the same partition . So corresponds to a cell in that is exactly the same as . For the same reason we can get the cell from (just note that ). So Lemma 1 holds for bridges.
Suppose is a loop in . In this case almost the same proof works, except that the condition must hold, and is connected. We have ; therefore we may define on in the same way as on , as well as for all . Thus our claim also holds for loops.∎
5 Computations in low genus
In this section we present some computational results on the polyhedral structure of tropical Hodge bundles of small genus. In order to describe all cones in we first list all cones in . Then for each cone, we compute its subdivision by the structure of . It turns out that already the two cases and show a surprisingly distinct behavior.
Proposition 4
Let be a tropical curve in . Then the combinatorial structure of is uniquely determined by the minimal model of . In other words, it is independent of the edge lengths in .
Proof
There are faces in as in (Chan_tropicalTorelli, , Figure 4). For faces among them, all edges are loops or bridges, so the claim follows from Lemma 1. For the ”theta graph” , by explicit computation we know that the canonical linear system is always a one-dimensional polyhedral complex with three segments, as in Figure 4.∎
The face lattice of is visualized in Figure 1 from the introduction.
Remark 2
The -vector of is , which is consistent with Theorem 1.1 (ii). The unique -dimensional face consists of the ”dumbbell” graph and a triangular cell in . In other words, any divisor in this cell is of the form , where and are distinct points in the interior of the bridge in the dumbbell graph.
When , the counterpart of Proposition 4 is no longer true. One counterexample consists of the -dimensional cone in parametrizing tropical curves whose minimal model is a complete metric graph . The following proposition characterizes the open chambers of regarding the structure of . It is a result of explicit computations using the algorithm described in (Lin_linearsystems, , Section 2.3).
Proposition 5
- (i)
There are open chambers in . For all metrics in the same chamber, the canonical linear system has the same set of cells. In that case, the polyhedral complex always has vertices, edges, and two-dimensional faces (* triangles and quadrilaterals). However, there are non-isomorphic combinatorial structures of .* 2. (ii)
Let be a metric. Consider the following four -subsets:
[TABLE]
Then belongs to an open chamber if and only if among the elements of each -subset in (3), the minimum is attained only once.
Remark 3 (The structure of for a generic metric)
If belongs to an open chamber, the canonical linear system always has the vertices in Figure 6.
Among them, the labeled vertices are all connected to an extra vertex that is the divisor . The remaining vertices come from copies of a sub-structure (we call a bat) attached at , , , and . Note that some edges in Figure 6 are subdivided by other vertices in the bats.
The distinct combinatorial types of come from different ways of attaching the bats. Since belongs to an open chamber, the minimum of appears only once. Suppose it is , then the bat at is attached along the edges towards and , as in Figure 6. Figure 7 shows the divisors and .
Remark 4 (The boundary of the open chambers)
The action of the permutation group on the vertices of induces orbits among the open chambers, with lengths , , , and . Each orbit corresponds to a combinatorial type of . Each open chamber is an open cone in , defined by homogeneous linear inequalities involving , , , , , and . The inequalities are displayed as the covers in a lattice. For example, covering means that the inequality holds.
6 The realizability problem
Let be an algebraically closed field carrying the trivial absolute value. In AbramovichCaporasoPayne_tropicalmoduli , expanding on earlier work (see e.g. Viviani_tropvscompTorelli ), the authors have constructed a natural (continuous) tropicalization map
[TABLE]
from the non-Archimedean analytic moduli space onto . Let us recall the construction of : A point parametrizes an algebraic curve over some non-Archimedean extension of . Possibly after a finite extension we can extend to a stable model over the valuation ring of . Denote by the weighted dual graph of the special fiber of , whose vertices correspond to the components of and in which we have an edge between two vertices and for every node connecting the two corresponding components and . The vertex weight function is given by
[TABLE]
where denotes the normalization of . Finally, around every node in we can find formal coordinates and of such that for some element in the base. Then the edge length of is given by .
Denote by the non-Archimedean analytification of the total space of the algebraic Hodge bundle .
Proposition 6
There is a natural tropicalization map that makes the diagram
[TABLE]
commute.
We expect that is also continuous, but refrain from investigating this question here, since such an investigation appears to be too technical for the nature of a contribution to this volume.
Proof
An element parametrizes a tuple consisting of a smooth projective curve over a non-Archimedean extension of together with a canonical divisor on . Then we may associate to the point \big{(}[\Gamma_{x}],\tau_{\ast}(K_{C})\big{)}, where
[TABLE]
denotes the specialization map constructed in (Baker_specialization, , Section 2.3) that is given by pushing forward to the non-Archimedean skeleton of . As shown in (Baker_specialization, , Section 2.3) and the references therein, this is well-defined and the commutativity of the above diagram is an immediate consequence of the definition. ∎
It is well-known that is surjective. By Theorem 1.1 (ii) we have that
[TABLE]
and therefore the analogous statement for appears to be false. This gives rise to the following problem.
Problem 1
Find a characterization of the locus in , the so-called realizability locus in .
In other words, given a (stable) tropical curve of genus together with a divisor that is equivalent to , find algebraic and combinatorial conditions that ensure that there is an algebraic curve over a non-Archimedean field extension of together with a canonical divisor on such that
[TABLE]
Since is surjective, we know already that every tropical curve can be lifted to a smooth algebraic curve . In the case of having integer edge lengths we can give a constructive approach to this problem: Consider a special fiber over whose weighted dual graph is , then apply logarithmically smooth deformation theory to find a smoothing of to a stable family with deformation parameters at each node (see e.g. (Gross_book, , Proposition 3.38)). If all , we may alternatively also proceed as in (Baker_specialization, , Appendix B).
Now let be a divisor on that specializes to the given canonical divisor on . Since , Clifford’s theorem (or alternatively Baker’s Specialization Lemma (Baker_specialization, , Corollary 2.11)) shows that the rank of is at most . If the rank of is smaller than it cannot be a canonical divisor. If, however, the divisor has rank , then, by Riemann-Roch, it is a canonical divisor. So the realizability problem reduces to finding a lift of the divisor of rank .
The existence of such a divisor would follow, for example, from the smoothness of a suitable moduli space of limit linear series (see e.g. EisenbudHarris_limitlinearseries and Osserman_limitlinearseries ). Unfortunately the machinery of limit linear series is not available in full generality (i.e. for nodal special fibers that are not of compact type). However, considerations undertaken from the point of view of compactifications of the moduli space of abelian differentials and its strata (see in particular BainbridgeChenGendronGrushevskyMoeller_abeliandifferentials ) treating the special case of limits of canonical linear systems seem to provide us with a very promising approach for future investigations into this question.
Acknowledgements.
This article was initiated during the Apprenticeship Weeks (22 August-2 September 2016), led by Bernd Sturmfels, as part of the Combinatorial Algebraic Geometry Semester at the Fields Institute. Both authors would like to acknowledge his input. Thanks are also due to the Max-Planck-Institute of Mathematics in the Sciences in Leipzig, Germany, for its hospitality. The second author, in particular, would like to thank Diane Maclagan for several discussion related to the topic of this note, as well as the Fields Institute for Research in Mathematical Sciences. Finally, many thanks are due to the anonymous referees for several helpful comments and suggestions.
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