Tropical momentum maps for toric log symplectic manifolds Gualtieri, Marco∗
Li, Songhao†
Pelayo, Alvaro‡
Ratiu, Tudor§
arXiv:1407.3300v1 [math.SG] 11 Jul 2014
We study a class of Poisson manifolds which are symplectic away from a collection of hypersurfaces forming a normal crossing configuration. We impose the analogue of toric Hamiltonian symmetries, which in this case means that there is a momentum map to a generalization of affine space called a log affine manifold. We also extend the Delzant construction to this setting, obtaining a classification of such manifolds in terms of decorated log affine polytopes.
Contents 1 Introduction
2
2 Log 2.1 2.2 2.3
4 4 5 9
affine manifolds Affine monodromy . . . . . . . . . . . . . . . . . . . . . . . . . . . Tropical domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . Welding of domains . . . . . . . . . . . . . . . . . . . . . . . . . .
3 Toric log symplectic manifolds 13 3.1 Trivial principal case . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.2 Nontrivial principal case . . . . . . . . . . . . . . . . . . . . . . . . 16 4 Polytopes and symplectic cutting 19 4.1 Log affine polytopes . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.2 Symplectic cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 5 Delzant correspondence
24
A Appendix A.1 Local normal forms for log A.2 Log symplectic reduction A.3 Symplectic uncut . . . . . A.4 Logarithmic cohomology .
27 27 29 30 36
affine manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Department of Mathematics, University of Toronto.
[email protected] Department of Mathematics, Washington University in St. Louis.
[email protected] ‡ Department of Mathematics, University of California, San Diego.
[email protected] § ´ Section de Math´ematiques and Bernoulli Center, Ecole Polytechnique F´ed´erale de Lausanne.
[email protected] †
1
1
Introduction
In this paper, we generalize the theory of toric symplectic geometry to a class of Poisson manifolds which are generically symplectic but degenerate along a normal crossing configuration of smooth hypersurfaces. Such Poisson structures may be viewed as logarithmic symplectic forms, in a way which is particularly useful in guiding intuition. The study of toric log symplectic manifolds was initiated in the paper of Guillemin, Miranda, Pires and Scott [8], who focused on the case where the degeneracy locus Z ⊂ M of the Poisson structure is a smooth hypersurface. Their main observation was that a collection of moment maps µi : Mi → t∗ , each defined in one component Mi of the nondegenerate locus M \ Z, may be extended over Z to define a global map µ : M → X taking values in a space which they construct by gluing the codomains of the µi together end-to-end. They also characterize the momentum image µ(M ) ⊂ X and extend the Delzant correspondence to this setting. By allowing the degeneracy locus to have normal crossing singularities, we show that a global tropical moment map µ : M → X may still be defined, but the geometry and topology of X may be significantly more involved than in the case where Z is smooth. In [8], the codomain X and momentum image µ(M ) are either contractible or homotopic to the circle, while in our case there is no a priori constraint on their topology. In Example 4.21, for instance, X is a genus 2 surface constructed from four copies of t∗ , with a momentum image given by the gluing of four convex polytopes forming a surface of genus 1 with boundary. Under a Hamiltonian assumption, we assemble the codomain X from elementary pieces called tropical domains, which are partial compactifications of affine spaces closely related to the extended tropicalizations of toric varieties defined by Kajiwara [10] and Payne [17]. In fact, our work was initially motivated by the problem of defining moment maps for Caine’s examples of real Poisson structures on complex toric varieties [2]. Such a variety may be blown up along its circle fixed point loci to produce a log symplectic manifold with corners, and in this way we may identify its tropical moment map with its tropicalization morphism. The generalization of momentum maps studied in this paper is similar in spirit to quasi-Hamiltonian moment maps [1] as well as to optimal momentum maps [16], though we shall not explore the relations here. The structure of the paper is as follows: In Section 2, we define log affine manifolds and provide a method for constructing a great variety of examples by welding together tropical domains. In Section 3, we define toric log symplectic manifolds and classify those with principal torus actions. In section 4, we describe the analogue of the Delzant polytope and provide several examples, including the key example 4.21. Finally, in Section 5, we establish a Delzant correspondence between log affine polytopes and toric log symplectic manifolds.
2
Acknowledgements: We thank Matthias Franz, Eckhard Meinrenken, David Speyer, and Rafael Torres for many insightful remarks. M.G. is partially supported by an NSERC Discovery Grant. A.P. is partially supported by NSF Grant DMS-1055897, the STAMP Program at the ICMAT research institute (Madrid), and ICMAT Severo Ochoa grant Sev-2011-0087. T.R. is partially supported by NCCR SwissMAP and grant 200021-140238, both of Swiss National Science Foundation. Notation and conventions: i) Manifolds with corners: Throughout the paper, manifolds have boundary and corners. Most convenient for our purposes is the definition in which each local chart has codomain given by an arbitrary union or intersection of coordinate half-spaces ±xi ≥ 0, in standard coordinates (x1 , . . . , xn ) on Rn . ii) Lie algebroids: A Lie algebroid is a vector bundle A over the manifold X, together with a Lie bracket [·, ·] on its smooth sections and a morphism of vector bundles % : A → T X, called the anchor, which is bracket-preserving and satisfies the Leibniz rule [a, f b] = f [a, b] + (%(a)f )b, for all a, b sections of A and f ∈ C ∞ (X, R). iii) Action Lie algebroid: Given an infinitesimal action of a Lie algebra g on a manifold X, i.e., a Lie algebra homomorphism ρ : g → C ∞ (X, T X), the action Lie algebroid g nρ X is the trivial bundle g × X → X, equipped with anchor map % defined by %(a) = (x, ρ(a)(x)) and Lie bracket on sections given by the unique extension of the Lie bracket on constant sections which satsifies the Leibniz rule. iv) Free divisors: A free divisor is defined to be a union of smooth closed hypersurfaces (i.e., real codimension one submanifolds) with the property that the sheaf of vector fields tangent to all hypersurfaces is locally free. Given a free divisor D on the n-manifold X, we let T X(− log D) denote the rank n vector bundle defined by the above sheaf. It is a Lie algebroid, with Lie bracket inherited from the Lie bracket of vector fields and anchor map defined by the natural inclusion of sheaves. Its dual bundle is denoted by T ∗ X(log D), and the Lie algebroid de Rham complex is usually written (Ω• (X, log D), d), and is called the logarithmic de Rham complex (See Appendix A.4). v) Normal crossing divisors: A convenient subclass of free divisors are those with only simple normal crossing singularities, meaning that each point x ∈ D, there is a chart (U, ϕ) of X, x ∈ U , such that ϕ(D) is a union of a subset of the coordinate hyperplanes in Rn intersected with ϕ(D). The smooth locus of D is defined to be the set of points in D which lie only on a single hypersurface of D. The singular locus of D is the complement in D of the smooth locus.
3
2
Log affine manifolds
2.1 A toric Hamiltonian symplectic manifold is equipped with a momentum map to the dual of the Lie algebra of the torus: the affine space t∗ . When generalizing the theory to log symplectic forms, one is led to replace the codomain t∗ by a log affine manifold, possibly with boundary and corners: Definition 2.2. A real log affine n-manifold is a manifold X of real dimension n, equipped with a free divisor D and an isomorphism ξ of Lie algebroids between T X(− log D) and an abelian action algebroid, i.e. ξ
T X(− log D)
/ Rn nρ X ,
(1)
where ρ : Rn → C ∞ (X, T X) is a Lie algebra homomorphism. We call a log affine manifold complete when the infinitesimal action of Rn integrates to a group action of Rn on X. The isomorphism (1) may be viewed equivalently as a closed logarithmic 1-form ξ ∈ Ω1 (X, log D) ⊗ Rn . (2) Remark 2.3. If X has boundary or corners and if D is a normal crossing divisor, then in order to be complete log affine, the divisor D must include the normal crossing boundary components of X. Remark 2.4. Saito’s criterion [18] provides a convenient way to characterize log affine manifolds. A simple consequence of the criterion is the following: if (X1 , . . . , Xn ) are commuting vector fields on the n-manifold X which are tangent to a normal crossings divisor D, and if their determinant X1 ∧ · · · ∧ Xn ∈ C ∞ (X, ∧n T X)
(3)
vanishes precisely on D and trasversely on its smooth locus, then T X(− log D) is an action Lie algebroid globally generated by the n sections (X1 , . . . , Xn ).
2.1
Affine monodromy
2.5 For any smooth point of the degeneracy locus D, there are coordinates (x1 , . . . , xn ) in a neighbourhood N ⊂ X and a basis (v1 , . . . vn ) for Rn such that the log affine structure in N is defined by the closed logarithmic form (see Proposition A.5) X ξ = x−1 dxi ⊗ vi . (4) 1 dx1 ⊗ v1 + i>1
The form ξ defines an affine structure in each of the two regions N ± = {±x1 > 0}, with affine coordinates (log |x1 |, x2 , . . . , xn ). Definition 2.6. The reflection through D is the map defined in N by (x1 , . . . , xn ) 7→ (−x1 , . . . , xn ), using coordinates as above.
4
The reflection operation has an intrinsic meaning, as follows. Though ξ is singular along D, its definite integral along any path γ : [0, 1] → N is welldefined via the Cauchy principal value if γ is transverse to D and γ(0) and γ(1) lie on opposite sides of D. The reflection of x ∈ N may be defined as the unique point R(x) ∈ N such that the integral of ξ from x to R(x) is zero in t∗ . 2.7 Assuming that D is a normal crossing divisor, we may use reflections as above to define an affine monodromy representation, as follows. We may represent any homology class in H1 (X) as a piecewise linear path consisting of a finite sequence of either translations by the action of Rn on X or of reflections across the components of D. Taking the sum of the required translations, we obtain a representation of H1 (X) in the additive group Rn . By the Poincar´e pairing, this affine monodromy may also be seen as the component in H 1 (X) ⊗ Rn of the logarithmic form [ξ] ∈ H 1 (X, log D) ⊗ Rn ∼ = (H 1 (X) ⊕
X
H 0 (Di )) ⊗ Rn ,
i
where we use the Mazzeo-Melrose decomposition (A.4). Definition 2.8. The affine monodromy of the log affine manifold (X, D, ξ) (where D has normal crossing type) is defined to be the component of [ξ] in H 1 (X) ⊗ Rn . The log affine manifolds we consider in this paper will always be assumed to have trivial affine monodromy, and the gluing construction of log affine manifolds presented in this section is designed to produce examples with trivial affine monodromy.
2.2
Tropical domains
2.9 Let X be an affine space for the real vector space U of dimension n, and fix a nonzero vector α ∈ U . We now define a real analogue of the symplectic cut construction which partially compactifies X to a manifold with boundary; the boundary is a copy of X/hαi located “at infinity” in the −α direction. Definition 2.10. Let (X, U, α) be as above. Then we define the manifold with boundary X α = (X × R+ ) / R, (5) where R+ = [0, ∞) and s ∈ R acts via the (free and proper) anti-diagonal action s · (x, λ) = (x − sα, es λ).
(6)
X is then included in X α via x 7→ [(x, 1)]. The choice of an affine hyperplane H ⊂ X transverse to α defines a slice for the action in (5), identifying X α with the half-space H+ = H + R+ α. The slice is given by / X × [0, ∞) , H+ (7) / (x⊥ , λ) x
5
where x = x⊥ + λα and x⊥ ∈ H. In this way we see that X α is isomorphic to the closed half-space H+ , and so is non-canonically isomorphic to X/hαi × R+ as a manifold with boundary. Proposition 2.11. The residual action of U on X α given by [(x, λ)] + u = [(x + u, λ)]
(8)
renders X α into a log affine manifold; that is, T X α (− log ∂X α ) is naturally isomorphic to the action algebroid U n X α as a Lie algebroid, in a way which sends the normal Euler vector field of the boundary to the translation α. The last statement may also be interpreted in the following way: as in Definition 2.2, the identification of T X α (− log ∂X α ) with U n X α defines a logarithmic U -valued 1-form ξ ∈ Ω1 (X α , log ∂X α ) ⊗ U, (9) and the residue of this form along the boundary ∂X α is α ∈ U . 2.12 The compactification introduced in Definition 2.10 may be applied to any complete log affine manifold X in the sense of Definition 2.2, so long as the R-action generated by α remains free and proper. Proposition 2.13. Let (X, D, U ) be a complete log affine manifold and let α ∈ U generate a free and proper action of R on X. Then the partial compactification X α of X defined by (5) is also a complete log affine manifold, and as in Proposition 2.11, the residue of (9) along its new boundary component is α. 2.14 We now apply the partial compactification several times to the affine space X, iterating over a k-tuple of vectors (α1 , . . . αk ) in U . For freeness and properness of each successive R-action, we assume that the vectors are linearly independent. The resulting log affine manifold (now with corners) does not depend on the ordering of the k-tuple, and we have the following description of it. Proposition 2.15. Let X be an affine space for the real vector space U of dimension n, and let A = {α1 , . . . , αk } be a set of linearly independent vectors in U . Then we define the manifold with corners
X A = X × Rk+ / Rk ,
(10)
where Rk acts via the (free and proper) anti-diagonal action (s1 , . . . , sk ) · (x, (λ1 , . . . , λk )) = (x −
k X
si αi , (es1 λ1 , . . . esk λk )).
(11)
i=1
The residual U –action, given by [(x, (λ1 , . . . , λk ))] + u = [(x + u, (λ1 , . . . , λk ))], renders X A into a complete log affine manifold with corners.
6
(12)
The space X A constructed above is diffeomorphic to the standard corner Rk+ × and we may describe each stratum of the manifold with corners as follows. The open dense stratum is a copy of the original affine space X, included in X A via the U -equivariant embedding x 7→ [x, (1, . . . , 1)]. There are k codimension 1 strata: for each αi ∈ A, we have the equivariant embedding Rn−k ,
X/hαi i
/ XA
x
/ [(˜ x, (1, . . . , 1, 0, 1, . . . , 1))]
(13)
where x ˜ ∈ X is any lift of x and 0 occurs in the ith position. In a similar way, all the strata are naturally identified with the quotient affine spaces X/hA0 i corresponding to the subsets A0 ⊂ A. As before, the identification of algebroids defines a logarithmic U -valued 1form ξ ∈ Ω1 (X A , log ∂X A ) ⊗ U, (14) which has residue along the boundary stratum X/hαi i given by αi ∈ U . 2.16 The compactification in Proposition 2.15 may be described informally as at taching a set of k hyperplanes, k2 codimension 2 planes, etc. to an n-dimensional affine space at infinity, in the configuration of a single k-corner. We now explain how several such corner attachments can be made to a single affine space. While it is possible to describe multiple corner attachments using a quotient construction as above, we find it simpler to use a gluing approach, as follows. Let A0 ⊂ U be a set of linearly independent vectors in U . Suppose that we have two enlargements A0 ⊂ A− and A0 ⊂ A+ , where each of A− , A+ is a set of linearly independent vectors in U , and the convex hulls of A− , A+ intersect precisely in the convex hull of A0 . According to the procedure described above, each of A0 , A− , A+ determines a partial compactification X A0 , X A− , and X A+ . It is clear from the construction that we have open embeddings of log affine manifolds as shown below X A− c ; i
X A0
X A+
(15)
j
The gluing of X A− to X A+ along X A0 , i.e. the fibred coproduct, is then welldefined as a log affine manifold X A− ∪A+ = (X A− )i qj (X A+ ),
(16)
and it achieves the goal of attaching both of the corners defined by A− , A+ to the affine space X. To organize the system of coproducts necessary to combine many such partial compactifications, it is natural to borrow from toric geometry the notion of a fan of strictly convex cones in U .
7
2.17 Let U be a real vector space of dimension n, and let F be a finite set of vectors in U . Let CF be a collection of linearly independent subsets A ⊂ F, each of which is closed under convex hull in the sense that the convex hull of A does not meet F \ A. The collection CF must have the property that if A ∈ CF then all subsets of A are also contained in CF . The collection CF then determines a simplicial fan ΣF consisting of the set of all strictly convex cones hAi+ ⊂ U , A ∈ CF , where hAi+ denotes the convex hull of A.
Figure 1: A fan associated to F = {(1, 0), (1, 1), (0, 1)} For example, we may choose a fan associated to F = {(1, 0), (1, 1), (0, 1)} consisting of one 2-dimensional cone and three 1-dimensional cones, as shown in Figure 1. Given the fan ΣF , we now define a partial compactification of any affine space X modeled on U , by attaching to X a collection of quotient affine spaces at infinity. Namely, for each cone hAi+ ∈ ΣF of dimension k, we attach the quotient space X/hAi at infinity, creating a codimension k stratum of a log affine manifold with corners. Definition 2.18. Let ΣF be the simplicial fan on U defined above, and view CF as a directed graph, ordered by containment. For any affine space X modeled on U , and any containment A ⊂ A0 , we have an open embedding X A → X A0 . The induced coproduct a X CF = XA (17) A∈CF
then defines a partial compactification of X to a log affine manifold with corners, which we call a tropical domain. The space X CF is compact if and only if the fan ΣF is complete, i.e. its cones cover U . 2.19 The notion of tropical domain defined above is closely related to the notion of extended tropicalization Trop(X ) of a complex toric variety X introduced by Kajiwara [10] and Payne [17] (see also [5] for similar ideas). More precisely, if the fan ΣF is chosen to be the fan defining a complex toric variety X , and the vectors in F are chosen to be primitive integral vectors, then the resulting tropical domain can be identified with the extended tropicalization of X . Indeed, it is precisely in this case that the tropical domain serves as the correct target for the tropical momentum map defined on Caine’s examples [2] of canonical real Poisson structures on complex toric varieties. To fit these examples into our formalism, we blow up their circle fixed point loci to produce a toric log symplectic manifold with corners.
8
2.3
Welding of domains
In the previous section, we explained how an affine space X modeled on the real vector space U may be partially compactified to a log affine manifold with corners X CF called a tropical domain. We now explain how tropical domains may be welded together, i.e. glued along their boundaries, in such a way that the result is a log affine manifold with corners and normal crossing degeneracy divisor. The welding is defined in such a way that the affine monodromy of the result is trivial. We call these log affine manifolds tropical welded spaces. Welding tropical domains along a matched pair of faces 2.20 Let X CF and X CF0 be tropical domains with interior affine space X and corresponding finite subsets F, F0 ⊂ U . To weld these together along the codimension 1 boundary strata corresponding to α ∈ F and α0 ∈ F, we require the following compatibility condition. Definition 2.21. For α ∈ F, let CFα ⊂ CF denote the collection of sets A ∈ CF which contain α. We say that α ∈ F and α0 ∈ F0 are a matched pair when 0 CFα = CFα0 as a set of subsets of U . In particular, α = α0 and the fans ΣF , ΣF0 share any cones containing α. Any β ∈ F which is present in any set in CFα is 0 said to be adjacent to α, and its copy in F0 is denoted ψαα (β).
α0
α
Figure 2: A matched pair α ∈ F, α0 ∈ F0 Restricting the coproduct (17) to CFα ⊂ CF , we obtain Uα =
a
X A,
(18)
A∈CFα
defining an open neighbourhood in X CF of the codimension 1 stratum corresponding to α. The matching of α ∈ F and α0 ∈ F0 ensures that the neighbourhoods 0 U α and U α of the corresponding boundary strata are isomorphic. 2.22 The welding is then defined as follows. Each of the log affine manifolds with corners X A constituting the coproduct (17) is canonically embedded in the log affine manifold (without corners)
e A = X × Rk / Rk , X
(19)
e A is via the where the Rk action is as in (11) and the embedding of X A in X k k standard inclusion of R+ in R as an orthant. This allows us to define a collar extension of the neighbourhood U α :
9
a
U α ⊂ Ueα =
eA. X
(20)
A∈CFα
The matching condition between α ∈ F and α0 ∈ F0 then gives rise to a canonical U -equivariant “reflection” isomorphism Ueα
ψα
/ Ueα0 ,
(21)
given as follows: for each A ∈ CFα , we write A = {α, α2 , . . . , αk } and define ψ α on e A ⊂ Ueα via X [(x, (λ1 , λ2 , . . . , λk ))]
ψα
/ [x, (−λ1 , λ2 , . . . , λk )] .
(22)
0
The isomorphism ψ α embeds U α into Ueα in such a way that it intersects U α exactly in the closure of the codimension 1 stratum corresponding to α.
0
Definition 2.23. Let X CF , X CF0 be as above and let α ∈ F, α0 ∈ F0 be a matched pair. The welding of X CF to X CF0 along (α, α0 ) is defined by gluing together collar neighbourhoods of the boundary strata corresponding to α, α0 , namely: X CF # X CF0 = (X CF (α,α0 )
a
Ueα )
a
(X CF0
a
0
.
Ueα ) (x ∼ ψ α (x)).
(23)
U α0
Uα
The result is again a complete log affine manifold with normal crossing degeneracy locus. General welding along a matched pair of faces 2.24 We assume by induction that a space X has been produced by welding together a collection of tropical domains {X CF1 , . . . , X CFn } according to a list of matched pairs L, and we explain how to weld an additional matched pair (αi , αj ) ∈ Fi × Fj \ L. Note that X need not be connected, so that this procedure includes the general welding of two tropical welded spaces. Definition 2.25. Let (αi , αj ) ∈ Fi × Fj \ L be a matched pair as above. The welding along (αi , αj ) is locally obstructed if there exists βi adjacent to αi such that either α
α
i) (βi , ψαij (βi )) ∈ L, that is, βi is already welded to ψαij (βi ), or α
ii) There exist βk , αl , βm such that (βi , βk ), (ψββik (αi ), αl ) and (ψαij (βi ), βm ) are all in L, or either of the conditions holds with i, j exchanged. 2.26 If a matched pair (αi , αj ) is locally unobstructed, it may give rise to coerced pairs, namely pairs adjacent to αi , αj which should be welded in order to maintain the smoothness of the resulting log affine manifold, as shown in Figure 3.
10
We say that (αi , αj ) is globally unobstructed when iteratively taking coerced pairs results only in matched pairs which are locally unobstructed.
βi
βl
αj αi
Figure 3: Locally unobstructed pair (αi , αj ) and coerced pair (βi , βl ) Definition 2.27. Let (αi , αj ) ∈ Fi × Fj \ L be a matched pair which is globally unobstructed. We then define the welding along (αi , αj ) by iteratively gluing as in Definition 2.23 along (αi , αj ) and all its resulting coerced pairs. 2.28 Using the welding method detailed above, we may construct any complete log affine manifold X with corners, with divisor D of normal crossing type, trivial affine monodromy, and such that each component of X \ D is identified with a standard affine space U . Examples We now present three illustrative examples of welding to produce nontrivial log affine structures on R2 , S 2 , T 2 and an orientable surface of genus two. In these examples we omit a detailed discussion of the resulting logarithmic one-form (9) and its residues or integrals. Example 2.29. Consider a complete log affine manifold diffeomorphic to R2 , with degeneracy along three curves with normal crossings as depicted in Figure 4.
7 6 5
2 1 4
3
Figure 4: Log affine structure on R2 with degeneracy along three skew lines To build the example, we use seven partial compactifications of the affine plane, labeled as in Figure 4 and with fans as shown in Figure 5. The set of matched pairs depicted in Figure 5 give rise, after welding, to a complete log affine structure on a contractible 2-dimensional manifold without boundary, with degeneration locus as described above.
11
1β α γ
3
2
4β
5 η
δ γ
ζ
6
θ
θ
η
ζ
7 α
ι
δ ι
Figure 5: Welding of seven log affine manifolds with corners Example 2.30. Expressing S 2 as the quotient of R3 \ {0} by the diagonal Raction t · (x, y, z) = et (x, y, z), (24) we see that S 2 carries a residual R2 action, and therefore a log affine structure with degeneracy locus given by the intersection with the coordinate hyperplanes, a normal crossing divisor which divides the sphere into 8 contractible regions.
Figure 6: Log affine structure on S 2 with degeneracy along three circles We may therefore describe it as a welding of 8 copies of a triangular compactification of R2 , with fans and matching pairs as described in Figure 7.
β
δ
α γ
ζ
α η
δ
κ
ζ
ι
β
ι
λ θ η
θ γ
µ
λ
µ
κ
Figure 7: Log affine structure on S 2 welded from eight triangles Example 2.31. The 2-torus S 1 × S 1 carries a standard affine structure but we may also equip it with a log affine structure given by the action of R2 generated by the vector fields V1 = (X, 0) and V2 = (0, X), where X is a vector field on S 1 with two nondegenerate zeros. The determinant V1 ∧ V2 then defines a normal crossing divisor consisting of four circles, dividing the torus into four square regions. We may therefore describe this example by a welding of four squares, as in Figure 8.
12
β α
γ
γ
β
α
θ
η ζ
δ
θ
η ζ
δ
Figure 8: Log affine structure on T 2 welded from four squares Example 2.32. Fix a complete hexagonal fan in R2 and weld four copies of this fan according to the matched pairs listed in Figure 9. The result is a complete log affine structure on the orientable genus 2 surface, corresponding to a tiling of the surface by four hexagons.
β
γ
β
c α
δ
α d
ζ
e
b
c
a δ
ζ
b
γ
a d
z
e
z
Figure 9: Log affine structure on genus 2 surface welded from four hexagons
3
Toric log symplectic manifolds
3.1 In analogy with the Poisson structure on the dual of any Lie algebra, there is a canonical Poisson structure on the total space of the dual bundle of any Lie algebroid [4]. In the case of the tangent bundle, this coincides with the canonical symplectic form on the total space of the cotangent bundle. Given an action of Rn on X generated by vector fields X1 , . . . Xn , let A = Rn n X be the corresponding abelian action Lie algebroid. If {p1 , . . . , pn } are standard vertical coordinates on N = tot(A∗ ), then the canonical Poisson structure is given by n X ∂
∂pk k=1
∧ Xk ∈ C ∞ (N, ∧2 T N ).
If X is taken to be a log affine manifold with free divisor D, then N ∼ = Rn × X is also log affine for the pullback divisor Z = Rn ×D. The above Poisson structure is then nondegenerate as a section of ∧2 (T N (− log Z)). If {ξ1 , . . . , ξn } is the basis of log forms dual to the basis {X1 , . . . , Xn } of T X(− log D), then we may write the inverse of the above Poisson structure as follows: ω=
n X
ξk ∧ dpk ∈ Ω2 (N, log Z).
(25)
k=1
This generalization of a symplectic structure is the focus of our study, and we may characterize it as follows.
13
Definition 3.2. Let M be a manifold with free divisor Z and corresponding Lie algebroid T M (− log Z). A logarithmic symplectic form is a closed logarithmic 2-form ω ∈ Ω2 (M, log Z) (26) which is nondegenerate in the sense that interior product defines an isomorphism ω : T M (− log Z) → (T M (− log Z))∗ . Remark 3.3. A log symplectic form, as defined above, may be viewed alternatively as a usual Poisson structure π which degenerates along Z in such a way that the top power π n is a reduced defining section for the free divisor Z. 3.4 The log symplectic form (25) is invariant under the global Rn action on N by fibrewise translations. We may therefore quotient by the lattice Γ = (2πZ)n ⊂ Rn and obtain a (trivial) principal T n bundle M equipped with a log symplectic form invariant under the torus, which we also denote by Ω. The bundle projection then induces a smooth submersion M
µ
/X.
Let t ∼ = Rn be the Lie algebra of T n and let ρ : t → C ∞ (M, T M ) be the infinitesimal principal action, which takes values in the sections of T M (− log π ∗ D). By contraction, we have a natural closed logarithmic 1-form with values in t∗ : iρ ω ∈ Ω1 (M, log Z) ⊗ t∗ . In the case that Z is of normal crossing type, we may use the Mazzeo-Melrose decomposition A.4 to express the logarithmic cohomology as follows, summing over the smooth components of Z: ∼ H 1 (M ) ⊕ H 1 (M, log Z) =
X
H 0 (Zi ).
(27)
i
Recall that as a log affine manifold, X carries a natural t∗ -valued logarithmic form (c.f. Equation 2) ξ, and so the map µ plays the role of a momentum map, in the sense that it satisfies iρ ω = −µ∗ ξ. (28) Definition 3.5. A toric log symplectic manifold is a log symplectic manifold (M, Z, ω) of dimension 2n, equipped with an effective action of the torus T n by symplectomorphisms, as well as a proper, T n -invariant smooth map µ : M → X to a log affine manifold (X, D, ξ) called a tropical moment map, satisfying condition (28). In the case that Z is a normal crossing divisor, we say that the toric log symplectic manifold is Hamiltonian when the logarithmic cohomology class of iρ ω has vanishing component in H 1 (M ) ⊗ t∗ .
14
Remark 3.6. The most significant aspect of the above definition is that while iρ ω is closed, we do not require it to be exact, that is, to be the differential of a Hamiltonian function. The “reason” for this is that such Hamiltonians would have logarithmic singularities along Z, and there is no need to introduce functions with singularities in the development of the theory. On the other hand, in the Hamiltonian case we do require that the component of [iρ ω] in H 1 (M ) ⊗ t∗ vanishes, though the full logarithmic class may not. This condition specializes to the traditional Hamiltonian assumption in the case of usual symplectic manifolds. Remark 3.7. One can show using Proposition A.19 that the vanishing of the component of [iρ ω] in H 1 (M ) ⊗ t∗ is equivalent to the same condition on [ξ], namely that its Mazzeo-Melrose component in H 1 (X) ⊗ t∗ vanishes. This means that the Hamiltonian assumption above is equivalent to the assumption that (X, D, ξ) has trivial affine monodromy.
3.1
Trivial principal case
3.8 In Paragraph 3.4 we described the simplest example of a toric Hamiltonian log symplectic manifold: a trivial principal T n bundle over a log affine manifold. Proposition 3.9. Let (X, D, ξ) be a log affine manifold, and let M be the trivial principal T n bundle over X. The log symplectic form (25) then defines a toric Hamiltonian log symplectic structure on M with momentum map µ given by the bundle projection and degeneracy locus Z = µ−1 (D). 3.10 This example may be deformed by adding a magnetic term. That is, if ω is the canonical symplectic form (25), then ω + µ∗ B is also a log symplectic form, for any closed form B ∈ Ω2 (X, log D). By an application of the Moser method, one can show that only the logarithmic cohomology class of B is relevant to the resulting symplectic structure up to equivariant symplectomorphism, yielding the following result. Theorem 3.11. Fix the log affine manifold (X, D, ξ) and the trivial principal T n bundle µ : M → X as above, and let ω0 be the canonical log symplectic form (25). If ω is another log symplectic form such that (M, ω, µ) is toric Hamiltonian, then the difference ω − ω0 is basic and closed. Furthermore, the map ω 7→ [ω − ω0 ] ∈ H 2 (X, log D) induces a bijection between equivalence classes of toric Hamiltonian log symplectic structures with fixed momentum map π and the vector space H 2 (X, log D).
15
Remark 3.12. The usual notions of invariant and basic forms on a principal bundle carry over to the logarithmic context above, because of the exact sequence relating logarithmic vector fields on the domain and codomain of π: 0
/ TM/X
/ TM (− log µ∗ D) µ∗ / µ∗ TX (− log D)
/0,
(29)
where TM/X is the vertical tangent bundle of the principal bundle M . This is an example of an algebroid submersion (see [6]) and gives rise to an injective cochain homomorphism from the log de Rham complex of (X, D) to the log de Rham complex of (M, µ∗ D). Remark 3.13. By an argument similar to that of Mazzeo-Melrose [13] (see Appendix A.4), one can show that for D ⊂ X a normal crossing divisor, the logarithmic cohomology groups may be expressed in terms of usual de Rham cohomology: H 2 (X, log D) = H 2 (X) ⊕
X
H 1 (Di ) ⊕
X
H 0 (Di ∩ Dj ),
where we sum over the components Di of D and their pairwise intersections. For example, the log affine structure on S 2 described in Example 2.30 has a divisor with three circular components intersecting in six points; as a result there is a 10-dimensional space of toric Hamiltonian log symplectic structures on the trivial T 2 -bundle.
3.2
Nontrivial principal case
3.14 We now show that certain nontrivial principal T n bundles over the log affine manifold (X, D, ξ) admit toric Hamiltonian log symplectic structures. Theorem 3.15. Let π : M → X be a principal T n bundle over the log affine manifold (X, D, ξ), with real Chern classes c1 (M ) = (c11 , . . . cn1 ) ∈ H 2 (X, R) ⊗ t. There exists a log symplectic form ω making (M, ω, π) a toric Hamiltonian space if and only if the following class vanishes: Tr(c1 (M ) ∧ [ξ]) =
n X
ck1 ∧ [ξk ] ∈ H 3 (X, log D).
(30)
k=1
Under this condition, the space of equivalence classes of log symplectic forms making (M, ω, π) toric Hamiltonian is an affine space modeled on the vector space H 2 (X, log D). Proof. Let (M, ω, π) be as above. Choosing a principal connection θ ∈ Ω1 (M )⊗t∗ , we may write ω=
n X
θk ∧ αk + β,
k=1
16
(31)
where αk , β are invariant and basic logarithmic forms. The momentum map condition (28) implies that αk = ξk . Taking derivatives, we obtain n X
F k ∧ ξk + dβ = 0,
k=1
where [F k ] = ck1 , so that (30) vanishes, as required. Conversely, choose local trivializations for the principal T n bundle M over an open cover {Ui } of X, and let ϕij : (Ui × T n )|Ui ∩Uj → (Uj × T n )|Ui ∩Uj be the transition isomorphisms satisfying the usual cocycle condition. On the k th k : U ∩ U → S 1 . Also, choose a circle factor, ϕij acts by multiplication by gij i j k principal connection, defined by connection forms Ai ∈ Ω1 (Ui , log D) satisfying k. the cocycle condition i(Aki − Akj ) = d log gij n We first equip each Ui × T with the standard log symplectic form given in (25), namely ω ˜i =
n X
ξk ∧ dpk .
k=1 k = exp(iτ k ) then we have If we write gij ij
ϕ∗ij ω ˜j − ω ˜i =
n X
ξk ∧ dτijk =
k=1
n X
ξk ∧ (Aki − Akj ).
(32)
k=1
If the class (30) vanishes, then there exists B ∈ Ω2 (X, log D) such that, over Ui , dB =
n X
F k ∧ ξk =
k=1
n X
dAki ∧ ξk .
(33)
k=1
Combining (32) and (33), we see that the modified symplectic forms ωi = ω ˜ i + µ∗ (B|Ui +
n X
ξk ∧ Aki )
k=1
define a global form ω on M rendering (M, ω, µ) toric Hamiltonian, as required. Due to the choice of the 2-form B, we see that the symplectic form is only uniquely determined modulo closed logarithmic 2-forms on X. Applying the Moser method, which uses the properness of µ, we therefore obtain a free and transitive action of H 2 (X, log D) on the set of equivalence classes of toric Hamiltonian log symplectic structures (M, ω, µ), as needed. Remark 3.16. For D ⊂ X a normal crossing divisor, we have the following expression for the third logarithmic cohomology: H 3 (X, log D) = H 3 (X) ⊕
X
H 2 (Di ) ⊕
X
H 1 (Di ∩ Dj ) ⊕
X
H 0 (Di ∩ Dj ∩ Dk ),
where we sum over double and triple intersections of the components Di of D. If we also have dim X = 2, then this group vanishes.
17
Example 3.17. For the log affine structure on S 2 described in Example 2.30, we have a 10-dimensional affine moduli space of toric Hamiltonian log symplectic structures on any principal T 2 bundle over S 2 , including for example the Hopf manifold S 3 × S 1 . Example 3.18. The log affine structure in Example 2.32 has a divisor with six components intersecting in six R2 fixed points. As a result, we obtain a 13dimensional affine moduli space of toric Hamiltonian log symplectic structures on any principal T 2 bundle over the orientable genus 2 surface. 3.19 Theorem 3.15 provides a classification of log toric Hamiltonian structures on a principal T n bundle which satisfies condition (30). Note, however, that the notion of equivalence is that of equivariant symplectomorphism ψ : M → M , in the sense that µ ◦ ψ = µ. In some cases, there are additional symmetries which do not commute with µ, and we describe these now. The main observation is that while toric Hamiltonian forms on such a principal bundle may be deformed by closed basic forms, the map on cohomology groups has a kernel: we have the exact Gysin sequence H 0 (X, log D) ⊗ t∗
c1 (M )
/ H 2 (X, log D)
µ∗
/ H 2 (M, log π ∗ D) .
(34)
As a result, we may expect that 2-forms in the kernel of µ∗ would act trivially on the symplectomorphism class of (M, ω). Indeed, let v = (v1 , . . . , vn ) ∈ t∗ and consider the class c1 (M ) · v =
n X
vk ck1 ∈ H 2 (X, log D).
k=1
Choosing a connection θ on M as in the proof of Theorem 3.15, the above class gives rise to a 1-parameter family of toric log symplectic structures ωt = ω + t
n X
vk F k .
k=1
The Moser argument then indicates that the flow of Xt = ωt−1 (v · θ) trivializes the above family. This vector field is invariant and projects to the vector field ρ(v) on the log affine manifold X. Therefore, Xt is complete (allowing the Moser argument to proceed) if and only if ρ(v) is complete, yielding the following result. Theorem 3.20. If ω, ω 0 are two toric Hamiltonian log symplectic structures on the principal T n -bundle (M, π) over the complete log affine manifold (X, D, ξ), and if [ω 0 −ω] is of the form c1 (M )·v for v ∈ t, then there is a symplectomorphism ψ : (M, ω) → (M, ω 0 ) such that µ ◦ ψ = Tv ◦ µ, where Tv : X → X is the automorphism given by translation by v.
18
4
Polytopes and symplectic cutting
In this section, (X, D, ξ) denotes a fixed complete log affine n-manifold, possibly with corners, and we assume that it is a tropical welded space of the type constructed in Section 2.3, so that all of its strata (i.e. Rn -orbits) are affine spaces and D is of normal crossing type.
4.1
Log affine polytopes
4.1 Each stratum of X is an affine space, where there is a clear notion of affine hyperplane. We extend this notion as follows. Definition 4.2. An affine linear hypersurface in the log affine manifold (X, D, ξ) is a connected and embedded hypersurface in X, not contained in D, and invariant under the action of a linear hyperplane in t. Definition 4.3. A log affine polytope is a closed equidimensional submanifold with corners of X each of whose codimension 1 boundary strata has closure which is either: • contained in D, and called a singular face, or • contained in an affine linear hypersurface. Such a stratum may intersect D, in which case we require it to intersect all possible intersections of components of D transversely (so its union with D is also of normal crossing type), and we call the stratum a log face. If the stratum does not intersect D, we call it an interior face. The polytope is called convex when its intersection with each component of X \ D is convex.
Figure 10: Convex log affine polytope with one singular, one interior, and three log faces 4.4 Suppose the hypersurface H ⊂ X is not contained in D, but is affine linear for the hyperplane tH ⊂ t. If H ∪D is normal crossing, then H inherits a log affine structure, with divisor DH = D ∩ H and trivialization ξH ∈ Ω1 (H, log DH ) ⊗ t∗H given by the projection to t∗H of the pullback of ξ to H. As a result of this, we see that the closure of any log face contained in H is itself a log affine polytope. Similarly, if C is a component of the degeneracy divisor D, then it inherits a log affine structure, with divisor DC = (D \ C) ∩ C and trivialization ξC ∈
19
Ω1 (C, log DC ) ⊗ t∗C given by the isomorphism of extensions RO
/ T C(− log DC ) O
/ T X(− log D)|C O ξ
sC
ξC
/t
/ tC
where sC is the 1-dimensional stabilizer of a generic point on C. As a result, the closure of any singular face contained in C is also a log affine polytope. Proposition 4.5. The closure of any stratum of a log affine polytope is itself a log affine polytope. 4.6 Any interior face F of a log affine polytope is an affine polytope in the usual sense, and if it has dimension k, its associated 1-form ξ ∈ Ω1 (F ) ⊗ Rk defines a volume form det ξ = ξ1 ∧ · · · ∧ ξk . As a result, any compact interior face has a well-defined volume in R+ . Even logarithmic volume forms may have well-defined integrals over compact manifolds, by extracting the Cauchy principal value: combining the results of [19] with Fubini’s theorem (and using the normal crossing assumption on D), we see that any compact and oriented log affine polytope has a well-defined volume, as long as none of its faces is contained in D. Definition 4.7. The regularized volume of a compact and oriented log affine polytope ∆ ⊂ (X, D, ξ) without singular faces is the real number defined by the iterated Cauchy principal value Z
det ξ.
Vol(∆) = P V ∆
Definition 4.8. An affine linear function on the log affine manifold (X, D, ξ) is a function f : X → R such that there exists a covector a ∈ (Rn )∗ (called the linear part of f ) such that for all u ∈ Rn , Lρ(u) f = a(u),
(35)
or, equivalently when X is complete, f (x + u) = f (x) + a(u) for all x ∈ X and translations u ∈ Rn . 4.9 let ∆ ⊂ X be a log affine polytope, and let F ⊂ ∆ be any nonsingular face. Then there is an affine linear function f defined in a tubular neighbourhood UF of F such that f |F = 0 and f ≥ 0 on ∆ ∩ UF . We call this a boundary defining function for the face F . Lemma 4.10. Let F be a log face with boundary defining function f . If F intersects a component Di of the divisor D with associated residue vi , then the linear part of f satisfies a(vi ) = 0. Proof. If p ∈ F ∩Di then it is fixed by the action of vi , and so Lρ(vi ) f = a(vi ) = 0, as needed.
20
Definition 4.11. An elementary log affine polytope is a convex log affine polytope ∆ with ∆ \ D connected. Lemma 4.12. Let ∆ be an elementary log affine polytope, let F be a nonsingular face of ∆ and let f be a boundary defining function for F with linear part a. If a component Di of the divisor D with associated residue vi meets ∆ but does not intersect F , then a(vi ) < 0. Proof. By convexity, each point p ∈ ∆ \ D lies on a unique straight line in the direction vi whose closure meets Di (at infinity in the −vi direction). Since f is positive on the interior of ∆, it follows that Lρ(vi ) f < 0. 4.13 Let ∆ be an elementary log affine polytope in (X, D, ξ), which we now assume is a tropical domain with associated simplicial fan ΣF in U = Rn , as in Definition 2.18. Uniquely defined up to a positive scalar, the boundary defining function fi for each of the nonsingular faces Fi , i ∈ I of ∆ has linear part ai which determines a half-space Hi = {u ∈ U : ai (u) > 0}. By the above Lemmas, Hi does not intersect ΣF . Compactness of ∆ may be expressed in terms of these half spaces, as follows: Proposition 4.14. The elementary log affine polytope ∆ is compact if and only if ΣF t (∪i∈I Hi ) = U. Example 4.15. Let X be the tropical domain (R2 )A , for A = {(1, 0), (1, 1)}. Let ∆ ⊂ X be the log affine polytope given by the closure of {f1 ≥ 0} ∩ {f2 ≥ 0}, for f1 (x, y) = −y,
f2 (x, y) = −x + y.
Then ∆ is a compact convex log affine manifold with two singular faces and two log faces. Figure 11 shows the union of the fan defining X with the pair of half-planes defined by the nonsingular faces of ∆ ⊂ X. H2 H1
Figure 11: Fan and half-spaces defined by the compact polytope ∆ 4.2
Symplectic cuts
4.16 Let ∆ ⊂ X be a log affine polytope and let F ⊂ ∆ be a nonsingular face with boundary defining function f . Let (M, Z, ω, π) be a principal toric Hamiltonian space over X, as in Theorem 3.15. If the linear part of f is a primitive integral
21
vector a ∈ t = Rn , then the Hamiltonian function π ∗ f generates the action of a circle subgroup Sa1 ⊂ T n on M and we may define a log symplectic cut of M along π −1 (F ), generalizing the usual symplectic cut [12] as follows. Note that while f is only defined in a neighbourhood of F , this is irrelevant to the symplectic cut procedure, in which we only modify M near π −1 (F ). We assume for simplicity that π ∗ f is defined on all of M . Theorem 4.17. Let (M, Z, ω) be a log symplectic manifold and let f : M → R be a Hamiltonian function generating an S 1 action. Suppose that the S 1 action is free along f −1 (0). Then M × C, equipped with the log symplectic form ω + idz ∧ d¯ z , has an 1 2 antidiagonal S action whose Hamiltonian µ = f − |z| gives rise to a smooth log symplectic reduction Mf = µ−1 (0)/S 1 (36) called the log symplectic cut. The Hamiltonian f descends to a function f˜ on Mf , defining a residual S 1 action which now fixes the log symplectic submanifold S = f˜−1 (0). As in the usual symplectic cut, we may identify S with the log symplectic reduction f −1 (0)/S 1 , whereas Mf \ S is isomorphic to M \ f −1 (0). Proof. Since the S 1 action is free along f −1 (0), the generating vector field ∂θ is nowhere vanishing there, and so by the nondegeneracy of ω, along this locus df = −i(∂θ )ω is a nowhere vanishing log 1-form, showing that df : T M (− log Z)|f −1 (0) → T0 R is surjective, verifying the transversality condition of Proposition A.8. This immediately implies the transversality condition for µ also holds. The same freeness assumption also implies that the antidiagonal S 1 action on µ−1 (0) is free, giving a smooth log symplectic quotient by Proposition A.8. The remaining statements follow from the same arguments given for usual symplectic cutting. 4.18 The above construction allows us, just as in the usual Delzant theory, to iteratively apply symplectic cuts along the nonsingular faces of ∆, under the usual Delzant condition that whenever k such faces meet, we may choose boundary defining functions whose linear parts define an integral basis for a maximal rank k sublattice of Zn . Definition 4.19. The log affine polytope ∆ satisfies the Delzant condition when, for any point p ∈ ∆, the collection of k nonsingular faces meeting p defines an integral basis for a maximal rank k sublattice of Zn ⊂ t. Corollary 4.20. Given any log affine polytope ∆ in the log affine manifold (X, D, ξ), and given Chern classes (c11 , . . . cn1 ) ∈ H 2 (X, R) ⊗ t such that (30) vanishes, we may construct a corresponding principal toric Hamiltonian log symf, Z, e ω e ) with Z of normal crossing type and momentum map plectic manifold (M to X as in Theorem 3.15.
22
f along If ∆ satisfies the Delzant condition, we may apply symplectic cuts to M all of the nonsingular faces of ∆ to produce another toric Hamiltonian log symplectic manifold (M, Z, ω), also with Z of normal crossing type, with an identical momentum map image. The toric orbit type stratification of (M, Z) then coincides with the stratification of ∆, via the moment map. If, in addition, ∆ has no singular faces, then (M, Z, ω) is a smooth log symplectic manifold without boundary.
Example 4.21. On the log affine manifold (X, D, ξ) constructed in Example 2.32, we choose an affine linear function h whose linear part vanishes on the vectors α = a = −δ = −d and is positive on = e and ζ = f . The function h defines a half-space in each of the four tropical domains of X, defining a compact log affine polytope ∆ with a single log face and no singular faces. In Figure 4.21, we show the fans corresponding to the strata of X which meet ∆, as well as the half-spaces H1 , . . . , H4 defined by the linear parts of h in each tropical domain. In Figure 4.21, we show the orientable genus 2 surface X together with its decomposition into four hexagonal domains, and we see that the log affine polytope ∆ is 2-dimensional submanifold of genus 1 with a single log face.
γ
β
c α
δ H1
β
γ a
δ
b α
d
H2
c
H3
b a
d H4
Figure 12: Fan and half-spaces defined by polytope ∆ in Example 4.21
Figure 13: Log affine polytope of genus 1 with single log face, sitting in log affine manifold of genus 2 obtained from welding four hexagons. This example demonstrates that log affine polytopes may have nontrivial topology and so toric Hamiltonian log symplectic manifolds need not be equivariantly diffeomorphic to usual toric symplectic manifolds, or even quasitoric manifolds. In fact, if we take the trivial principal T 2 bundle over the polytope ∆ above and perform a symplectic cut along the unique log face, we conclude by Proposition 1 in [9] that the resulting manifold is diffeomorphic to S 1 × ((S 1 × S 2 )#(S 1 × S 2 )), and by Lemma 2.3 of [14], this manifold, although
23
it is almost complex, does not admit any symplectic structure, by an argument involving the vanishing of Seiberg-Witten invariants. Nevertheless, it admits a log symplectic structure by our construction.
5
Delzant correspondence
5.1 In this section, we establish an analogue of the Delzant correspondence for a class of toric Hamiltonian log symplectic manifolds. The correspondence differs from Delzant’s in several important ways: first, the analog of the Delzant polytope is a log affine polytope ∆ sitting in a tropical welded space, and need not be contractible; second, the n-dimensional polytope ∆ is decorated by an n-tuple of integral classes in H 2 (∆); finally, with the above data fixed, the equivariant isomorphism class of the log symplectic form is unique only up to addition of a magnetic term coming from the logarithmic cohomology of ∆. 5.2 We prove an analogue of the Delzant classification for compact orientable log symplectic manifolds (M, Z, ω), under the assumption that Z is a real divisor of normal crossing type and such that ∂M ⊂ Z. We assume that there is an effective action of T n preserving the structure and which is Lagrangian, meaning that the infinitesimal action ρ : t → C ∞ (M, T M ) satisfies ω(ρ(a), ρ(b)) = 0 for all a, b ∈ t. Theorem 5.3. Let (M, Z, ω) be an orientable compact and connected log symplectic manifold with Z of normal crossing type and ∂M ⊂ Z, equipped with an effective Lagrangian T n action. Then the following hold: 1. The quotient M/T n defines a compact and connected Delzant log affine polytope (∆, D, ξ) (with D of normal crossing type). Furthermore, the quotient map µ : M → ∆ is a tropical momentum map. 2. The equivariant isomorphism class of (M, Z, ω, µ) is determined, up to translation by H 2 (∆, log D), by the equivalence class of the log affine polytope ∆ f, the symplectic uncut of M . and the n-tuple of Chern classes of M If, in addition, the action is Hamiltonian in the sense of Definition 3.5, so that (∆, D, ξ) has trivial affine monodromy, then we also have: 3. ∆ is equivalent to a convex log affine polytope in a tropical welded space as in Section 2.3. In particular, (M, Z, ω) is equivariantly isomorphic to a toric Hamiltonian log symplectic manifold of the type constructed in Corollary 4.20. Proof. Part 1: The notion of symplectic uncut was introduced in [15] to give a simpler proof of the Delzant correspondence for toric symplectic manifolds with proper momentum maps. It was studied carefully in [11] in order to characterize noncompact toric symplectic manifolds. We modify the construction slightly in Appendix A.3 in order to apply it to log symplectic manifolds.
24
f, Z, e ω e ), as described in Definition A.20, is a modificaThe symplectic uncut (M tion of (M, Z, ω) along its submaximal orbit strata which renders the Lagrangian T n action into a principal action. It follows from Lemma A.14 that ∆ is a compact manifold with corners and that D = Z/T n is a normal crossing divisor. Since the infinitesimal action ρ is Lagrangian, we have that the logarithmic 1-form f, log Z) e ⊗ t∗ e ∈ Ω1 (M ιρ ω
is basic, defining a closed logarithmic 1-form ξ ∈ Ω1 (∆, log D) ⊗ t∗ which endows ∆ with the structure of a log affine manifold. To see that ∆ is a log affine polytope, note that each component of ∂∆ not f. contained in D is the T n quotient of an exceptional boundary component of M By the normal form given by Lemma A.16, and by the computation in Example A.18, we see that each exceptional boundary component is the zero locus of an affine linear function with linear part given by a primitive generator of the integral lattice in t. Finally, Proposition A.17 implies that ∆ satisfies the Delzant condition required by Definition 4.19. Part 2: The uncut construction is such that (M, Z, ω) may be obtained from f, Z, e ω e ) by symplectic cutting along the nonsingular faces of ∆. By Theo(M rem 3.15, the equivariant isomorphism class of the uncut is determined, up to translation by H 2 (∆, log D), by the equivalence class of ∆ and the n-tuple of f. Since equivariant isomorphisms descend under symplectic Chern classes of M cutting, we obtain the result. Part 3: We first construct a tropical welded space (X, DX , ξX ) which will contain the polytope (∆, D, ξ). We construct X by welding a collection of tropical domains {t∗j } corresponding to the connected components {Mj } of M \ Z. Each tropical domain t∗j is constructed as in Definition 2.18, from a set Fj of vectors and a fan Σj defined as follows. Each component Zi ⊂ Z of the normal crossing divisor has an induced normal crossing divisor Z|i = (Z \ Zi )∩Zi , and the residue of ω defines a form ResZi (ω) ∈ Ω1 (Zi , log Z|i ) which is T n -invariant, so that diρ ResZi (ω) = 0. This implies that ιρ ResZi (ω) ∈ Ω0 (Zi , log Z|i ) ⊗ t∗ is a constant: vi = iρ ResZi (ω). Each of the vectors vi is nonzero, since T n acts effectively on Zi , as shown in the proof of Lemma A.14. The set Fj of vectors used for the compactification t∗j is then the set of all vi such that M j ∩ Zi 6= ∅. The associated fan Σj in t∗ consists of all cones generated by subsets {vi1 , vi2 , . . . , vik } of Fj with the property that Mj ∩ Zi1 ∩ Zi2 ∩ . . . ∩ Zik 6= ∅. Such subsets are linearly independent by Proposition A.5.
25
Each component of Z which separates Mj from Mj 0 gives rise to a matched pair in Fj × Fj 0 , and in this way we obtain a list of globally unobstructed matched pairs which defines a tropical welded space (X, DX , ξX ) as in Definition 2.27. We now construct an embedding ι of ∆ into X as a log affine submanifold; the embedding is unique up to an overall translation. Choose a basepoint x0 in a component ∆0 of ∆\D, and choose y0 in the interior of the corresponding tropical domain t∗0 ⊂ X. We define ι(x0 ) = y0 , which fixes the translation ambiguity. For any other point x ∈ ∆\D, we choose a path γ from x0 to x which, for convenience, is transverse to each component of D. We may cover γ by open sets in each of which we have coordinates such that ξ is in normal form (47), and by applying Theorem A.6 we obtain a unique extension of ι to the union Uγ of this cover of γ, ιγ : Uγ → X,
(37)
which is a local diffeomorphism onto its image and preserves log affine structure. We write y = ιγ (x) ∈ X. We now show that the construction above is path-independent. Let γ 0 be another such path from x0 to x, yielding a possibly different point y 0 = ιγ 0 (x) ∈ X. Then ιγ 0 (γ 0 ) · ιγ (γ −1 ) is a path from y to y 0 . Because ∆ has trivial affine monodromy, the loop γ 0 · γ −1 has zero regularized length, implying Z
Z ιγ 0 (γ 0 )·ιγ (γ −1 )
ξX =
γ 0 ·γ −1
ξ = 0.
(38)
But by the construction of X, y and y 0 lie in the same affine stratum A of X, so we may choose a path δ ⊂ A from y to y 0 , and then using the affine structure on A, we have Z Z y0 − y =
ξA = δ
ιγ 0 (γ 0 )·ιγ (γ −1 )
ξX = 0,
where ξA is the 1-form defining the affine structure on A, obtained from ξ by reduction and pullback as in Paragraph 4.4. In this way, we obtain a local diffeomorphism of log affine manifolds onto its image ι : ∆ → X.
(39)
Because ∆ is compact, (39) is a covering map, so it suffices to show that it is a 1-sheeted cover. Take y0 ∈ ι(∆) ∩ (X \ DX ) and x0 , x00 ∈ ι−1 (y0 ). Because x0 and x00 are in the same component of ∆ \ D, we may choose a path λ from x0 to x00 contained in that component. Since ∆ has trivial monodromy, it follows that x00 − x0 =
Z
as required.
26
ξ = 0, λ
A A.1
Appendix Local normal forms for log affine manifolds
A.1 Let (X, D, ξ) be a log affine manifold with D of normal crossing type. If x ∈ X is such that precisely k components D1 , . . . Dk of D meet x, then we may choose coordinates (y1 , . . . , yn ) in a neighbourhood U centered at x such that D is the vanishing locus of the monomial y1 · · · yk . The coordinates also provide a natural trivialization of the algebroid T X(− log D), given by
T X(− log D) = y1 ∂y1 , . . . , yk ∂yk , ∂yk+1 , . . . , ∂yn .
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Proposition A.2. There exist coordinates (x1 , . . . , xn ) such that D is the vanishing locus of x1 · · · xk and ξ=
k X
x−1 i dxi ⊗ vi +
i=1
n X
dxi ⊗ vi ,
(41)
i=k+1
for (v1 , . . . vn ) a constant basis of Rn . Proof. Begin with coordinates (y1 , . . . , yn ) in the neighbourhood U as above, so that D is given by y1 · · · yk = 0 and ξ=
k X
yi−1 dyi
⊗ αi +
i=1
n X
dyi ⊗ αi ,
i=k+1
where αi : U → Rn are smooth vector-valued functions. It follows from dξ = 0 that for each i ≤ k, αi is constant along the hypersurface yi = 0. Denoting the basis (α1 (0), . . . , αn (0)) by (v1 , . . . , vn ), we conclude that n k X X dyi ⊗ vi = dF, ξ − yi−1 dyi ⊗ vi + i=1
i=k+1
for F : U → Rn smooth and F (0) = dF (0) = 0. We now expand F in terms of the basis (v1 , . . . , vn ): F =
n X
fi vi ,
i=1
and we define new coordinates on a possibly smaller neighbourhood: (
xi =
yi efi yi + fi
i≤k i > k.
In these coordinates we have the required expression (41).
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A.3 In the coordinates provided by Proposition A.2, the Rn -action on X may be written as ∗
∗
∗ u · (x1 , . . . xn ) = (ev1 (u) x1 , . . . evk (u) xk , xk+1 + vk+1 (u), . . . , xn + vn∗ (u)),
(43)
where (v1∗ , . . . vn∗ ) is the dual basis to (v1 , . . . , vn ). A.4 We see from (41) that for each i ≤ k, vi is the residue of ξ along Di , and so is an invariant of the log affine manifold. We also see that these residues are linearly independent, as they form part of the basis (v1 , . . . , vn ). The remaining basis elements {vi : i > k}, however, are not invariant and may be adjusted by a change of coordinates, as follows. If (v1 , . . . , vk , wk+1 , . . . , wn ) is any other basis, define {rij , sij } via vj =
X
rij wi +
i>k
X
sij vi , for all j > k.
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for i ≤ k for i > k.
(45)
i≤k
We then make the coordinate change (
x ˜i =
P
xi exp( j>k sij xj ) P j>k rij xj
As a consequence, we obtain k X i=1
x−1 i dxi
⊗ vi +
n X
dxi ⊗ vi =
k X
xi x ˜−1 i d˜
i=1
i=k+1
⊗ vi +
n X
d˜ xi ⊗ wi .
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i=k+1
Proposition A.5. Let (X, D, ξ) be a log affine n-manifold and let x ∈ X meet exactly k components D1 , . . . Dk of the normal crossing divisor D. The residues vi ∈ Rn of ξ along Di then form a linearly independent set, and for any extension (v1 , . . . , vk , vk+1 , . . . vn ) to a basis, there are coordinates near x such that ξ=
k X
x−1 i dxi ⊗ vi +
i=1
n X
dxi ⊗ vi .
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i=k+1
Theorem A.6. Let (X, D, ξ) and p ∈ U ⊂ X be as above. If we choose a basepoint q ∈ U \ D and map it to any point φ(q) ∈ Rn , then there is a unique extension of φ to a map to the log affine manifold constructed in (19): g n) , φ : U → (R A
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where A = {v1 , . . . , vk } is the set of residues of ξ in U . This map is an isomorphism of log affine manifolds onto its image. Proof. Choose local coordinates (x1 , . . . , xn ) in U so that ξ is given by (41), and n ) , so that its log 1-form ] choose similar adapted coordinates (y1 , . . . , yn ) for (R A ξA has the form ξA =
k X
yi−1 dyi
⊗ vi +
i=1
n X i=k+1
28
dyi ⊗ wi ,
(49)
for a basis (v1 , . . . , vk , wk+1 , . . . wn ) of Rn . By Proposition A.5, we may change coordinates so that ξA =
k X
n X
y˜i−1 d˜ yi ⊗ vi +
i=1
d˜ yi ⊗ vi .
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i=k+1
As a result, the map φ, which must satisfy φ∗ ξA = ξ, is given by (
y˜i =
e ci x i xi + ci
for i ≤ k for i > k,
(51)
where the constants {ci ∈ R} are fixed by φ(p) = q, as required.
A.2
Log symplectic reduction
A.7 Let (M, Z, ω) be a log symplectic manifold with an S 1 action generated by the vector field ∂θ , and suppose it is Hamiltonian, in the sense that there is a function µ : M → R such that i∂θ ω = −dµ.
(52)
f = µ−1 (0), the comWe impose the following transversality condition: along M f position dµ of dµ with the anchor map T M (− log Z) → T M
T M (− log Z)|M e
e dµ
/ µ∗ T0 R
(53)
must be surjective. In particular, this implies that 0 is a regular value of µ and f ⊂ M is a smooth hypersurface. If Z is a normal crossing divisor, (53) so M is surjective if and only if 0 is a regular value, not only for µ, but also for the f is restriction of µ to any stratum of Z. As a result, the intersection Ze = Z ∩ M f, and the kernel of the morphism (53) is the Lie a normal crossing divisor in M f e algebroid T M (− log Z). f, we obtain a If we assume further that S 1 acts freely and properly on M 1 e f/S 1 , and normal crossing divisor Z0 = Z/S in the quotient manifold M0 = M f the quotient map π : M → M0 induces a Lie algebroid morphism and an exact f sequence over M 0
/ h∂θ i
f(− log Z) e / TM
π∗
/ T M0 (− log Z0 )
/0
(54)
As in the usual Marsden-Weinstein symplectic reduction, the corank 1 subalgef(− log Z) e ⊂ T M (− log Z) is coisotropic with respect to the log symbroid T M plectic form, whose restriction has rank 1 kernel generated by ∂θ . Therefore, the f is basic relative to π, and may be expressed as the pullback pullback of ω to M of a unique log symplectic form ω0 ∈ Ω2 (M0 , log Z0 ), defining the logarithmic symplectic reduction. We state a slight extension of this result for arbitrary free divisors.
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Proposition A.8. Let (M, Z, ω) be a log symplectic manifold, let µ be a Hamiltonian generating a circle action via (52), and assume that (53) is surjective, which implies that µ−1 (0) is smooth, with free divisor Ze = µ−1 (0) ∩ Z. Assume the S 1 action on µ−1 (0) is free and proper, with quotient M0 containing the free divisor e 1 . Then the pullback of ω to µ−1 (0) is basic in the sense of (54), and Z0 = Z/S defines a logarithmic symplectic structure (M0 , Z0 , ω0 ), called the log symplectic quotient.
A.3
Symplectic uncut
Let (M, Z, ω) be a compact orientable log symplectic manifold with corners and with Z of normal crossing type, equipped with an effective T n action. We conf, Z, e ω e ) over the quotient M/T n , struct a log symplectic principal T n bundle (M called the symplectic uncut (Definition A.20), by applying the compressedblowup operation to M along its submaximal orbit type strata, following an idea of Meinrenken [15]. This operation is inverse to the symplectic cut introduced in f along Z e to recover M . Section 4.2, which may be applied to M Compressed blow-up To define the compressed blow-up operation (Definition A.11), we first recall the blow-up operation and the boundary compression construction. A.9 For M a manifold and K a submanifold of codimension 2, we denote the real oriented blow-up of M along K by BlK (M ). We denote the exceptional divisor by E, which is a boundary component of BlK (M ). The blow-down map p : BlK (M ) → M
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restricts to a diffeomorphism on BlK (M )\E and expresses E as an S 1 fibre bundle over K. A.10 We now define a compression operation closely related to the unfolding defined in [3]. Let M be a smooth manifold with corners and let Z be a smooth component of the boundary. Let U ∼ = Z × [0, ) be a tubular neighbourhood of Z, and let V = M \ Z. Then M is the fibered coproduct, or gluing, of U and V along U ∩ V . Consider the open embedding e = Z × [0, 1 2 ), ι:U ∩V ∼ = Z × (0, ) → U 2
(p, r) 7→ p, 21 r2 .
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e and V along U ∩ V is a smooth manifold with corners The fibered coproduct of U f. The identity map on V extends to a smooth homeomorphism q : M → M f M e such that the restriction to Z is a diffeomorphism onto its image Z = q(Z). The f, Z, e q) is called the compression of (M, Z). triple (M
We now compress the blow-up along its exceptional divisor to obtain the following.
30
Definition A.11. Let M be a manifold with corners and let K ⊂ M be a submanifold of codimension 2. Then the compressed blow-up of M along K is the fK , E, e q) of (BlK (M ), E). compression (M If S ⊂ M is a submanifold that intersects K cleanly, the compressed proper transform of S is the submanifold fK . Se = q ◦ p−1 (S \ K) ⊂ M
A.12 We may now apply the compressed blow-up operation to the submaximal strata of an effective T n action. Let M be a 2n-dimensional compact orientable manifold with corners equipped with an effective T n action. By [7, Corollary B.48], the fixed point set of a circle subgroup H ⊂ T n is a disjoint union of closed submanifolds of codimension at least 2. Let H1 , H2 , . . . , Hm ⊂ T n be the circle subgroups such that the fixed point set M Hi contains at least one codimension 2 component (there are finitely many Hi , because M is compact, and therefore has finitely many orbit type strata). Let Ki be the union of the codimension 2 components of M Hi . fK be the compressed blow-up of M along K1 . The compressed proper Let M 1 transform f2 ⊂ M fK K 1 fK , so we may apply the compressed blow-up is a codimension 2 submanifold of M 1 f f operation to MK1 along K2 . Continuing in this way, we obtain the following.
Definition A.13. Let M be a 2n-dimensional compact orientable manifold with f is the corners with an effective T n action. The iterated compressed blow-up M manifold with corners obtained from M by successively applying the compressed blow-up operation to the fixed point sets {K1 , K2 , . . . , Km }. f receives the folding map The iterated compressed blow-up M f q : BlK (M ) → M
where BlK (M ) is the iterated real oriented blow-up of M along {K1 , K2 , . . . , Km }. Lemma A.14. Let M be a 2n-dimensional compact orientable manifold with corners endowed with an effective T n action. The iterated compressed blow-up f in Definition A.13 is a principal T n bundle over the quotient ∆ = M/T n , M and therefore ∆ is a compact orientable manifold with corners. Furthermore, the following diagram commutes BlK (M )
p
/M
(57)
q
f M
/∆
and the blow-down and folding maps p, q are T n -equivariant. In addition, if Z is a T n -invariant normal crossing divisor in M , then the f, and D = Z/T n compressed proper transform Ze is a normal crossing divisor in M is a normal crossing divisor in ∆.
31
Proof. Because M is orientable, T n acts freely on the principal orbit stratum, and every point with non-trivial stabilizer is contained in some Ki . For any point x ∈ Ki1 ∩ Ki2 ∩ . . . ∩ Kik , there exists a T n -invariant neighbourhood Ux that is equivariantly isomorphic to a neighbourhood of (x, 0) ∈ (Ux ∩ Ki1 ∩ Ki2 ∩ . . . ∩ Kik ) × Ck , where Ck is endowed with the standard U (1)k action. The preimage q ◦ p−1 (x) is equivariantly isomorphic to a k-torus T k , and q ◦ p−1 (Ux ) is equivariantly isomorphic to a neighbourhood of q ◦ p−1 (x) ∼ = {(x, 0)} × T k ∈ (Ux ∩ Ki1 ∩ Ki2 ∩ . . . ∩ Kik ) × Rk+ × T k , f is free and proper, where R+ = [0, ∞). Therefore, the induced T n action on M and diagram (57) commutes. f is Since M is orientable, it follows that the iterated compressed blow-up M orientable, and therefore ∆ is orientable. Let Z1 , Z2 , . . . , Zl be components of Z. The intersection
L=
\
Zi
i=1,...,l
is T n -invariant. By induction on l, the T n action on L is effective. The isotropy action of Hi on Tx M descends to the normal space Nx Ki . Up to a linear transformation, Hi acts on Nx Ki ∼ = C by the U (1) action. Because the n T action on L is effective, it follows that the T n -invariant subspace Tx L ⊂ Tx M f and D is a is transverse to Tx Ki . Therefore Ze is a normal crossing divisor in M normal crossing divisor in ∆. Local normal form Given a log symplectic manifold (M, Z, ω) equipped with an effective T n action, we show that intersections among the codimension 2 circle fixed point sets {K1 , K2 , . . . , Km } are log symplectic submanifolds. We also show the existence of local normal forms for neighbourhoods of points on such submanifolds. We begin with a convenient definition of log symplectic submanifold. Definition A.15. Let (M, Z, ω) be a log symplectic manifold. An embedded submanifold i : N ,→ M is called a log symplectic submanifold if ι∗ : T N → T M is transverse to the anchor map T M (− log Z) → T M and ω is nondegenerate on the resulting intersection. If Z is of normal crossing type, then the above transversality condition is equivalent to the condition that N be transverse to all possible intersections among components of Z. This means that ZN = N ∩ Z is a normal crossing divisor in N , and then N is log symplectic when ω is nondegenerate when pulled back to the subbundle T N (− log ZN ) ,→ T M (− log Z).
32
Lemma A.16. Let (M, Z, ω) be a 2n-dimensional compact orientable log symplectic manifold with Z of normal crossing type, equipped with an effective T n action. Let K be a codimension 2 component of the fixed point set M H of a circle subgroup H ⊂ T n . Then K is a log symplectic submanifold. For every point x ∈ K, there exists a T n -invariant neighbourhood Ux such that Ux is isomorphic to a neighbourhood of (x, 0) ∈ M H × C. (58) where C is equipped with the symplectic structure i dz ∧ dz 2 and H acts on C by the standard U (1) action. Proof. Let Z1 , Z2 , . . . , Zl be components of Z. As shown in the proof of Lemma A.14, K is transverse to the intersection \
Zi .
i=1,...,l
This implies K ∩ Z is a normal crossing divisor in K. For every point x ∈ K, the log tangent space Tx (M, − log Z) is a symplectic vector space, and the H-invariant subspace Tx (K, − log(K ∩ Z)) is a symplectic subspace. This shows that K is a log symplectic submanifold. Taking a smaller neighbourhood if necessary, we may assume that Ux is contractible and Ux ∼ = (Ux ∩ K) × D where D ⊂ C is an open disk. We use the Moser method to show that (Ux , ω) is equivariantly symplectomorphic to (Ux ∩K, ωK )×(D, σ) where ωK is the induced log symplectic structure on Ux ∩ K and D ⊂ C is equipped with the symplectic structure σ = 2i dz ∧ dz and the U (1) action. We write ωt := (1 − t)ω + t(ωK ⊕ σ),
0 ≤ t ≤ 1.
Because K t (Zi ∩ Zj ), for small enough Ux , we have M i,j
H 0 (Ux ∩ Zi ∩ Zj ) =
M
H 0 (K ∩ Ux ∩ Zi ∩ Zj ),
i,j
and therefore H 2 (Ux , log Z) = H 2 (K ∩ Ux , log(K ∩ Z)).
33
This implies [ω0 ] = [ωt ] ∈ H 2 (Ux , log(Ux ∩ Z)), and so ω0 − ω1 = tdτ 0 for some τ 0 ∈ Ω1 (Ux , log(Ux ∩ Z)). Choose a chart centered at x such that τ 0 |x =
l X i=1
ci
2n X dxi + ci dxi . xi i=l+1
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We use the right hand side of (59) to define a closed logarithmic 1-form τ0 on Ux . Consequently, ω0 − ω1 = tdτ 0 = td(τ 0 − τ0 ) = tdτ. where τ := τ 0 − τ0 has the property that τ |x = 0. We may average τ over T n , rendering τ T n -equivariant. The usual Moser method yields the desired result. By Lemma A.16, we may deduce that K is a compact orientable log symplectic submanifold of dimension 2n − 2 with an induced effective action by the torus T n /H, so we obtain the following result by induction on dimension. Proposition A.17. Let (M, Z, ω) be a 2n-dimensional compact orientable log symplectic manifold with Z of normal crossing type, equipped with an effective T n action. Let {K1 , . . . , Km } be the submaximal orbit type strata as described in A.12. Then N = Ki1 ∩ Ki2 ∩ . . . ∩ Kik is a log symplectic submanifold with the induced action by the torus T n / Hi1 Hi2 · · · Hik . For a point x ∈ N , there exists a T n -invariant neighbourhood Ux that is equivariantly isomorphic to a neighbourhood of (x, 0) ∈ (N ∩ Up ) × Ck .
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where Ck is equipped with the symplectic structure i (dz1 ∧ dz 1 + dz2 ∧ dz 2 + . . . + dzk ∧ dz k ) 2 and each Hip acts only on the p-th factor of Ck by the standard U (1) action. Example A.18. Let S 1 act on
i C, σ = dz ∧ dz 2
by the standard U (1) action. Then, the blow-up of C along 0 n
Bl0 (C) ∼ = (r, θ) r = |z| ∈ [0, ∞), θ = arg(z) ∈ S 1
34
o
is endowed with the induced presymplectic structure rdr∧dθ, and the compressed blow-up ( ) 2 |z| 1 e0 = ∼ (t, θ) t = C ∈ [0, ∞), θ = arg(z) ∈ S 2 e = dt ∧ dθ. The Hamiltonian is endowed with the induced symplectic structure σ function e 0 → R, t:C (t, θ) 7→ t
e 0 . The symplectic cut of C e 0, σ e with respect generates the induced S 1 action on C to the function t is equivariantly isomorphic to (C, σ).
Symplectic uncut We use Proposition A.17 to show that there is an induced T n -invariant log symf. By Lemma A.14, M f defines a plectic structure on the compressed blow-up M n log symplectic principal T bundle, which we call the symplectic uncut. Proposition A.19. Let (M, Z, ω) be a 2n-dimensional compact orientable log symplectic manifold with corners where Z is of normal crossing type, equipped with an effective T n action. Let ∆ = M/T n and D = Z/T n . f, Z) e in Lemma A.14, as a principal Then the iterated compressed blow-up (M n T bundle over (∆, D), is endowed with a unique T n -invariant log symplectic e such that the pullback of ω e to BlK (M ) coincides with the pullback of structure ω ∗ ∗ e in diagram 57. ω to BlK (M ), i.e. we have p ω = q ω Proof. For a point x ∈ N = Ki1 ∩ Ki2 ∩ . . . ∩ Kik ⊂ M, let Ux ⊂ M be a T n -invariant neighbourhood that is isomorphic to a neighbourhood of (x, 0) ∈ (N ∩ Ux ) × Ck . as in Proposition A.17. In the proof of Lemma A.14, the preimage of q◦p−1 (Ux ) ⊂ f is equivariantly isomorphic to a neighbourhood of M
(x, 0) × T k ⊂ (N ∩ Ux ) × R+ × S 1
k
where R+ = [0, ∞). In these coordinates, the T n -invariant symplectic form e = ωN ⊕ (dt1 ∧ dθ1 + dt2 ∧ dθ2 + . . . + dtk ∧ dθk ) ω e. satisfies p∗ ω = q ∗ ω
Definition A.20. Let (M, Z, ω) be a compact orientable log symplectic manifold with corners with Z of normal crossing type, equipped with an effective T n action. f, Z, e ω e ), as a log symplectic principal T n bundle over (∆, D), is called The triple (M the symplectic uncut of (M, Z, ω).
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A.4
Logarithmic cohomology
A.21 Let (M, Z) be a manifold with real codimension 1 closed hypersurface. Let T M (− log Z) be the log tangent algebroid. Then the residue exact sequence of complexes 0
/ Ωk M
/ Ωk (log Z) Res / Ωk−1 M Z
/0
leads to a long exact sequence with zero connecting homomorphism, which therefore splits into the following short exact sequence in each degree k: 0
/ H k (M )
/ H k (M, log Z)
/ H k−1 (Z)
/0,
recovering the well-known isomorphism of Mazzeo-Melrose [13]: H k (M, log Z) ∼ = H k (M ) ⊕ H k−1 (Z).
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A.22 If Z is the degeneracy locus of a log symplectic structure, then Z has a foliation F by codimension 1 symplectic leaves, and the Poisson Lie algebroid is isomorphic to T M (− log Z, F ), the sheaf of vector fields tangent to Z and to F . A local computation shows that the morphism of complexes dualizing the morphism of algebroids T M (− log Z, F ) → T M (− log Z), namely Ω•M (log Z) → Ω•M (log Z, F ), is a quasi-isomorphism and so the Poisson cohomology coincides with the logarithmic cohomology of the hypersurface Z. A.23 In the case that there are several smooth hypersurfaces Z1 , . . . , Zk which intersect in a normal crossing fashion, for example the boundary of a manifold with corners, the Mazzeo-Melrose theorem easily generalizes to give (for Z = Z1 + . . . + Zk ) H k (M, log Z) ∼ = H k (M ) ⊕
X
H k−1 (Zi ) ⊕
i
X
H k−2 (Zi ∩ Zj ) ⊕ · · · .
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i
