On the conditional measures on the orbits of the complex torus
aa r X i v : . [ m a t h . C V ] F e b On the conditional measures on the orbits of the complextorus
Szymon Myga
Abstract
We explore the structure of invariant measures on compact K¨ahler manifoldswith Hamiltonian torus actions. We derive the formula for conditional measureson the orbits of the complex torus and use it to prove a conditional statementabout uniqueness of solutions to the g -Monge-Amp`ere equation. Let (
X, ω ) be a compact K¨ahler manifold of real dimension 2 n , i.e. X is a complexmanifold and one can find a hermitian metric on it whose fundamental form ω is closed,thus making ( X, ω ) into a symplectic manifold. We assume that there is a real torus T k acting smoothly and effectively on it by symplectomorphisms. This action can beextended to the holomorphic action of complex torus T k C ≃ ( C ∗ ) k . Finally, we assumethat the action is Hamiltonian, so there is a momentum map: an action invariant map m : X → ( R k ) ∗ such that for every element t of t ≃ R k – the Lie algebra of T k , wehave that − d h m ( p ) , t i = ω p ( t , · ) , with t being the vector field generated by t and h m ( p ) , t i the value of the linear form m ( p ) at t .By the classical result of Atiyah [2], Guillemin and Sternberg [11] the image m ( X ) ⊆ R k is a compact convex polytope. Let us denote it by ∆ and mentionthat it is an invariant of the action and cohomology group of ω .Suppose we take an invariant quasiplurisubharmonic function φ and look at ameasure µ := ( ω + i∂ ¯ ∂φ ) n . We normalize ω , so that µ is a probability measure. There is an open and dense subset X ⊆ X that is a principal T k C -fibration over some manifold W . For now we assumethat φ is C and ω + i∂ ¯ ∂φ >
0, hence µ has a positive density. We look at the orbits ofcomplexified action, which coincide with fibers of X . In appropriate coordinates thepotential of the form ω + i∂ ¯ ∂φ is convex along the orbits of the complex torus. Take m := n − k and let ( x, y, w ) be those coordinates, namely ( x, y ) ∈ R k × ( R / Z ) k and w ∈ D ⊆ C m . Let u be the potential that locally satisfies i∂ ¯ ∂u = ω + i∂ ¯ ∂φ and let1 µ ( w ) be the density of µ averaged along the orbits, thus naturally defined on W andpositive from our assumption. Then we prove the following Theorem 1.1.
In the setting sketched above let η w ( x ) dx be the conditional measureof µ along the orbit π − ( w ) of the complex torus in X . It has a density η w ( x ) = ( − m σ m ( m u , w )ˆ µ ( w ) det D x u ( x, w ) , where m u is a momentum map for the symplectic form ω + i∂ ¯ ∂φ and σ m ( p, w ) =det D w u ∗ with p ∈ ∆ , is the Hessian of the (locally defined) Legendre transform of u ,with respect to w variables. Remark.
In the case when k = n , and hence X is a toric variety, the result is trivialas there is only one open orbit, and the conditional measure is simply µ restricted tothat orbit. That determines µ completely as it does not put mass on pluripolar sets,so in particular it does not put mass on the singular orbits. For this reason we assumefrom now on that k < n .The Legendre transform u ∗ in the theorem above is defined locally, but its corre-sponding (1 , w variables is defined globally and in fact for fixed p is thereduced symplectic form coming from Mardsen-Weinstein reduction. If in turn we fix w and vary p , the top form σ m ( p, w ) becomes the Duistermaat-Heckman measure ofthe particular T k C -orbit, namely it is a measure on ∆ that comes from transporting η w ( x ) dx through m u .From that perspective we might consider a more general equation for the potentials φ . Instead of φ solving the Monge-Amp`ere equation, we assume it solves the g -Monge-Amp`ere or transport Monge-Amp`ere equation, namely we take a positive continuousfunction g defined on ∆ and assume that φ solves M A g ( φ ) = g ( m φ )( ω + i∂ ¯ ∂φ ) n = µ. (*)This operator was first considered by Berman and Witt Nystr¨om in [5] in relation toK¨ahler -Ricci flow and optimal transport problem. In this preprint the authors provedthe existence of solutions to the problem and their uniqueness for the measures offinite energy. It was done by adapting the pluripotential methods developed for thevariational approach to the complex Monge-Amp`ere equation in [4].We also notice that some of the assumptions can be weakened though we still wantto use the formula for the conditional measures. For this we only need the positivity ofthe form on X . Thus we might assume that µ is of the form f ω n , with f positive on X but possibly singular on small orbits. Of course there also must be a normalizationassumption, i.e. R X f ω n = R X g ( m ) ω n .If we consider the densities of the conditional measures with respect to transportMonge-Amp`ere operator the theorem above holds mutatis mutandis with the densityof η w ( x ) being ( − m g ( m u ) σ m ( m u ,w )ˆ µ ( w ) det D x u ( x, w ).With the formula for conditional measures in hand we can say something aboutthe uniqueness of the transport Monge-Amp`ere equation.2 heorem 1.2. Suppose we have two solutions to the problem (*) with µ = f ω n and f as above, then if they both produce the same momentum map, they are equal up toan additive constant. The equation (*) comes up in the study of K¨ahler -Ricci solitons on Fano varieties,which are projective varieties that admit an ample line bundle. A K¨ahler -Ricci solitonis a K¨ahler form ω that solves the equationRic( ω ) − ω = L V ω, with Ric( ω ) being the Ricci form and L V the Lie derivative along some holomorphicvector field V . By the results of [5] the existence of such a form corresponds to solutionof (*) for some function g V .The upshot of our result is that we do not rely on the variational methods andinstead connect the potentials of K¨ahler forms with the symplectic geometry of theunderlying manifold. That way our result is valid even for the measures of infiniteenergy. Of course, on the other hand we still have a strong assumption about thepositivity on X .The structure of the paper is following. In the Section 2 we introduce the toricsetting in some more detail, in the short Section 3 we present the results from thetheories of optimal transportation and disintegration od measures that we will uselater on. Finally in Section 4 we present the proofs of the results. Those are ratherstraightforward as the difficulties lie mainly in making sure that all the tools worktogether. The last Section is devoted to proving a result about permuting the minorsof a matrix. In this section we will take a closer look at the symplectic geometry of Hamiltoniantorus actions on K¨ahler manifolds. The results are classical and can be found in themany excellent textbooks (e.g. [3], [16], [10])We assume that the real torus T k ≃ ( S ) k acts on ( X, ω ) as a smooth Lie groupin a holomorphic, isometric and effective way. Specifically, there is a map T k × X ∋ ( t, x ) −→ t · x ∈ X, smooth in both variables that commutes with the group operations of T k . We willnot make a distinction between the elements of the torus and automorphisms of X generated by those elements. We assume that those automorphisms are in fact sym-plectomorphisms, thus t ∗ ω = ω for each t ∈ T k . The holomorphicity of the actionmeans that the complex structure J of ( X, ω ) is preserved, i.e. each dt commutes with J . By ’effective’ we mean that the action understood as a homomorphism of T k intothe isometries of ( X, ω, J ) is injective.The action lifts to the mapping from the Lie algebra of the torus t ( ≃ R k ) to theholomorphic vector fields on X . We denote this by R k ∋ Y → Y . This vector field3an be simply computed at every point from the formula Y x = ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 exp( tY ) · x, and its called the fundamental vector field associated to Y . This map is a monomor-phism of algebras where the multiplication of vector fields on X is the usual Lie bracketof vector fields. Since the Lie algebra of a commutative Lie group has a trivial bracket,that implies that the fundamental vector fields of the torus action must commute too.We also assume that the action is Hamiltonian, which means that there is a map m that maps X to the dual algebra t ∗ in a way that is equivariant and at each x ∈ X satisfies d h m ( x ) , Y i = − ω x ( Y , · )for every Y ∈ R k . Since the group is commutative, equivariance here simply meansthat m is torus invariant. For fixed Y the function h m , Y i : X → R will be denotedby m Y and called the Hamiltonian function of Y . Let us finally state one simpleproperty of the momentum map that we will heavily rely on Fact 2.1.
The action of T k at x is locally free (i.e. the stabilizer subgroup is finite) ifand only if d m x is surjective. By the fundamental results in the theory of Lie group actions, since the action iseffective there is an open dense subset where the action is free. Orbits in this subsetare called the principal orbits, on the other hand, orbits that have a stabilizer group ofpositive dimension are called singular. Moreover, since the target manifold is compact,there are only finitely many types of singular orbits and the holomorphicity impliesthat those orbits form an analytic subset of X . Indeed, each such orbit must be anull set of some real holomorphic vector field. Let us denote by X the principal orbitsubset of X , i.e. the subset of X where the action is free and by X the complement ofsingular orbits. That way X , aside from X can also contain the exceptional orbits,i.e. orbits with finite stabilizer subgroups. Complexification.
Due to classical Bochner-Montgomery [6] result, the holomor-phic action of T k can be extended to the holomorphic action of its complexification,although not by isometries anymore. Moreover, this extension can be made somewhatexplicit. Since the complexification of T k is ( C ∗ ) k we can represent its elements as( e x + iy , . . . , e x k + iy k ) with Lie algebra isomorphic to R k ⊕ i R k . Now its a matter ofcomputation to see that the action of elements from { y = . . . = y k = 0 } will induce avector field of the form J Y for Y in t . Let us denote the imaginary part i R k by H .The complexified vector fields can be computed from the momentum mapping.The fundamental observation is the following: let g be the Riemannian metric of ω and ∇ g f := g ij ∇ j f be the metric gradient. Then ∇ g m Y = J Y . Indeed, let A beany tangent vector, then g ( ∇ g m Y , A ) = d h m , Y i ( A ) = − ω ( Y , A ) = g ( J Y , A ) . eduction. In general, producing the space of orbits of a group acting on a man-ifold by taking the quotient
X/G will be of little value. One can readily see this byconsidering the action of R on the two dimensional real torus T . The line acts bytranslations on R with t · ( x, y ) = ( x + t, y + αt ) and α any irrational number. Thisaction projects to a free action on torus with each orbit being a dense subset, hencethe orbit space will fail to even be Hausdorff. On the other hand there is the QuotientManifold Theorem, which says that the orbit space of a free and proper action is amanifold. Unfortunately, there is no hope for this to hold in the setting describedabove as the action of the torus must have fixed points. It turns out though that inthe case of Hamiltonian actions one can get relatively close to the orbit space beinga manifold, moreover the resulting space will also inherit the symplectic and complexstructures of original manifold. Theorem 2.2 (Mardsen-Weinstein reduction [14]) . Suppose ( M, ω ) is a symplecticmanifold with Hamiltonian action of compact group G and a momentum map m . Sup-pose 0 is a regular value of m and G acts freely on m − (0) then:1. M red := m − (0) /G is a manifold called the reduced space,2. π : m − (0) → M red is a principal G -fibration,3. M red inherits the symplectic structure of M , namely there is a symplectic struc-ture σ on M red such that i ∗ ω = π ∗ σ, where i : m − (0) → M is the inclusion map. Remark.
In the case of commutative actions there is nothing special about the nullvalue and the theorem will hold for any regular value of the momentum map.Let us pick a level set Z := m − ( ξ ) of some regular value ξ . It must lie inside X , but not necessarily in X , thus the reduction theorem is not directly applicable.However, since by invariance the action of T k on Z is almost free, Z/ T k will have astrucutre of an orbifold. Indeed, it is the canonical way of producing effective orb-ifolds [1]. Let us denote this orbifold by W . Moreover, if we perform the quotientoperation on Z ∩ X , we notice that W has a structure of a symplectic orbifold, withthe fundamental form defined outside its orbifold singularities. One can also check bycomputation that W will also inherit the complex structure of X . Thus, we are dealingwith a K¨ahler orbifold.Instead of considering the orbits of T k one might also look at the action of thecomplex torus T k C . The point x ∈ X is called semistable iff the closure of T k C -orbitthrough x intersects the level set m − (0). The point is called stable if the intersectionhappens at the point where d m is of full rank. The sets of stable and semistable pointof X will be denoted by X s and X ss respectively. Note that in the case of the torus,the choice of specific value is arbitrary and we may replace it by any regular value.Moreover, for any choice of regular value the notions of stable and semistable sets willcoincide.Now we may look at the quotient X ss / T k C . It turns out that X ss / T k C ≃ W , thusoutside of an analytic subset X can be thought of as an orbifold principal fiber bundle5ver W , in the sense that it becomes a bona fide principal bundle outside of the singularpoints of W .The principal fiber bundle structure ensures that there are local trivialisations of X . Let ( z, w ) = ( z , . . . , z k , w , . . . , w m ) be local holomorphic coordinates on X ( m + k = n ) coming from such a bundle trivialisation. That means that T k C acts on z ’s by ( t + is ) · z j → z j e t j + is j . In particular, changing the coordinates by z j = e x j + iy j we have the complex torus C k / π Z k acting on ( x, y )’s by translations.Let φ be a local potential for the form ω on such a coordinate patch, then ω = i k X i,j =1 ∂ φ∂z i ∂ ¯ z j dz i ∧ d ¯ z j + i = k,j = m X i,j =1 ∂ φ∂z i ∂ ¯ w j dz i ∧ d ¯ w j + i = m,j = k X i,j =1 ∂ φ∂w i ∂ ¯ z j dw i ∧ d ¯ z j + m X i,j =1 ∂ φ∂w i ∂ ¯ w j dw i ∧ d ¯ w j . Changing z ’s to ( x, y )’s and keeping in mind that φ is invariant in y the following termsin ω change i φ z i ¯ z j dz i ∧ d ¯ z j = i φ x i x j ( dx i ∧ dx j + idy i ∧ dx j − idx i ∧ dy j + dy i ∧ dy j ) i φ z i ¯ z i dz i ∧ d ¯ z i = 14 φ x i x i dx i ∧ dy i i φ z i ¯ w j dz i ∧ d ¯ w j = i φ x i ¯ w j ( dx i ∧ d ¯ w j + idy i ∧ d ¯ w j ) i φ w i ¯ z j dw i ∧ d ¯ z j = i φ w i x j ( idy j ∧ dw i − dx j ∧ dw i ) . The terms with derivatives in w ’s only will stay the same and the mixed terms dx i ∧ dx j and dy i ∧ dy j will cancel with the terms coming from dz j ∧ d ¯ z i , thus leaving14 φ x i x j ( dx i ∧ dy j + dx j ∧ dy i ) . In those coordinates, the momentum map m has the form ∇ x φ , i.e. the gradient of φ with respect to x variables only. Indeed, in that case, the real torus acts linearly y → y + t , so the fundamental vector fields are of the form a i ∂∂y i for some constants a i . It is enough to check the momentum map condition on some basis of R k , it mightas well be the canonical basis. Thus one needs to prove the following: − dφ x i = ω (cid:18) ∂∂y i , · (cid:19) , which follows easily form the previous formulas.Suppose we take another invariant symplectic form ˜ ω in the cohomology class [ ω ].The ∂ ¯ ∂ -lemma implies that there exists a global quasiplurisubharmonic function ψ such that ˜ ω = ω + i∂ ¯ ∂ψ [17, chapter 15]. We stay in ( x, y, w ) coordinates and look atthe properties of this Hamiltonian action. The moment map will again be expressed6ocally as ∇ x ( φ + ψ ). Moreover, there exists an equivariant symplectomorphism be-tween the two symplectic forms, let us call it s – this is a simple consequence of theMoser’s stability theorem [16] adapted to the equivairnat setting. It is now a matterof calculation to show that in that case the momentum maps are also related by s ,namely that ∇ x ( φ + ψ ) = ( ∇ x φ ) ◦ s . That implies that the level sets are equivariantlysymplectomorphic, hence they are isomorphic as principal fiber bundles, which in turnimplies that W depends only on the cohomology class. Equivariant cohomology.
We would like to connect the cohomology of W to thatof X . To this end we need to use the notion of equivariant cohomology. We will notintroduce this set of ideas in detail as we only need a handful of results, instead wewould like to recommend some references ([12], [7],[10, Appendix C]). The only factsfrom this theory we would like to use will be contained in this paragraph. The inclusionmap i : Z ξ → X , with Z ξ being as before a level set of m for some regular value ξ ,induces the equiviariant cohomology mapping i ∗ : H T k ( X ) −→ H T k ( Z ξ ) . It turns out that H T k ( Z ξ ) = H ( W ), with standard, say de Rham cohomology of W .In this setting the de Rham cohomology of W is nothing more than the cohomologyof T k -invariant forms on Z ξ , which in fact is isomorphic to the real cohomology of W as a topological space [1]. We are only really interested in second cohomology groupsand in this case H T k ( X ) has an interpretation that shall look familiar – it consists ofpairs ( ω, Φ) of closed invariant two-forms and equivariant smooth functions from X to R k such that for each Y ∈ R k ω ( Y , · ) = − d Φ Y . In that case, i ∗ called the Kirwan map, sends an invariant symplectic form ω on X toits reduction on the level set of appropriate momentum mapping ω → σ ( ξ ) . It is not hard to see, that it will be exactly the form given by the Mardsen-Wiensteinreduction. As any two invariant symplectic forms in the same de Rham cohomol-ogy class are also equivariantly cohomologous, we conclude that the reduction is welldefined on the level of cohomology classes.
Compact K¨ahler manifolds with torus action.
Let us gather everything to-gether now to describe our setting. First let us state the refinement of the convexitytheorem proved by Atiyah in [2].
Theorem 2.3.
Suppose there is a Hamiltonian action of a torus on a compact K¨ahlermanifold X . Let Y be an orbit of T k C through some point. If Z j are critical points of m that intersect Y let c j = m ( Z j ) , then1. m ( Y ) is the convex polytope ∆ ′ with vertices c j ,2. for each open face ζ of ∆ ′ , the preimage m − ( ζ ) ∩ Y is a single T C orbit, . m induces a homeomorphism of Y / T k onto ∆ ′ . Remark.
Let us note here that if dim X = 2 k then ∆ ′ might happen to be of thesame dimension as the full polytope ∆ = m ( X ) and yet be a proper subset of it.Thus putting everything together we arrive at the following picture. The momentpolytope divides into open convex chambers of regular values. The preimages of thosechambers are ( C ∗ ) k -principal fiber bundles over K¨ahler orbifolds W that possibly differchamber to chamber.Suppose we would like to use the information that is contained on each orbit. Letus restrict ourselves to a single chamber ∆ ′ in ∆. We can use the momentum map todefine the map m + on X – a bundle map between ( C ∗ ) k -bundle over W and ∆ ′ -bundleover the same base. We simply write the map down locally m + : ( C ∗ ) k × U ∋ ( z, w ) → ( m ( z, w ) , w ) ∈ ∆ × U, for any local trivialistion over U ⊆ W and notice that it is well defined by the fact that m is a global map. What is worth noting here is that m + ’untangles’ the fibration X into simple cartesian product. We say that the function T : R n → R n transports probability measure µ to probabilitymeasure ν if for any Borel set A the following equality holds ν [ A ] = µ [ T − ( A )] . Alternatively we say that T pushes µ forward to ν and denote the push-forward measureby T µ .In general there will be a lot of such maps, so it is natural to put some optimalityconstraints on them. The best understood constraint and in some cases the natural oneis minimizing the quadratic cost, i.e. the transport map should minimize the followingfunctional Z R d | x − T ( x ) | dµ. In general there might not be a solution and if it exists it might not be unique, someregularity assumptions for the measures must be added. For example, one can assumethat the measures have finite second moments and µ is absolutely continuous. In thatcase the solution exists and has a form of T = ∇ φ for some convex function φ . Forthorough discussion of this problem, the reader might consult [19].Supposing that a solution exists, by the transport condition we get Z χ A dν = Z A dν = Z ( ∇ φ ) − ( A ) dµ = Z χ A ◦ ∇ φ dµ. f ∈ C b ( R n ) Z f dν = Z f ◦ ∇ φ dµ. (1)Here C b ( R n ) denotes the set of continuous and bounded functions on R n .Suppose now that dν = g ( x ) dx for some density g ( x ) and φ is a C function. Bychange of variables formula we get that Z f ( ∇ φ ( x )) dµ = Z f ( ∇ φ ( x )) g ( ∇ φ ( x )) det D φ dx and that provides one with a notion of solution to the transported Monge-Amp`ereequation M A R g ( φ ) := g ( ∇ φ ( x )) det D φ = µ as long as the optimal transport map exists. There is a rich theory of regularity resultsfor this equation. When the solution φ becomes singular, its gradient ∇ φ turns intothe subgradient ∂φ defined by p ∈ ∂φ ( x ) iff ∀ z, φ ( z ) ≥ φ ( x ) + h p, z − x i . As we mentioned, for any two probability measures the optimal transport solu-tion might not exist. However, under a mild regularity assumption it is still possibleto transport one to another through a subgradient of convex function, so that thecondition (1) is still satisfied. This is the content of the following important theorem.
Theorem 3.1 (McCann [15]) . Let µ, ν be probability measures on R n and suppose that µ vanishes on Borel subsets of R n of Hausdorff dimension at most n − . Then thereexists a convex function ψ on R n whose subgradient ∂ψ pushes µ forward to ν . ∂ψ isuniquely determined µ -almost everywhere. Of course the assumption on the null sets of µ can not be abandoned. For exampleif µ = δ x and ν is not a point measure, then if A is such a set that 0 < ν [ A ] < φ , ν [ A ] = µ [( ∂φ ) − ( A )] since the latter must alwaysbe either 0 or 1. Suppose Z is a compact metric space equipped with a Borel probability measure µ .Let P be a partition of Z into measurable sets, it is then straightforward to define ameasure space structure on P . Namely if π : Z → P is a projection sending a pointto the (unique) partition set containing it, then we define a probability measure b µ , bysimply averaging µ over the preimages of π b µ ( Q ) = Z π − ( Q ) dµ. Definition.
Disintegration of µ with respect to P or a system of conditional measuresof µ with respect to P is a family of probability measures η p for p ∈ P such that9. η p ( p ) = 1 for b µ -a.e. p ∈ P ,2. for every continuous function f : Z → R the function P ∋ p → Z p f dη p ismeasurable and Z f dµ = Z (cid:18)Z p f dη p (cid:19) d b µ ( p ).The question for which spaces and their partitions the systems of conditional mea-sures exists arises immediately. For that we need one more definition. Definition.
A partition P of Z is called measurable if there exists a countable familyof { E i } ⊆ Z and a subset W ⊆ Z of full measure such that for each q ∈ P q ∩ W = E ∗ ∩ E ∗ ∩ . . . ∩ W, where E ∗ j is either E j or Z \ E j .Equipped with that notion we can state the theorem of Rokhlin [18] Theorem 3.2. If P is a measurable partition of ( Z, µ ) then µ disintegrates with respectto P . Moreover the conditional measures are unique b µ -a.e. The induced form on the level orbifolds.
Since the results are local let us fixsome open convex chamber ∆ and some local trivialisation of m − (∆) in the ( x, y, w )variables mentioned before. Let φ be the local potential of the K¨ahler from, as wementioned it is convex in x . It will be worthwhile to study the relationship betweenthe potential and its Legendre transform defined by φ ∗ ( p, w ) = sup x ∈ R k { x · p − φ ( x, w ) } . This is not a function on W × ∆ since it is not globally defined. On a local atlas of Wφ ∗ is convex in p ∈ ∆ with fixed w ∈ W and plurisuperharmonic in w for fixed p [13].If φ ∗ happens to be differentiable in p its derivative is well defined globally and as suchis an inverse of m + between (cid:0) m − (∆) ∩ X (cid:1) / T k and ∆ × W .In fact on each orbit O of T k C in X the momentum map m will be a diffeomorphismof O/ T k to ∆ . Indeed, by the definition of X , (2.1) and the constant rank theoremwe get that m | O is a local diffemorphism. That it is a global diffemorphism followsfrom the fact that T k C act transitively on O and the function m Y is strictly increasingalong the flow of J Y . Note that by the properties of the Legendre transform [19,2.1.3] the inverse of m | O will be exactly ∇ p φ ∗ ( · , w ), with w the parameter of the orbit O in W . Hence, from this and the Implicit Function Theorem we imply the followingidentities ∇ x φ ( ∇ p φ ∗ ( c, w ) , w ) = c p φ ∗ w i = − ( D x φ ) − ∇ x φ w i . Using those we can compute the local expression for the reduced form with little effort,namely
Proposition 4.1.
For any regular value q the reduced symplectic form can be locallyexpressed by σ φ ( q ) = − i X φ ∗ w i ¯ w j ( q, w ) dw i ∧ d ¯ w j . Proof.
Indeed, on the one hand φ ∗ w ( p, w ) = − φ w ( φ ∗ p ( p, w ) , w ) . It holds since for any convex function u , u ( x )+ u ∗ ( p ) = x · p iff p ∈ ∂u ( x ) iff x ∈ ∂u ∗ ( p ).Taking another derivative we get φ ∗ w i ¯ w j = − φ w i ¯ w j − ∇ x φ ¯ w j · ∇ p φ ∗ w i = − φ w i ¯ w j + ∇ x φ ¯ w j · ( D x φ ) − ∇ x φ w i . On the other hand we can parametrize the level set by { x = φ ∗ ( q, w ) } , with q beingconstant, that allows us in turn to directly compute the form i ∗ ω using the localexpressions for ω form Section 2. Finally, after noticing that all summands involving dy disappear we get the desired formula.It is a good place to note here that the Kirwan map mentioned above reduces inthis coordinates to taking the Legendre transform. Indeed, suppose that ω has a localpotential f . Then the local potential on ∆ × W will be the Legendre transform f ∗ asfollows from the above computations. Moreover, if u is a global quasiplurisubharmonicfunction then there exists ˜ u such that locally f + u −→ ( f + u ) ∗ = f ∗ + ˜ u will be the potential for the form σ u reduced to W from ω + i∂ ¯ ∂u at some fixed regularvalue.Let us now try to derive the formula for det φ ∗ w ¯ w , which at any fixed q will be thelocal density of the symplectic volume of the quotient orbifold. Theorem 4.2.
In the setting as above, the following relationship between volume den-sities holds det D φ ( x, w ) = ( − m det D w φ ∗ ( ∇ x φ ( x, w ) , w ) det D x φ ( x, w ) , with D x and D w understood as restricting the derivation to toric and transveral vari-ables respectively.Proof. First let us notice that det D x φ ( x, w ) will be nonzero by the discussion at thebeginning of this section and hence will always be invertible. Let us now prove theformula. Since we don’t see any clever way to arrive at it we will just calculate. Wealso would like to recollect a well known fact that the inverse of an invertible matrix A can be represented by formula A − = C T / det( A ) where C is the cofactor matrix of A ,11amely a matrix whose ( i, j )-th entry is the determinant of A with i -th row and j -thcolumn removed multiplied by ( − i + j . Since we are dealing with symmetric matrices,we can forget the transpose in the formula.Let us denote the Hessian matrix of x variables as H , by D its determinant and by D αβ the minor of H with row α and column β removed. Similarly we define D α ,...,α k β ,...,β k .The minor of the matrix with all rows and columns removed is defined to be 1. Tomake the notation a little more readable we will denote real derivatives by Greek lettersand complex ones by Latin letters.Let us look the determinantdet( σ ) = X t ∈ S sgn ( t ) Y i φ ∗ i, ¯ t ( i ) = X t ∈ S sgn ( t ) Y i ( − φ i,t ( i ) + ∇ x φ i · H − · ∇ x φ ¯ t ( i ) )Now we expand each product and look at the number of terms with H − in each factor.Let us denote this number by l . If l = 0 then we get a summand( − m det( φ w ¯ w ) . If l = 1 we get the following summands( − m D sgn( t ) X i,α,β ( − α + β − φ i,α D αβ φ β, ¯ t ( i ) Y j = i φ j,t ( j ) and save for division by D each of those appears in ( − m det( D φ ) with the exactsign sgn( t )( − α + β − since it’s one addtional transposition from φ i, ¯ t ( i ) φ β,α .Now suppose that l > L a multi-index of length l and by L c itscompletion of length m − l . For each pair of those we get the following summands( − m Y k ∈ L X α,β ( − α + β − l φ k,α D αβ φ β, ¯ t ( k ) Y i ∈ L c φ i,t ( i ) for some fixed permutation t . Comparing those between permutations we notice thatproducts in which α or β appear at least twice cancel each other. Indeed, if determi-nants D α · appear next to e.g. φ j,x α and φ k,x α , then they will appear with opposite signin summands with t ( j ) and t ( k ) reversed. For that reason, if m > k , then the productsfor l > k will cancel out since some α and β will have repeat itself in each product.Armed with that observation we fix some w indices K and w indices K with fixedorder and factor out the product Q i ∈ K φ i,t ( i ) ( t being the bijection from j -th elementof K to j -th element of K ). The remaining derivatives of φ will range over all thepermutations of K c . Thus in front of Q i ∈ K φ i,t ( i ) there will be a following multiplier( − m X u ∈ S ( K c ) sgn( t + u ) X α l ,β l ( − | α | + | β |− l X r ∈ S ( α L ) sgn( r ) Y i ∈ K c φ i,r ( α i ) φ β i u ( i ) ! D − l X s ∈ S k sgn( s ) l Y i =1 D α i β s ( i ) . t + s ) we understand the sign ofpermutation that comes from concatenating t and u , that is the permutation thatapplies t to indices from K and u to indices from K c . The symbols α l and β l denotemultiindices of length l , with | α | = α + . . . + α l .Finally, the Proposition 5.1 proven at the end of the paper implies that this mul-tiplier is in fact equal to( − m D X u ∈ S ( K c ) sgn( t + u ) X α l ,β l ( − | α | + | β |− l X r ∈ S ( α L ) sgn( r ) Y i ∈ K c φ i,r ( α i ) φ β i u ( i ) !(cid:16) D α ,...,α k β ,...,β k (cid:17) . Each of those terms will appear exactly once in ( − m det( φ w ¯ w ) /D . Indeed, applyingthe Laplace expansion to D from the highest index in α l downward we will eventu-ally arrive at the term ( − | α | + | β | Q φ β i ,α i D α ,...,α k β ,...,β k . It will be multiplied by the termsgn( s ) Q φ i,s ( i ) with s = t + u , understood as above. Now we switch i with β i for each i ∈ K c and that will result with additional ( − l = ( − l factor, and finally permute α l to get additional multiplier sgn( r ).Finally, we couclude thatdet( σ ( p, w )) = ( − m det D φ ( ∇ p φ ∗ ( p, w ) , w )det D x φ ( ∇ p φ ∗ ( p, w ) , w )and the claim follows from the fact that p = ∇ x φ ( x, w ) for some x .Armed with the above formula we can prove the first theorem. Let us take ameasure µ and a solution u to g -Monge-Amp`ere problem. Since the complement of X is a pluripolar set, we can safely restrict our considerations to X . It is a set offull measure that is also a principal fibration over a compact orbifold (which amongother things is second countable and locally compact), so the Rokhlin’s theorem clearlyapplies in this setting. Let measures η w ( x ) dx be the conditional measures on orbitswith the average ˆ µ . The conditionals are unique ˆ µ -a.e. Moreover, since the base of thefiber bundle W is in a sense a space of fibers, the average measure can be naturallyunderstood as a measure on W .The formula for the determinant of the reduced form allows us to define the con-ditional measures immediately as η w ( x ) = ( − m g ( m u ) σ m ( m u , w )ˆ µ ( w ) det D x u ( x, w ) . With ˆ µ ( w ) being simply the local density of ˆ µ and serving as a normalizing factor.To prove the conditional statement about the uniqueness of solutions to the trans-port Monge-Amp`ere equation let us first mention a result that will be of use in theproof Theorem 4.3 ([9]) . Let ( W, σ ) be a compact K¨ahler orbifold, then for any smooth f the equation ( σ + i∂ ¯ ∂φ ) n = e f σ n always has a unique solution, provided both measures have the same total mass. g -Monge-Amp`ere problemwe can ask for a family of solutions, one for each orbit that satisfies the formula provedabove. That is the content of the Proposition below. Proposition 4.4.
Suppose u and v are two solutions to the g -Monge-Amp`ere problem.Then then following statements are equivalent:1. The conditional measures have ˆ µ -a.e. unique solutions orbit-wise, i.e. on eachorbit form the set of ˆ µ -full measure of W there is only one (up to additive con-stant) convex function F that satisfies η w ( x ) = ( − m g ( ∇ x F ) σ mu ( ∇ x F, w )ˆ µ ( w ) det D x F ( x ) . σ u = σ v , ˆ µ -a.e..Moreover, any of the above statement implies global uniqueness of solutions.Proof. Suppose the second assertion holds. Then at ˆ µ -almost every orbit the potentialrestricted to this orbit solves the equation1ˆ µ ( w ) g ( ∇ u ) σ m ( ∇ u, w ) det D x u = η w ( x ) , with w serving as an orbit parameter. By McCann’s result the solution to this problemis unique up to an additive constant, thus the difference between u and v can onlydepend on w .Suppose the first assertion holds. Thus the moment map restricted to the orbitserves as an optimal transport map between η w and gσ mu (disregarding the normal-ization). Since those maps are unique, that implies that the target measures must beequal, which in turn implies that σ mu = σ mv , ˆ µ -a.e..Suppose there is a set A ⊆ W , such that ˆ µ [ A ] = 0 and on this set σ mu ( p, A ) = σ mv ( p, A ) for some p . If the p ’s for which this holds form a set of non-zero measurethen taking the average over ∆ one gets that ˆ µ [ A ] = 0, thus for a.e. p we must havethat σ mu = σ mv a.e.. For those p ’s we can solve the Monge-Amp`ere equation on theorbifold W . Indeed, recall that going from the manifold to the reduced space inducesa well defined map in cohomology, thus the potentials corresponding to u and v willbe the potentials in the same cohomology class. Let us denote them by u ∗ and v ∗ .By the solution of Monge-Amp`ere equation on orbifolds with right-hand σ mv we getthat σ u = σ v and u ∗ − v ∗ = c . Moreover, W × ∆ being compact implies that u ∗ − v ∗ differ only by a function of p , say f ( p ). Together with the first statement it implies f ( p ) = C .The main result follows easily as corollary from the above theorem Corollary 4.5.
If two solutions provide the same momentum map they must differ bya constant. roof. The proof is trivial - the measure is determined by the conditionals and theaverage, thus both functions must satisfy the first point of the above proposition withthe same σ m , since the optimal transport maps are unique. The only thing left to prove is the following
Proposition 5.1.
Suppose we are given a n × n matrix M . Let us denote the deter-minant of the matrix by D , and by D αβ the determinant of the matrix with α -th rowand β -th column removed. Then for every k ∈ { , . . . , n } and for every multi-index ( α , . . . , α k ) with α < . . . < α k and ( β , . . . , β k ) with β < . . . < β k the followingequality holds X s ∈ S k sgn ( s ) k Y i =1 D α s ( i ) β i = D k − D α ,...,α k β ,...,β k Remark.
The formula goes back probably to Muir. In [8] one can see the proof ofsort of ‘dual‘ identity, where the rows are added instead of removed. As before, wedefine the determinant of the matrix with all rows and columns removed as 1.
Proof.
The proof goes by induction on n and k . For n = 2 and k = 2 the formula issimply the definition of 2 × n = k the formula is just thedefinition of determinant of the cofactor matrix of M , which is indeed D k − .Before we start the proof there is one more notational remark: we will often usesummation over some symmetric group of incomplete set of indices. For example,suppose the lower set of indices misses some j and upper misses i . By permutation ofsuch set we understand renumbering both sets and then permuting. With that out ofthe way we are ready to prove.Assume now that the formula holds for some k − k up to n −
1, we wouldlike to prove the formula for n and k .First, let us deal with the left-hand side. We may assume that for α there is atleast one β i such that D α β i = 0. Indeed, if not the proof becomes trivial since thenthe left hand-side vanishes and elementary proof shows that the right-hand side mustvanish too. Thus we may assume that there is some i such that D α β i = 0. Since itdoes not effect the proof’s mechanics and makes sign computation little less tiresomewe will assume that i = 1. Then we have that X s ∈ S k sgn( s ) k Y i =1 D α s ( i ) β i = X i ( − i − D α i β X t ∈ S k − sgn( t ) Y j> D α t ( j ) β j = X i ( − i − D α i β (cid:16) D k − D α \ α i β ,...,k (cid:17) , where the first equality comes from factoring out the number of inversions involving i and the second comes from the induction hypothesis on k −
1. Now assume that there15s another j such that D α β j = 0. If there is not, then again the proof simplifies sincewe are left with the following claim D α β (cid:16) D k − D α ,...,α k β ,...,β k (cid:17) = D k − D α ,...,α k β ,...,β k and by the assumption on the nonvanishing of D α β and induction assumptions D k − D α ,...,α k β ,...,β k = D k − P t ∈ S k − sgn( t ) Q j D α ,α t ( j ) β ,β j ( D α β ) k − = D P t ∈ S k − sgn( t ) Q j DD α ,α t ( j ) β ,β j ( D α β ) k − = D P t ∈ S k − sgn( t ) Q j D α β D α t ( j ) β j − D α β j D α t ( j ) β ( D α β ) k − = DD α β X t ∈ S k − sgn( t ) Y j D α t ( j ) β j = D α β (cid:16) D k − D α ,...,α k β ,...,β k (cid:17) . Thus we might suppose that there is an index j > D α β j = 0. For the samereason as before we will take j = k . Finally, the left hand-side equals D α β (cid:16) D k − D α ,...,α k β ,...,β k (cid:17) + X i> ( − i − D α i β D k − P t ∈ S k − sgn( t ) Q j D α ,α t ( j ) β k ,β j ( D α β k ) k − ! . With the right-hand side we use the same tricks D k − D α ,...,α k β ,...,β k = D k − P t ∈ S k − sgn( t ) Q j D α ,α t ( j ) β k ,β j ( D α β k ) k − . In each product above there is an element with second pair α t (1) β , we factor it outtogether with ( − t ( i ) , which is the number of inversions for this element, since α isnot counted. As before we multiply this element by one of D ’s and use hypothesis for n = 2 to get D k − X i ( − i (cid:16) D α β D α i β k − D α i β D α β k (cid:17) P t ∈ S k − sgn( t ) Q j D α ,α t ( j ) β k ,β j ( D α β k ) k − . We notice that the terms with D α i β D α β k are exactly those from the left-hand side sumexcept the first one, thus we are left with D k − X i ( − i D α β D α i β k P t ∈ S k − sgn( t ) Q j D α ,α t ( j ) β k ,β j ( D α β k ) k − = D α β X i ( − i D α i β k P t ∈ S k − sgn( t ) Q j DD α ,α t ( j ) β k ,β j ( D α β k ) k − ! = D α β X i ( − i D α i β k P t ∈ S k − sgn( t ) Q j (cid:16) D α β j D α t ( j ) β k − D α β k D α t ( j ) β j (cid:17) ( D α β k ) k − . D α β k ) k − . Indeed, suppose we have a product with α i removed from possible permu-tations and a factor D α β j D α t ( j ) β k in this product, then there will be the same productwith t ( j ) and i reversed. We can suppose without losing any generality that t ( j ) > i .The signs of those two permutations will differ by t ( j ) − i − i to t ( j ), thusboth products will cancel and we are left with D α β X i ( − i + k − D α i β k X t ∈ S k − sgn( t ) Y j D α t ( j ) β j . But then obviously ( − k + i − = ( − ( k − − ( i − and by induction hypothesis on k weend up with D α β (cid:16) D k − D α ,...,α k β ,...,β k (cid:17) . To finish the proof we must prove the formula for k = 2. Assume then that it holdsfor some n −
1. Let α < γ and β < δ , then D D α,γβ,δ = X i ( − α + i m α,i D αi D α,γβ,δ = ( − α + β m α,β D αβ D α,γβ,δ + X i<β ( − α + i m α,i X j<α ( − β + j +1 m j,β D α,jβ,i + X j>α ( − β + j m j,β D α,jβ,i D α,γβ,δ + X i>β ( − α + i m α,i X j<α ( − β + j m j,β D α,jβ,i + X j>α ( − β + j +1 m j,β D α,jβ,i D α,γβ,δ just by using the Laplace expansion twice. Now by the induction hypothesis we have D α,jβ,i D α,γβ,δ = ( − sgn( γ − j )sgn( δ − i ) D αβ D α,γ,jβ,δ,i + D α,γβ,i D α,jβ,δ and we plug that into the sum above.On the other hand we have D αβ D γδ = D αβ X i<δ ( − α + i m α,i D α,γδ,i + X i>δ ( − α + i +1 m α,i D α,γδ,i ! =( − α + β m α,β D αβ D α,γβ,δ + D αβ X i<β ( − α + i m α,i X j<α ( − β + j +1 m j,β D α,γ,jδ,β,i + X α
Acknowledgement
The author would like to thank S lawomir Dinew for his encouragement. The authorwas supported by Polish National Science Centre grant 2018/29/N/ST1/02817.
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E-mail address: [email protected]@im.uj.edu.pl