Pre-homogeneous spaces and stratifiedKahler spaces

J. Huebschmann

USTL, UFR de MathematiquesCNRS-UMR 8524

59655 Villeneuve d’Ascq Cedex, FranceJohannes.Huebschmann@math.univ-lille1.fr

Bielefeld July 22, 2007

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Abstract

A representation of a reductive Lie group is a pre-homogeneousspace provided it has a Zariski-open orbit. A stratified Kahlerspace is a stratified symplectic space together with a complex an-alytic structure which is compatible with the stratified symplecticstructure; in particular each stratum is a Kahler manifold in anobvious fashion. The notion of stratified Kahler space establishesan intimate relationship between nilpotent orbits, singular reduc-tion, invariant theory, reductive dual pairs, Jordan triple systems,symmetric domains, and pre-homogeneous spaces. The purposeof the talk is to illustrate the geometry of stratified Kahler spaces.

Examples of stratified Kahler spaces abound. The closure ofa holomorphic nilpotent orbit carries a normal Kahler structure.Symplectic reduction carries a Kahler manifold to a normal strat-ified Kahler space in such a way that the sheaf of germs of po-larized functions coincides with the ordinary sheaf of germs ofholomorphic functions. Projectivization of holomorphic nilpo-tent orbits yields exotic stratified Kahler structures on complexprojective spaces and on certain complex projective varieties in-cluding complex projective quadrics, Scorza varieties and theirsecant varieties. Physical examples are reduced spaces arisingfrom angular momentum.

The appearance of singularities in classical phase spaces is therule rather than the exception. While within canonical quantiza-tion the implementation of singularities is far from being clear thepossible impact of classical phase space singularities on quantumproblems can be explored via quantization on stratified Kahlerspaces.

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Contents

1 Origins and motivation 4

2 Vinberg’s theorem 5

3 Singularities, stratified Kahler spaces 6

4 Lie algebras of hermitian type 8

5 The closure of a holomorphic nilpotentorbit 10

6 Scorza varieties and their secant vari-eties 15

7 Quantization in the presence of singu-larities 18

8 Description of Severi and Scorza vari-eties 20

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1 Origins and motivation

Origin of the notion of stratified Kahler space:Quantization in the presence of singularities

Message today:

Stratified Kahler spaces establish an intimaterelationship between pre-homogeneous spaces,nilpotent orbits, singular reduction, invarianttheory, reductive dual pairs, Jordan triplesystems, symmetric domains.

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2 Vinberg’s theorem

Given a graded Lie algebra

g = g−1 ⊕ g0 ⊕ g1,

g1 acquires a G0-pre-homogeneous structureAttractive mathematics behindPurpose of my talk: Unveil some of thisattractive mathematicsGuiding example,illustration of Vinberg’s theorem:

sp(`, R) = k⊕ p+, k = u(`), p+ ∼= S2C[C`]

g = sp(`, C) ∼= p− ⊕ gl(`, C)⊕ p+

p− ∼= S2C[C`]

p+ ∼= S2C[C`] a pre-homogeneous space, the

GL(`, C)-action on S2C[C`] being given by

x · S = xSxt, x ∈ GL(`, C), S ∈ S2C[C`]

second symmetric power of defining represen-tation of GL(`, C)Lurking behind grading: Jordan triple sys-tem

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3 Singularities, stratified Kahler spaces

Issue of singularities of true classical physicalphase spaces not academic:Singularities rule rather than the exception.Simple mecanical systems and solution spacesof field theories come with singularities.Examples:— ` particles in Rs with total angular momen-tum zero;reduced classical phase space: space of com-plex symmetric (`×`)-matrices of rank at mostequal to min(s, `).— Special case s = 3 our solar system.— ` harmonic oscillators in Rs with total an-gular momentum zero and constant energy:exotic projective varietyDefinition. A stratified Kahler space con-sists of a complex analytic space N , with(i) a stratification, finer than complex analytic(ii) a stratified symplectic structure

(C∞N, {·, ·})6

compatible with complex analytic structure.That is: (C∞N, {·, ·}) is a Poisson alge-bra of continuous functions on N ; on eachstratum, an ordinary symplectic Poisson alge-bra which combines with the complex analyticstructure to a Kahler structure.C∞N an algebra of continuous functions, notnecessarily ordinary smooth functions, referredto as a smooth structure on N .

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4 Lie algebras of hermitian type

(Semisimple) Lie algebra of hermitian type:pair (g, z) real semisimple Lie algebra gwith Cartan decomposition g = k⊕ pand central element z of k , referred to as anH-element , such that:Jz = ad(z)

∣∣p a complex structure on p

Reductive Lie algebra of hermitian type: re-ductive Lie algebra g together with an elementz ∈ g whose constituent z′ in the semisimplepart [g, g] of g an H-element for [g, g]Notation:G Lie group having g = Lie(G)K ⊂ G compact connected subgroup havingk = Lie(K)(g, z) of hermitian type equivalent to G

/K

a (non-compact) hermitian symmetric spacewith complex structure induced by z.complex structure Jz on p necessarily K-in-variantE. Cartan’s infinitesimal classification of ir-

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reducible hermitian symmetric spaces:complete list of simple hermitian Lie algebras

su(p, q) (An, n ≥ 1, n + 1 = p + q)

so(2, 2n− 1) (Bn, n ≥ 2)

sp(n, R) (Cn, n ≥ 2)

so(2, 2n− 2) (Dn,1, n > 2)

so∗(2n) (Dn,2, n > 2)

e6(−14)

e7(−25)

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5 The closure of a holomorphic nilpotent orbit

g = k⊕ p hermitianAdjoint orbit O ⊆ g pseudoholomorphic:projection from g to p, restricted to O, a dif-feomorphism onto its imageA pseudoholomorphic orbit O inherits— a complex structure from the embeddinginto p— viewed as a coadjoint orbit by means of (apositive multiple of) the Killing form, O in-herits a symplectic structure, the Kostant-Kirillov-Souriau-formthe two combine to a (not necessarily positive)Kahler structureChoose positive multiple of the Killing formDefine: pseudoholomorphic orbitO to be holo-morphic:Kahler structure on O positiveGiven a holomorphic nilpotent orbit O,C∞(O) algebra of Whitney smooth functionson topological closure O of O (not Zariski clo-

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sure) resulting from embedding of O into g∗

Lie bracket on g passes to Poisson bracket{·, ·} on C∞(O)Poisson bracket turns O into stratified sym-plectic space

Theorem. Given a holomorphic nilpotentorbit O, the closure O is a union of finitelymany holomorphic nilpotent orbits. More-over, the diffeomorphism from O onto itsimage in p extends to a homeomorphismfrom the closure O onto its image in p,this homeomorphism turns O into a com-plex affine variety, and the complex ana-lytic structure, in turn, combines with thePoisson structure (C∞(O), {·, ·}) to a nor-mal stratified Kahler structure.

Let r be the real rank of g. There are r + 1holomorphic nilpotent orbits O0, . . . ,Or, andthese are linearly ordered in such a way that

{0} = O0 ⊆ O1 ⊆ . . . ⊆ Or = p

Cartan decomposition induces decomposi-

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tiongC = p− ⊕ kC ⊕ p+

of complexification gC of g, p+ and p− be-ing the holomorphic and antiholomorphic con-stituents, respectively, of pC

Thus we are back in situation of Vinberg’stheorem.

Theorem. The projection from Or to pis a homeomorphism onto p, and the G-orbit stratification of Or passes to the KC-orbit stratification of p ∼= p+. Thus, for1 ≤ s ≤ r, restricted to Os, this home-omorphism is a K-equivariant diffeomor-phism from Os onto its image in p+, andthis image is a KC-orbit in p+.

Guiding example

sp(`, R) = u(`)⊕ S2C[C`], r = `,

Os space of complex symmetric (`×`)-matricesof rank s ≤ `;Os space of complex symmetric (`×`)-matricesof rank at most s ≤ `

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Summary: O holomorphic nilpotent orbit— O ⊆ g∗: stratified symplectic structure— O ⊆ p: complex analytic structurecombine to a stratified Kahler structureProjectivization: Given the hermitian Liealgebra g, with Cartan decompositiong = k⊕ p, the ascending sequence

{0} = O0 ⊆ O1 ⊆ . . . ⊆ Or = p

of closures of holomorphic nilpotent orbits de-scends to an ascending sequence

Q1 ⊆ . . . ⊆ Qr = P(p)

of projective varieties where P(p) is the projec-tive space on p and where each Qs (1 ≤ s ≤ r)arises fromOs by projectivization. The strati-fied Kahler structures on the Os descend tostratified Kahler structures on the Qs, andall stratified Kahler structures in sight arenormal.We refer to a stratified Kahler structure ofthis kind on a projective variety as being ex-otic since this kind of structure can not be

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induced from a Fubini-Study structure ona projective space.Example of ` harmonic oscillators in Rs withtotal angular momentum zero and constantenergy:exotic projective variety arising from space ofcomplex symmetric (`×`)-matrices of rank atmost equal to min(s, `) by projectivization

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6 Scorza varieties and their secant varieties

The above ascending sequence of the projec-tivizations of the closures of holomorphic nilpo-tent orbits admits an attractive interpretationin terms of Severi and Scorza varieties.

Theorem. Let k ≥ 2, let X be a k-Scorzavariety, and let PmC be the ambient com-plex projective space. This projective spacecarries an exotic normal stratified Kahlerstructure with the following properties:(1) The closures of the strata constitute anascending sequence

Q1 ⊆ Q2 ⊆ . . . ⊆ Qk+1 = PmCof normal stratified Kahler spaces where theclosed stratum Q1 coincides with the givenScorza variety and where, complex algebrai-cally, for 1 ≤ ρ ≤ k, Qρ is a projectivedeterminantal variety in Qk+1 = PmC; inparticular, in the regular case, Qk is a pro-jective degree k + 1 hypersurface.(2) For 1 ≤ ρ ≤ k, Qρ+1 is the ρ’th secant

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variety Sρ(Q1) of Q1 in Qk+1 = PmC.(3) For 2 ≤ ρ ≤ k + 1, Qρ−1 is the singularlocus of Qρ, in the sense of stratified Kahlerspaces.(4) The exotic stratified Kahler structure onPmC restricts to an ordinary Kahler struc-ture on Q1 inducing, perhaps up to rescal-ing, the standard hermitian symmetric spacestructure.

The regular rank 3 case is somewhat special.This case corresponds to the critical Sev-eri varieties and includes the exceptional caseof the Severi variety arising from the octo-nions .

Addendum. For m = 5, 8, 14, 26, the com-plex projective space PmC carries an exoticnormal stratified Kahler structure with thefollowing properties:(1) The closures of the strata constitute anascending sequence

Q1 ⊆ Q2 ⊆ Q3 = PmC

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of normal stratified Kahler spaces where,complex algebraically, Q1 is a (critical) Sev-eri variety and Q2 a projective cubic hyper-surface, the chordal variety of Q1.(2) The singular locus of Q3, in the sense ofstratified Kahler spaces, is the hypersurfaceQ2, and that of Q2 (still in the sense ofstratified Kahler spaces) is the non-singularvariety Q1; furthermore, Q1 is as well thecomplex algebraic singular locus of Q2.(3) The exotic stratified Kahler structure onPmC restricts to an ordinary Kahler struc-ture on Q1 inducing, perhaps up to rescal-ing, the standard hermitian symmetric spacestructure.

In the case involving the octonions, the term“determinantal variety” refers to Freuden-thal’s determinant.

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7 Quantization in the presence of singularities

“Singular points in a quantum problem are aset of measure zero so cannot possibly be im-portant.”Emmrich-Romer: Wave functionsmay congregate near singular points.Ignoring singularities: inconsistencies(1) Can we unveil a structure on the quantumlevel which has the classical phase space sin-gularities as its shadow?(2) Do the strata carry physical information?Concerning (1):Schrodinger quantization failsDeveloping Kahler quantization on a strat-ified Kahler space I isolated the concept ofa costratified Hilbert spaceExploiting this notion:Answer to (2): for K = SU(2) tunneling prob-abilities among strata [HRS]Beginning of a huge research programStratified Kahler spaces arising from pre-

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homogeneous spaces yield interesting specialcases of quantization in the presence of singu-larities.

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8 Description of Severi and Scorza varieties

Let m ≥ 2. A Severi variety is a non-singular variety X in complex projective m-space PmC having the property that, for somepoint O 6∈ X , the projection from X to Pm−1Cis a closed immersion. Let X be a Severi va-riety and n the dimension of X . The criticalcases are when m = 3

2n + 2. Zak proved thatonly the following four critical cases occur:X = P2C ⊆ P5C (Veronese)X = P2C× P2C ⊆ P8C (Segre)X = G2(C6) = U(6)

/(U(2) × U(4)) ⊆ P14C

(Plucker)X = Ad(e6(−78))

/(SO(10, R) · SO(2, R)) ⊆

P26C.These varieties arise from the projective planesover the four real normed division algebras(reals, complex numbers, quaternions, octo-nions) by complexification.Scorza varieties generalize the critical Sev-eri varieties (Zak).

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Given a non-singular variety X in PmC, for0 ≤ k ≤ m, the k’th secant variety Sk(X)is the projective variety in PmC which arisesas the closure of the union of all k-dimensionalprojective spaces in PmC that contain k+1 in-dependent points of X . Then S0(X) = X andS1(X) is the ordinary secant variety, referredto as well as chordal variety . For k ≥ 2, ak-Scorza variety is defined to be a non-singular complex projective variety of maximaldimension among those varieties whose (k−1)-secant variety is not the entire ambient projec-tive space.The last Severi variety is a very special 2-Scorza variety. Let k ≥ 2. We now list theother k-Scorza varieties.

X = PkC ⊆ Pk(k+3)

2 C (Veronese)

X = PkC× PkC ⊆ Pk(k+2)C (Segre)

X = PkC× Pk+1C ⊆ Pk2+3k+1C (Segre)

X = G2(C2(k+1)) ⊆ Pk(2k+3)C (Plucker)

X = G2(C2k+3)) ⊆ P2k2+5k+2C (Plucker)

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The critical Severi varieties are exactly the2-Scorza varieties that are regular in a sensenot made precise here.When the chordal variety of a non-singularprojective variety Q in P5C is a hypersurface(and not the entire ambient space) the projec-tion from a generic point gives an embeddingin P4C. A classical result of Severi says thatthe Veronese surface is the only surface (notcontained in a hyperplane) in P5C with thisproperty. This is the origin of the terminology“Severi variety”. The terminology “Scorza va-riety” has been introduced by Zak to honorScorza’s pioneering work on linear normal-ization of varieties of small codimension.The above two results exhibit an interestinggeometric feature of Scorza varieties, in par-ticular of the critical Severi varieties, in theworld of singular Poisson-Kahler geome-try.There are exactly four simple regular rank 3hermitian Lie algebras over the reals; these re-

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sult from the euclidean Jordan algebras ofhermitian (3 × 3)-matrices over the four realnormed division algebras by the superstruc-ture construction; and each critical Severivariety arises from the minimal holomor-phic nilpotent orbit in such a Lie algebra.The resulting geometric insight into the fourcritical Severi varieties has been made ex-plicit in the Addendum above. For higherrank, that is, when the rank r (say) is at leastequal to 4, there are exactly three simple reg-ular rank r hermitian Lie algebras over the re-als; these result from the euclidean Jordanalgebras of hermitian (r×r)-matrices over thethree associative real normed division alge-bras by the superstructure construction.Thus we see in particular that the classificationof (regular) Scorza varieties parallels that ofregular pre-homogeneous spaces. The singu-lar Poisson-Kahler point of view exhibitsinteresting additional geometric features.

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