ANALYTIC GEOM & CALC
ANALYTIC GEOM & CALC MATH 171
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EXTERIOR DIFFERENTIAL SYSTEMS LIE ALGEBRA COHOMOLOGY AND THE RIGIDITY OF HOMOGENOUS VARIETIES 32 WFWPquot ugtoa U H 52 53 54 6 61 62 63 JM LANDSBERG ABSTRACT These are expository notes from the 2008 Srni Winter School They have two purposes I to give a quick introduction to exterior diHerential systems EDS which is a collection of techniques for determining local existence to systems of partial diHerential equations7 and 2 to give an exposition of recent work joint with C Robles on the study of the Fubini Gri 39iths Harris rigidity of rational homogeneous varieties which also involves an advance in the EDS technology CONTENTS Introduction Overview Examples of rational homogeneous varieties Notational conventions Acknowledgements Related projects Representation theory and computational complexity Sphericality and tangential varieties Vogelia Cartan Killing classi cation via projective geometry Projective differential geometry and results The Gauss map and the projective second fundamental form Second order rigidity History of projective rigidity questions Rigidity and exibility of adjoint varieties From PDE to EDS The Cartan algorithm to determine local existence of integral manifolds to an DS Linear Pfaf an systems Prolongations and the Cartan Kahler theorem Flowchart and exercises For fans of bases Moving frames for submanifolds of projective space Adapted frame bundles Fubini Forms Second order Fubini systems Date September 247 2008 O Doooo mcncncncncnybwww H 2 JM LANDSBERG 64 An easier path to rigidity 17 7 Osculating gradings and root gradings 17 71 The osculating ltration 18 72 The root grading 18 73 Examples of tangent spaces and osculating ltrations of homogeneous varieties 19 8 Lie algebra cohomology and Kostant s theory 9 From the Fubini EDS to Filtered EDS 22 91 Problem 1 Osculating vs root gradings 22 92 Problem 2 Even the systems de ned by the root grading do not lead to Lie algebra cohomolo y 93 The Fix for problem 2 Filtered EDS 24 10 Open questions and problems 26 References 26 1 INTRODUCTION Let G be a complex semi simple Lie or algebraic group and let V VA be an irreducible G module The homogeneous variety GP Gv C lP V is the orbit of a highest weight line For example let W be a complex vector space V AkW and let G SLW then GP Ck W C lP AkW is the Grassmarmian of k planes through the origin in W in its Plucker embedding A long term program with my collaborators Laurent Manivel Colleen Robles and Jerzy Weyman is to study relations between the projective geometry of GP C lPV especially its local differential geometry and the representation theory of G More than just the geometry of GP we are interested in the geometry of its auxiliary varieties for example the tangential variety rGP C lPV which is the union of all points on all embedded tangent lines to GP and the r th secant variety of GP 07GP C lPV which is the Zariski closure of all points on all secant lP T l s to GP The auxilary varieties are all G varieties ie preserved under the action of G and thus one can study their ideals coordinate rings etc as G modules 11 Overview These notes are focused on the local projective differential geometry of homogeneously embedded rational homogeneous varieties GP C lPV Speci cally they address the question how much of the local geometry is needed to recover GP We begin by describing many examples of rational homogeneous varieties in 12 The main question we deal with is rigidity but before discussing rigidity questions we give descriptions of related projects in 2 to give context to this work The rigidity results and questions are described in 3 In 4 and 5 we give a crash course on exterior di erential systems EDS Roughly speaking EDS is a collection of techniques for determining the space of local solutions to systems of partial differential equations The techniques usually involve extensive computations that can be simpli ed by exploiting group actions when such are present as with the rigidity questions that will be the focus of this paper In 6 we describe moving frames for submanifolds of projective space and a set of rigidity EDS that are EDS LIE ALGEBRA COHOMOLOGY AND RIGIDITY 3 natural from the point of View of projective differential geometry We also describe exibility results obtained using standard EDS techniques A different method for resolving certain EDS associated to determining the rigidity of compact Hermitian symmetric spaces CHSS was introduced by Hwang and Yamaguchi in 18 that avoided lengthy calculations by reducing the proof to establishing the vanishing of certain Lie algebra cohomology groups At rst it appeared that their methods would not extend beyond the CHSS cases but the machinery was nally extended in 39 This extension is the central point of these lectures Several problems had to be overcome to enable the extension the problems and their solutions are discussed in detail in in 9 The rst problem is that the EDS natural for geometry is not natural for representation theory once one moves beyond CHSS This problem is partially resolved in 91 with the introduction of the systems 17 JP which are natural for representation theory The next problem is that even these natural systems do not lead one to Lie algebra cohomology except in the case of CHSS However a re ned version of the 17 JP systems the ltered systems 1 9 do This is explained in 92 which then leads to our main theorem Theorem 910 Before discussing these systems we describe and compare for GP C lP V the ltration of V induced by the osculating sequence and a ltration induced by the Lie algebra in 7 and brie y review Lie algebra cohomology in 8 12 Examples of rational homogeneous varieties 121 Generalized oominvsovle varieties The simplest rational homogeneous varieties are the generalized oominvsovle varieties which are the homogeneously embedded compact Her mitian symmetric spaces In addition to the Grassmannians mentioned above the oominvs ovle varieties which are the irreducible CHSS in their minimal homogeneous embeddings are 0 the Lagrangian Grassmannians Gwn W CnPn C lP A Ww A ATL ZW where W is a 2n dimensional vector space equipped with a symplectic form to C A2W On is the group preserving the form and Gwn W C Cn W are the n planes on which to restricts to be zero Note that we may use to to identify W with W so to A An zW makes sense 0 the Spinor varieties Sn DnPn DnPn1 which are also isotropic Grassmanni ans only for a symmetric quadratic form where W again has dimension 2n Their minimal homogeneous embedding is in a space smaller than lPA W the quadrio hypersvrfaoes Qn l GQ1 W C lPW which are BmPl and DmPl depending if n 2m 1 or n 2m the Cayley plane lP2 E6P6 E6P1 C lP J3Ugt which are the octonionic lines in 2 embedded as the rank one elements of the exceptional Jordan algebra J3Ugt see eg 33 for details the Frevdenthal variety E7P7 C P55 which may be thought of as an octonionic Lagrangian Grassmanian Gw 3 06 see 33 122 Products of homogeneous varieties An elementary but important generalized comi nuscule variety is the Segre variety Let V W be vector spaces the Segre variety SeglPV x lPW C lPV W as an abstract variety is simply the product of two projective spaces It is embedded as the set of rank one elements of V W In general if GP C lP V and 4 JM LANDSBERG G P C lPV we may form the product SegGP x G P C lPV V which is of course a subvariety of SeglPV x lP V 123 Veronese re embeddz39ngs of homogeneous udrletles Considering SdV as the space of homogeneous polynomials of degree d on V we can consider the variety of d th powers inside lP SdV this is isomorphic to lP V via the map ud lP V a lP SdV gt gt pd called the Veronese embedding If X C lP V is a subvariety we can consider udX lts linear span udXgt C SdV has the geometric interpretation of the annihilator of IdX C SdV the ideal of X in degree d In particular if X GP C lP VA is homogeneous then udGPgt Vd the d th C drtdn power of V 124 Generalized flag udn39etles Given two Grassmannians GkV and Cl V with say k lt l we may form the incidence variety FlagwV E F C GkV x GlV 1E C F Ofcourse FlagwV c PAkV A V Write Akv Vwk Then in fact FlagwVgt Vw w giving a geometric realization of the Cartan product of the two modules Vwk and sz This generalizes to arbitrary Cartan products as follows The cominuscule varieties are special cases of generalized Grassmannians that is vari eties GP where P is a maximal parabolic Such varieties always admit interpretations as subvarieties of some Grassmannian usually given in terms of the set of k planes annihilated by some tensors Given two such for the same group GPi C lPVw and GPj C lP ij we may form an incidence variety GPM and again we will have CIPM Vwi dj Thus C drtdn powers and products of modules can be constructed geometrically 125 Adjolnt udn39etles After the generalized cominuscule varieties the next simplest ra tional homogeneous varieties are the adjoint udrletles where V is taken to be g the adjoint representation of G We write GP X i C ng to denote adjoint varieties The adjoint varieties can also be characterized as the homogeneous compact complex contact manifolds It is conjectured see eg 42 22 that they are essentially the only compact complex contact manifolds other then projectivized cotangent bundles Many of these have simple geometric interpretations o Xg zW Flagln1W is the variety of ags of lines in hyperplanes in the n dimensional vector space W o Xg WQ GQ2W C lP A2W ED50W is the Grassmannian of isotropic 2 planes in W 0 X3 Gmu2 17710 is the Grassmannian of two planes in the imaginary octonions on which the multiplication is zero It may also be seen as the projectivization of the set of rank two derivations of O or as the set of six dimensional subalgebras of U see 36 Theorem 31 o ngww 122lPW C lP SZW chn is the variety of quadratic forms of rank one Note that other than the pathological groups Am On all adjoint representations are funda mental Also note that the adjoint variety of tn is generalized cominuscule for umhl 13 Notational conventions We work over the complex numbers throughout all func tions are holomorphic functions and manifolds are complex manifolds although much of the theory carries over to R with some rigidity results even carrying over to the Cquot0 setting In EDS LIE ALGEBRA COHOMOLOGY AND RIGIDITY 5 particular the notion of a general point of an analytic manifold makes sense which is a point off of a nite union of analytic subvarieties We use the labeling and ordering of roots and weights as in For subsets X C PV X C V denotes the corresponding cone For a man ifold X TmX denotes its tangent space at m For a submanifold X C PV TmX TPX C V denotes its af ne tangent space and p C i Lm In particular TmX 2 X If Y C PW then Y C W denotes its linear span We use the summation convention throughout indices occurring up and down are to be summed over If G is semi simple of rank r we write P P1 C G for the parabolic subgroup obtained by deleting negative root spaces corresponding to roots having a nonzero coef cient on any of the simple roots 0 i9 6 I C 1r 14 Acknowledgements It is a pleasure to thank the organizers of the 2008 Srni Winter School especially A Cap and J Slovak I also thank C Robles for useful suggestions 2 RELATED PROJECTS 21 Representation theory and computational complexity These projects with Manivel and Weyman address questions about G varieties motivated by problems in com puter science and algebraic statistics speci cally the complexity of matrix multiplication and the study of phylogenetic invariants For a survey on this work see 29 22 Sphericality and tangential varieties For work related to Joachim Hilgert7s lec tures 17 recall that a normal projective G variety Z is G spherical if for all degrees at CZd the component of the coordinate ring of Z in degree d is a multiplicity free G module see Note that this property for Z rX a priori depends both on G and the embedding of X Theorem 21 41 Let X GP C PV be a homogeneously embedded rational homoge neous uariety Then TX is G spherical i X admits the structure of a C HSS and no factor of X is GgPl In 41 we also show that if GP is cominuscule then rGP is normal with rational singularities and give explicit and uniform descriptions of the coordinate rings for all cases in the spirit of the project described in 23 below An interesting class of rGP s occurs for the subeIceptional series the third row of F reudenthal s magic chart SegllD1 x P1 x P1 Gw36 G36 DaPa Sa E7P7 Gw 3 06 where rGP is a quartic hypersurface whose equation is given by a general ized hyperdeterminant See 33 for details The equations of these varieties will play an important role in what follows as the Fubini quartic forms for X 5 when G is an exceptional group see 62 23 Vogelia This project joint with Manivel is inspired conjectural categorical general izations of Lie algebras proposed by P Deligne for the exceptional series 12 13 and P Vogel for all simple super Lie algebras 46 It has relations Pierre Loday s lectures 24 because both conjectures appear to inspired by operads Let g be a complex simple Lie algebra Vogel derived a universal decomposition of 52g into possibly virtual Casimir eigenspaces 52g C 63 Y2 63 1269 Y2 which turns out to be a decomposition into irreducible modules If we let 2t denote the Casimir eigenvalue of 6 JM LANDSBERG the adjoint representation with respect to some invariant quadratic form these modules respectively have Casimir eigenvalues 4t 7 204 4t 7 23 4t 7 2y which we may take as the de nitions of oz 39y Vogel showed that t 04 B 39y For example for 50n we may take 043y 724n 7 4 and for the exceptional series 508f426 2728 we may take 043y 72 m 4 2m 4 where m O 1 2 48 respectively Vogel then went on to nd Casimir eigenspaces Y3 Y3 Y3 C 53g with eigenvalues 6t 7 604 6t 7 63 6t 7 6y which again turn out to be irreducible and computed their dimensions a r 275W 7 275W i 275 dimg 7 a v dimY 5 WW 2t5t yt3a 7 2t 2 azwa 7 may 7 y dimyg QTmm lW 2t tvtt 7atvia5a72t ag a Ma WM WM V In 35 37 we showed that some of the phenomena observed by Vogel and Deligne persist in all degrees For example let 6 denote the highest root of g here we have xed a Cartan subalgebra and a set of positive roots Let Yk be the k th Cartan power gm of g the module with highest weight k6 Theorem 22 37 Use Vogel s parameters oz 39y as aboue The h th symmetric power of g contains three uirtual modules Yk Y4 Yk with Casimir eigenvalues 2kt7 h2 7 ha 2kt7 kzgkm 2kt7h27k39y Using binomial coe icients de ned by 1 yyl we aue 7amp7 k i2t7 k 12t7 k 7t7k7o ak2ak1akll dimY 7 k HE 7 7Ik 7171k 2 ak ak 7 and dim Y4 dim Yk are respectiuely obtained by exchanging the role ofa with B resp 39y The modules Yk Yk are described in 37 This dimension formula is also the Hilbert function of Xg aa h 24 Cartan Killing classi cation Via projective geometry le Ck W C lP AkW then the variety of tangent directions to lines through a point E C X is Y SeglPE x C lP E WE lPTEX Moreover one can recover X from Y as the image of the rational map lPT C 7 lPN given by the ideals in degree r 1 of the varieties 07Y multiplied by a suitable power of a linear form coming from the C factor to give them all the same degree In 31 we showed that the same is true for any irreducible cominuscule variety This enabled us to give a new constructive proof of the classi cation of CHSS without having to rst classify complex simple Lie algebras Moreover a second construction constructs the adjoint varieties and gives a new proof of the Killing Cartan classi cation of complex simple Lie algebras without classifying root systems Here is the construction for adjoint varieties Let Y C W lPTl be a generalized cominuscule variety De ne Y to be admissible if the span of the embedded tangent lines to Y as a subvariety of the Grassmannian has codimension one in A2T1 For generalized cominuscule varieties this condition is equivalent EDS LIE ALGEBRA COHOMOLOGY AND RIGIDITY 7 to Y being embedded as a Legendm39an variety In particular 7Y is a quartic hypersur face for the exceptional series7 it is the quartic hypersurface described in 22 Linearly embedded T1 C C C C 1 respectively as the hyperplanes mn 0 and mo 0 and consider the rational map 1 quot9 W C Ms ch x07 mn gt gt 3 mng max 312Yll T17 mgzan 7 013TYsmg lP T1mngL 7 I4739Y7 lPTl In 31 we showed that the image is an adjoint variety and that all adjoint varieties arise in this way Here are the Legendrian varieties Y and the Lie algebras of the ng that they produce Y C llm z g M 1 C P3 92 P1 x Qm 4 C sz 5 50m Ga 37 C P13 f4 G3 6 c IP19 e6 S6 C P31 27 Cw 037 06 C P55 28 The two exceptional ie7 non fundamental cases are P n P C W stk C sz l 5pm See 317 34 for details The varieties Y C W4 are the asymptotic directions BUIXEd C llDTmX d de ned in the next section 3 PROJECTIVE DIFFERENTIAL GEOMETRY AND RESULTS 31 The Gauss map and the projective second fundamental form Let X C lP V be an n dimensional subvariety or complex manifold The Gauss map is de ned by 39yX We Gn1V m gt gt TEX Here TmX C V is the af ne tangent space to X at m it is related to the intrinsic tangent space TmX C TmlPV by TmX TmXi X C Vi X where i C V is the line corresponding to z E lPV Similarly NmX TmlPVTmX f VTmX The dashed arrow is used because the Gauss map is not de ned at singular points of X7 but does de ne a rational map Now let x C Xsmooth and consider 1 TmX a TEXGm 1 V TEXV VTmX Since7 for all u C TmX7 i C ker d y v where d y v TmX a VTmX7 we may quotient by i to obtain 1 e TX TmXif VTmX TgX 2 NmX 8 JM LANDSBERG In fact essentially because mixed partial derivatives commute we have dim e 52TX NmX and we write Hm dim the projectiue second fundamental form of X at z Hm describes how X is moving away from its embedded tangent space to rst order at m One piece of geometric information that Im encodes is the following Think of lP TmX C lPTmUPV as the set of tangent directions in TmlPV where there exists a line having contact to X at z to order at least one Then Balm lPu C TmX IIuu 0 often called the set of asymptotic directions is the set of tangent directions where there exists a line having contact to X at z to order at least two To study the macroscopic geometry of X we may study the smaller variety Balm and ask What does Balm tell us about the geometry of X Note that Balm is usually the zero set of codimX quadratic polynomials and thus we expect it to have codimension equal to codim X lPV assuming the codimension of X is suf ciently small otherwise we expect it to be empty Now let X GP C lPV be a homogeneous variety In particular we have IIXm IIXy for all z y C X so we will simply write IX UXW To what extent is X characterized by 11X Aside If the ideal of a projective variety X C lPV is generated in degrees at most at then any line having contact with X to order at at a point must be contained in X By an unpublished theorem of Kostant the ideals of rational homogeneous varieties are generated in degree two so BUIGP corresponds to the tangent directions to lines through a point Thus for example when X Ck V BUIE SeglPE x 32 Second order rigidity For the Segre variety BII959P2XP2 C P3 is the union of two disjoint lines Pl s The Segre has codimension four and normally the common zero set of four quadratic polynomials on P3 is empty This prompted Grif ths and Harris to conjecture Conjecture 31 15 Let X SeglP2 x P2 C MCS C3 Let Z4 C lPV be a uariety such that at z E denemj IZJ 11X then Z is projectiuely equiualent to the Segre Theorem 32 26 27 The conjecture is true moreouer the same result holds when X is any rank two cominuscule uariety ezceptfor Q C Fwd and 55901 X lP m C MCZ X Cm11 One can pose more generally the question Giuen a homogeneous uariety GP C lPV an unknown uariety Z C lPW and a general point z E Z how many deriuatiues must we take at z to conclude Z GPf2 33 History of projective rigidity questions The problem of projective rigidity dates back 200 years when Monge showed 122lP1 C P2 the conic curve in the plane is character ized by a fth order ODE ie it is rigid at order ve More recently about 100 years ago 14 F ubini showed that in dimensions greater than one quadric hypersurfaces are rigid at order three ie characterized by a third order system of PDE A vast generalization of Theorem 32 was obtained by Hwang and Yamaguchi Theorem 33 18 Let X C lP V be an irreducible homogeneously embedded C HSS other than a quadric hypersurface or projectiue space with osculating sequence of length Then X is rigid at order EDS LIE ALGEBRA COHOMOLOGY AND RIGIDITY 9 See 71 for the de nition of the osculating sequence Even more exciting than the theorem of Hwang and Yamaguchi are the methods they used to prove it More on this in 64 If one changes the hypotheses slightly7 one gets a second order result Corollary 34 28 Let X C lP V be a cominuscule uariety other than a quadric hypersur face Let Y C lP W be an unknown uariety such that dim Y dimV and such that for y E Ygeneml Ii2 11X Then Y is projectiuely equiualent to X The proof of this result uses two facts that the higher fundamental forms of cominuscule varieties are the full prolongations of the second7 and that any variety with such funda mental forms must be the homogeneous model which follows from Theorem 33 See 28 for details 34 Rigidity and exibility of adjoint varieties For the adjoint varieties7 it is easy to see that order two rigidity fails see 307 even though they have osculating sequence of length two These lectures will be centered around the proof of the following theorem Theorem 35 39 For simple groups G the adjoint uarieties X d C lP g other than C A1 are rigid at order three In the case G Al7 Xj f 122lP1 which Monge showed to be rigid at order ve but not four Robles and I originally wrote a brute force77 proof of this theorem in December 20067 although we had been attempting to use the methods of Hwang and Yamaguchi Finally7 when A Cap visited us in June 20077 in what can only be described as an incredible syncronicity7 we made the breakthrough needed7 in parallel with Cap making a breakthrough in his work on BGG operators with maximal kernel In 8 I describe the methods7 which involve a reduction to a Lie algebra cohomology calculation7 and which should be useful for other EDS questions I conclude this section with the description of a result that was obtained using traditional EDS techniques The adjoint varieties are the homogeneous models for certain parabolic geometries a much discussed topic at this conference In particular they are equipped with an intrinsic geometry that includes a holomorphic contact structure All the intrinsic geometry is visible at order two including the distinguished hyperplane except for the contact structure This inspires the modi ed question Assume Z C lPV is such that at z E ngeml we have ZZZ IIXEd and the resulting hyperplane distribution is contact7 can we conclude Z X i Of course for G Al7 we know the answer is no7 thanks to Monge Theorem 36 39 IfG 7 A17A2 then YES If G SL3 A2 then NOl there exist functions worth quot of impostors Remark 37 Although the results are formulated in the holomorphic category the exact same result holds in the real analytic category The second conclusion has interesting consequences for geometry A 3 manifold M equipped with a contact distribution which has two distinguished line sub bundles is the 10 JM LANDSBERG path space for a path geometry in the plane Such structures have two curvature 7 func tions call them J1 J2 which are differential invariants that measure the difference between M and the homogeneous model which is Xg z3 Flagl2C3 This geometry has been well studied by many authors including E Cartan For example if J1 E 0 then the paths are the geodesics of a projective connection See 19 Chapter 8 for more Theorem 38 39 The general impostor above has J1 J2 nonzero although they do satisfy di erentz39al relations This is interesting because it is dif cult to come up with natural restrictions on the invariants J1 J2 short of imposing that one or the other is zero A classical analog of this situation where the condition of being extrinsically realizable gives rise to a natural system of PDE is the set of surfaces equipped with Riemannian metrics that admit a local isometric immersion into Euclidean 3 space such that the image is a minimal surface Ricci discovered that for this to happen the Gauss curvature K of the Riemannian metric of the surface must satisfy the PDE Alog7K 4K where A is the Riemannian Laplacian See 11 19 for details 4 FROM PDE TO EDS Exercise show that any system of PDE can be expressed as a rst order system Hint add variables Thus we only discuss rst order systems We want to study them from a geometric perspective that of submanifold geometry Let R have coordinates ml mi and R coordinates ul um let 41 Fmiuapg 0 1 r g R be a system of equations in n m nm unknowns We view this as a system of PDE by stating that a map f R a R m gt gt u f is a solution of the system determined by 41 if 41 holds when we set a faz and a 3 a u 1 To armezphrase slightly let J1R Rm R x R x Rm have coordinates 214219 Consider the differential forms 0 1 du ipgdmi e 91J1R Rm 1 a g m Then we have the following correspondences Graphs of maps 1 R a R immersions i M a J1R Rm such that lt gt Pf C R x R 2W0 0 and idz1 A A dz is nonvanishing Graphs of maps 1 R a R immersions i M a E C J1R Rm Pf C R x R satisfying lt gt such that 09 0 and idm1 A A dz is the PDE system determined by41 nonvanishing where E is the zero set of 41 Now we are ready for EDS EDS LIE ALGEBRA COHOMOLOGY AND RIGIDITY 11 De nition 42 A Pfa ian EDS with independence condition on a manifold E is a sequence of sub bundles I C J Q T il Write n rankJI An integral manifold of I J is an immersed n dimensional submanifoldi M gt 2 such that iI 0 and iJI TM In the motivating example we had I 0 1 and J 0a dml Thus we have transformed questions about the existence of solutions to a system of PDE to questions about the existence of submanifolds tangent to a distribution We next show how to determine existence But rst here are a few successes of EDS 0 Determination of existence of local isometric embeddings of analytic Riemannian manifolds into Euclidean space and other space forms eg Cartan Janet theorem see eg 1 5 Proving the existence of Riemannian manifolds with holonomy G2 and Spin7 Bryant 4 Rigidity exibility of Shubert varieties in Grassmannians and other symmetric spaces Bryant Proving existence of special Lagrangian and other calibrated submanifolds Harvey and Lawson 16 5 THE CARTAN ALGORITHM To DETERMINE LOCAL EXISTENCE OF INTEGRAL MANIFOLDS To AN EDS The essence of the Cartan algorithm is to systematically understand the additional con ditions imposed by a system of PDE by the fact that mixed partial derivatives commute In the language of differential forms this is the statement i0 0 3 id0 0 V0 6 I For example in 4 id0 0 forces 8p 3m7 dpgdzi On integral manifolds pf Qua835i 51 Linear Pfa ian systems Among Pfaf an systems there are those where the set of integral elements through a point forms an af ne space the linear systems De nition 51 A Pfa ian EDS is linear if the map I a A2TEJ t9 gt gt 10 mod J is zero To simplify the exposition we will restrict to linear Pfaf an systems This is theoretically no loss of generality see 19 Chapter 5 Although some of the theory is valid in the Cquot0 category see eg 48 we will work in the real or complex analytic category where the theory works best and in the applications of this paper we will actually work in the holomorphic category In particular it makes sense to talk of a general point of an analytic manifold where general is with respect to the EDS on it eg points where the system does not drop rank where the derivatives of the forms in the system don t drop rank etc 12 JM LANDSBERG Fix z 6 295 To determine the integral manifolds through x we work in nitesimally and reduce to problems in linear algebra as one does with most problems in mathematics De nition 52 An n plane E C TmE is called an integral element if 0m E 0 and d0m E 0 for all 0 E I Let VIm C Cn TmE denote the set of all integral elements at m As remarked above if I J is linear then VIm is an af ne space Set V 11 W I Fix a splitting TE Jm 63 Jg and de ne a bundle map LV4 AV 01 H ddm mod I J0 we may consider this map as a tensor T C W A2V which we call the apparent torsion of I J at m Since the apparent torsion changes if we change the splitting we instead consider 53 T e W A2V called the torsion of the system at m which is well de ned The equivalence N is precisely over the different choices of splittings and is made explicit in 55 below Since on the one hand we are requiring I to vanish on integral elements but JI to be of maximal rank if T 7 0 there are no integral elements over z ie VIm Q If this is the case we start over on the submanifold analytic subvariety 2 C 2 de ned by T 0 Now consider the bundle map given by exterior di erentiation 0 gt gt 10 mod I a compo nent of which is I a TilJ X J Pointwise this is a linear map W a T iEJm X V which we may consider as a linear map TilJ a W X V Let ACW V denote the image of this map at m which is called the tableau of I J at m For linear Pfaf an systems A corresponds to VIm where we transform the af ne space to a linear space by picking a base integral element 9 7r 0 where the 9 give a basis of Im and 7139 give a basis for a choice of J5 The quantity dimA gives us an answer to the in nitesimal version of the question How many integral manifolds of I J pass through m Example 54 Say J TE which is the situation of the Frobenius theorem Then the tableau A is zero which corresponds to the uniqueness part of the theorem There exist integral manifolds if and only if T T 0 We may think of the tableau A as parametrizing the choices of admissible rst order terms in the Taylor series of an integral manifold at z expressed in terms of a graph From this perspective the next question is What are the admissible second order terms in the Taylor series At the risk of being repetitive the condition to check is that Mixed partials commute EDS LIE ALGEBRA COHOMOLOGY AND RIGIDITY 13 52 Prolongations and the CartanK hler theorem Let mW W WHW MW denote the skew symmetrization map De ne A0gtka6lA VA iHmwr ssz the prolongation of A We may think of Ail as parametrizing the admissible second order terms in the Taylor series At this point we can make explicit the equivalence in 53 It is m meW Mwwm W Now we know how to determine the admissible third order Taylor terms etc7 but should we keep going on forever When can we stop working The answer is given by the following theorem Theorem 56 Cartan7 Cartan Kahler see eg 57 19 Let I7 J be an analytic linear Pfa an system on 2 let m E Egeneml Assume Tz 0 Choose an A generic flag V V0 D V1 D 3 V7 1 3 0 Let Aj A W V7 Then dimA1 dimA dimA1 dimA 1 If equality holds then we say I7 J is involutive at m and then there exist local integral manifolds through m that depend roughly on dim ATA74 functions of r uariables where r is the unique integer such that A74 7 AT ATM 53 Flowchart and exercises Here 9 E WUI encodes the independence condition Inpun Prolong iiei7 start over linear Pfaf an system on a larger space E Rename 2 as E 17 J 011 E 7 rename E as E calculate d1 mod I and new system as 17 ls tableau involut ive Restrict to 2 C 2 de ned by T and 9 l2 0 Done there are no integral manifolds local existence of integral manifolds Exercises 57 Set up the EDS and perform the Cartan algorithm in the following problems 1 The Cauchy Riemann equations um vy7uy 712m Work on a codimension two submanifold of J1R27 R2 14 JM LANDSBERG 2 Find all surfaces M2 C E3 such that every point is an umbillic point 3 Determine the local existence of special Lagrangian submanifolds of R2 2 C 4 For the more ambitious Pick your favorite G C SO p q and determine local existence of pseudo Riemannian manifolds with holonomy Q G 5 After you read 6 show that Segll 2 x W C P8 is rigid to order two Then roll up your sleeves to show that SeglP1 x P is exible at order two 54 For fans of bases Here is a recap in bases take a local coframing of 2 adapted to the ag I C C Ti ie write I 0 1 1g a rankI J 0awl1 i rankJI Til 0awl 7139s 1 g e rank TilJ Then there exist functions A H such that 10 Agi n396 A If T524111 A wj Ezts n396 A 71396 Flgob A w aged A we H5005 A 00 Since we only care about 10 mod I we ignore the second row The system is linear iff E55 O The apparent torsion is T T5421 A v7 810 C A2V X W The tableau is A Agivi wa l 1 e rank T2J c V W The torsion is T Tgwa X vi A Uj mod Ageef 7 AZeQwa X vi A Uj l e E lF e W A2V6A V Here lF R or C 6 MOVING FRAMES FOR SUBMANIFOLDS OF PROJECTIVE SPACE Let U CNH Let C Q GLU have Maurer C mtan form Lug C 91Gg Recall that the Maurer Cartan form has the following properties 0 left invariance LZw w where Lg G a G is the map a gt gt ya 0 Law THE A g is the identity map 0 dw 7w A w or equivalently dw 7411411 Maurer Cartan equation Here lw7nlv7w 3 lwv7nwl lww7nvl 61 Adapted frame bundles We want to study the geometry of submanifolds Y C lPU from the perspective of Klein that is we consider Y N Z if there exists 9 C GLU such that gY Z In order to ef ciently incorporate the group action we will work upstairs on GLU Consider the projection map 7r GLU a lPU 50 eN gt gt so where we view the 57 as column vectors Fixing a reference basis we may identify GLU with the set of all bases of U We will restrict ourselves to submanifolds of GLU consisting of bases adapted to the local differential geometry on C lPU First consider f0 7r 1Y the O th order adapted frames bases Let n dim Y Next consider 51 f 50 eN 6 f3 1 They 50 e EDS LIE ALGEBRA COHOMOLOGY AND RIGIDITY 15 the frames adapted to the ag i C TmY C U over each point called the rst order adapted frame bundle Write L 32 T TmYi N UTmY Adopt index ranges 1 g a 3 g n dimeY n 1 uz dimU 71 Write gU LGBTGBNV LGBTGBN and let for example wLe T denote the component of to taking values in L X T C U X U gU Write 0 0 0 0 JV wu L wre L wm L 61 wgU WS wfa w wmsr wT T WN T w w wlf WL N wT N WN N Write i f a GLU as the inclusion We have fang iwL N 0 Note that at each 1 6 f we actually have a splitting U L63 T N 62 Fubini Forms Now anytime you ever see a quantity equal to zero Di erentz39ate it We have iw5 0 3 2 dw5 0 which using the Maurer Cartan equation tells us that f w A Lug 0 note use of summation convention We are assuming that the forms iw3 are linearly independent as they span the pullback of TY by our choice of adaptation so we must have ex qzmw for some functions qg f3 AC Moreover exercise qg qga for all 043 this is often called the Cartan Lemma The functions qg vary on the ber but they do contain geometric information If we form the tensor eld F2 qg wg o of e0 29 5 mod 23 6 HR 7rSZTY NY a short calculation shows that F2 is constant on the bers ie F2 7rII for some tensor H C PYSZTY NY H is indeed the projective second fundamental form de ned as the derivative of the Gauss map in 3 Unlike with the case of the Gauss map where it was not clear how to continue here it is we have a quantity equal to zero w 7 qg wg so we differentiate it From now on we drop the i when describing pullbacks of differential forms to simplify notation The result is that there exist functions rgm fil H C such that 161 7 qg wg q iaw q ewfa 155 TSng which gives rise to a tensor eld F3 6 HR 7r53TY 29 NY This tensor called the Fubini cubic form does not descend to be well de ned on Y but it does contain important geometric information 16 JM LANDSBERG 63 Second order Fubini systems Fix vector spaces L T N of dimensions 1 7101 and x an element F2 6 SZT N L Let U LGBTGBN and let w E 01GLUgU denote the Maurer Cartan form Writing the Maurer Cartan equation component wise yields for example de T 7wL T AwLlt2zgtL 7 wT T AwLlt2zgtT 7 wN T AwLlt2zgtN Given F2 6 L SZT N the second order Fubini system for F2 is 11111132 WL N7WT N 7 F2WL T7 JFub2 IF ub27WL T lts integral manifolds are submanifolds f2 C GLU that are adapted frame bundles of submanifolds X C lPU having the property that at each point z E X the projective second fundamental form F2Xm is equivalent to F2 The tautological system for frame bundles of arbitrary n dimensional submanifolds is given by I my N J IwL T Let R C GLL x GLT x GLN denote the subgroup stabilizing F2 and let t c U L ea T T ea N N gU0 denote its subalgebra These are the elements of gU0 annihilating F2 The motivation for the notation gU0 is explained in 71 Assume t is reductive so that we may decompose gU0 r69 tL as an t module In the case of homogeneous varieties GP F2 6 SZT N X L will correspond to a triv ial representation of the Levi factor of P which we denote Go For example let GP G2 M C lPAZM be the Grassmannian of 2 planes Then R G0 GLE x GLF T E X F N A2E X AZF and we have the decomposition SZT AZE X A2F 63 52E X 52F and F2 6 SZT X N corresponds to the trivial representation in AZE X A2F X AZE X A2F In the notation of 5 V L T W L N T N A 2 ti with L N C W in the rst derived system That tL C V X W may be seen as follows dwT N 7 F2wL T 7 anEL AwU N 7 meET AwT N 7 wT N AwN N F2wL L AwL T F2wL T AangT wL N AwNlt2zgtT 62 wTMsT AFzWngT 7 F2wL T AwNnguv F2 WL L AWL T 7 wL T AWT T mod I E Lug r F2 AwU T mod I E th F2 AwU T mod I To understand the last two lines we r F2 denotes the action of the gU07Valued compo nent wo of the Maurer Cartan form on F2 6 SZT X N Recall that t is the annihilator of this action By de nition wo F2 wt l mu F2 th F2 For the Cartan algorithm we need to calculate Au ker6 where 6ti v HW A2W EDS LIE ALGEBRA COHOMOLOGY AND RIGIDITY 17 One can check directly that A is never involutive for any F2 system One has not yet uncovered all commutation relations among mixed partials This is essentially because we have yet to look at the entire Maurer Cartan form Thus we need to prolong introducing elements of Au as new variables and differential forms to force variables representing the elements of AG to behave properly just as the W s forced the pf s to be derivatives in 4 Before doing so we simplify our calculations by exploiting the group action to normalize Au N F3 as much as possible Write gU1 r T X L N X T Consider the linear map 6gtltUgt1 a U M w w A de ned as the transpose of the Lie bracket gU1 x L T a ti c gU0 Now L T C gU1vr gUj Then we de ne 1 ker6A V HW A2V A 39 lmage6 gU1 r a A V39 One can calculate directly that if X is a rank 2 CHSS in its minimal homogeneous embed ding other than a quadric or llD1 x P and F2 IIX then A718 0 In these cases we begin again with a new system f Lth on GLU Again one can check that A is never involutive but that Ami 0 Finally one de nes faring which turns out to be F robenius in the case of rank 2 CHSS ie 0 O which implies rigidity 64 An easier path to rigidity A better way to obtain the same conclusion is to observe that A3182 looks like the graded Lie algebra cohomology group H11ggi de ned in 8 In the CHSS case it indeed is this cohomology group but in all other cases it is not In the next few sections we will see that the corrrespondence is exact in the CHSS case and how it fails in all other cases it fails in two ways but none the less with the introduction of certain ltered EDS the use of Lie algebra cohomology can be recovered 7 OSCULATING GRADINGS AND ROOT GRADINGS As mentioned above for homogeneously embedded CHSS the osculating ltration and a ltration induced by the Lie algebra coincide but that these two differ for all other homogeneous varieties In this section we explain the two ltrations 18 JM LANDSBERG 71 The osculating ltration Given a submanifold X C lP U and z E X the osculatz39ng ltration at m UOCU1CCUU is de ned by U0 92 U1 Tlva U2 U1 F2L SZTmX U U1 FL T 1 STTmX We may reduce the frame bundle f to framings adapted to the osculating sequence by restricting to e 50 50 5M ew E f such that 50 C X TEMX span 50504 and Uk span 50 50 542 eMC The indices 04 and M respectively range over 1 n and dim Uj1 1 dim Uj From now on we work on this reduced framebundle denoted 39r 1 X C fX At each point of B we obtain a splitting of U This induces a splitting 9W EBBKUM The asterisk above is a place holder for a second splitting given by the representation theory when X GP that we de ne in 72 The osculating ltration of U determines a re nement of the F ubini forms Let Nk U1 U1 and de ne F N a L8 H 29 SsTgX by restricting F9 6 NM 29 L H 29 SSTmX to Ni Although the F ubini forms do not descend to well de ned tensors on X the fun damental forms FM do By construction FM L k 1 SkTmX a thX is surjective 72 The root grading Let g be a complex semi simple Lie algebra with a xed set of simple roots 041 War and corresponding fundamental weights w1 wr Let I C 1 r and consider the irreducible representation 1 g a gU of highest weight 261 kiwi Set g Mg and let MG C GLU be the associated Lie group so that GP C lPU is the orbit of a highest weight line Write P P1 C G for the parabolic subgroup obtained by deleting negative root spaces corresponding to roots having a nonzero coef cient on any of the simple roots 04 i C I Since g is reductive we have a splitting gU g eagi where gL is the submodule of gU complementary to g Let w E 01GLUgU denote the Maurer Cartan form of GLU and let rug and wg1 denote the components of w taking values in g and gi respectively The bundle f P admits a reduction to a bundle fgP MG On this bundle the Maurer Cartan form pulls back to take values in g that is wg1 0 Conversely all dim G dimensional integral manifolds of the system I wg1 are left translates of MG Let Z Z1 C t be the grading element corresponding to 2561 0 The grading element Zi for a simple root oz has the property that Zi04j In general Z 2561 Zis Thus EDS LIE ALGEBRA COHOMOLOGY AND RIGIDITY 19 if 0 1 denotes the inverse of the Cartan matrix then given a weight 1 2 117 71 Zn Z View l i r iseI The grading element induces a Z grading of g Ekgk To determine k in the case P P0 is a maximal parabolic let 64 Z mjozj denote the highest root then k mi The module U inherits a Z grading U Uzo 69 UZ71 69 W 69 UZ7f The Uj are eigen spaces for Z This grading is compatible with the action of g p Uj C Uij We adopt the notational convention of shifting the grading on U to begin at zero The component U0 formally named UZW is one dimensional and corresponds to the highest weight line of U and GlPU0 GP C lPU The labeling of the grading on gU U X U is independent of our shift convention Note in particular that the vector space ldGPld gp is graded from 71 to 7k The osculating grading on U induces gradings of gU g and gi ln Examples 73 and 74 the summands in TmGP appearing are in order from 71 to 7k We write 9W 9Usj where the rst index refers to the osculating grading 71 induced by GP C lPU and the second the root grading We adopt the notational convention EMU EBBKUM so if there is only one index it refers to the root grading Note that the grading of gU is indexed by integers if f 73 Examples of tangent spaces and osculating ltrations of homogeneous vari eties Example 72 Consider GkV C lP AkV lPU Fix E E GkV Then the osculating sequence is U0 ME c Ak lE A V c Ak ZE A AZVCCA1E A Ak 1VC AkV U Remark The only nonzero Fubini forms of a homogeneously embedded CHSS are the fun damental forms For the adjoint varieties the only nonzero F ubini forms are F23 F23 F14 One de nition of GP C lP V being cominuscule is that TIdGP is an irreducible P module Here are some examples describing tangent spaces and osculating sequences of non cominuscule varieties 20 JM LANDSBERG Example 73 For orthogonal Grassmannians GQk V C PAkV assume k lt dimV TEWQW V71 E X ELE TEGQk7 V72 AZEE where the l refers to the Q orthogonal complement gr to the associated graded vector space of the ltered vector space TEGQk V Note that E C EL because E is isotropic Example 74 For the 89 dimensional variety EsP3 C llDVm3 P6696999 T1 U A2W T4 MW 13 U A6W n4 W where U C2 is the standard representation of A1 and W C7 the standard representation of A6 Remark For those familiar with Dynkin diagrams it is possible to obtain T4 and Tf pictorially where Tf is the last summand For simplicity assume P is maximal take the Dynkin diagram for g delete the node for P and mark the adjacent nodes with the multiplicity of the bond assuming an arrow points towards the marked note otherwise just mark with multiplicity one 0 o o o o o gt 0 0 0 0 o O X GQ4 12 T4 c4 A2c4 O The last ltrand Tf is obtained by marking the nodes associated to the adjoint representation of g and taking the dual module in the new diagram These coincide iff GP is CHSS 731 Symplectic Grassmannians Here is the full osculating sequence and some details for the symplectic Grassmannians taken from 30 where many other cases with P maximal may be found as well Let Gwk2n CnPk denote the Grassmanian of k planes isotropic for a symplectic form lts minimal embedding is to Vwk AltkgtV AkVQAAk ZV the k th reduced exterior power of V C2 where Q E AZV denotes the symplectic form on V induced from w E A2V Let E E Gwk V and write U EiE A straightforward computation shows that Vwk has the following decomposition as an H SLE x SpU module Vw Altkgtv EB AbE AabE Altagt U ab Here dimE k and dim U 2n 7 2k Note that U is endowed with a symplectic form induced by the symplectic form on V C2 EDS LIE ALGEBRA COHOMOLOGY AND RIGIDITY 21 Proposition 75 30 LetE E Gwk 2n let EL 3 E denote the Q orthogonal complement to E and let U EiE Then the tangent space and normal spaces of Gwk 2n are as Go modules T1 E 29 U T4 5219 N14 ME 29 Alt2gtU N173 521E U N174 Szin Np EB AltagtU szb12E abcp EB AdU SzgldE dep SWE is the irreducible GLE module associated to the partition 7r Here 5le corre sponds to the partition with a 2 s and b 1 s In particular the length of the osculating sequence is equal to k 1 the last non zero term being Nk AWCGBU Corollary 76 30 BltHG2Egt new ea e l e e Em u 6 won This set of asymptotic directions contains an open dense P orbit the boundary of which is the union of the two disjoint closed H orbits Y1 Pk l x P2 2k1 c M11 and Y2 agar c M12 732 The bigrading for adjoint uarieties For adjoint varieties TmX d has a two step l tration with the hyperplane being the rst ltrand and the osculating sequence is simply i C T C U The induced bi grading on gUOSCalg is indicated in the table below 32 11 12 NJ N3 NJ 9 00 11 12 22 23 24 T4 11 00 01 11 12 13 T72 12 01 00 10 11 12 N72 22 11 10 00 01 02 N3 23 12 11 01 00 01 N4 24 13 12 02 01 00 In all cases T71 T72 may be determined by the remark above T72 is the trivial module as there is no node left to mark and N72 2Y N73 3TY5mg and N74 is triVial corresponding to the quartic generating I4rY 22 JM LANDSBERG 8 LIE ALGEBRA COHOMOLOGY AND KOSTANT S THEORY Let be a Lie algebra and let P be an module De ne maps 87 Ai P a A71 P in the only natural way possible respecting the Leibniz rule This gives rise to a complex and we de ne HkP ker aklmage ak l We will only have need of 80 and 81 which are de ned explicitly as follows if X C P and v w E then 80XU UX and if a X 6 A1 8P then 81 811 X U A w av wX avwX 7 awUX Now let be a graded Lie algebra and P a graded module The chain complex and Lie algebra cohomology groups inherit gradings as well Explicitly 13 i7i 1114 H jgm L7 A me Pdijim Kostant 23 shows that under the following circumstances one can compute HkP combinatorially 1 n C p C Q is the nilpotent subalgebra of a parabolic subalgebra of a semi simple Lie algebra g 2 P is a g module Under these conditions letting go C p be the reductive Levi factor of p H701 F is naturally a go module Kostant shows that for any irreducible module P it is essentially trivial to compute H1n P one just examines certain simple re ections in the Weyl group However in our situation where we need to compute H1ggi there may be numerous components to gi and moreover we would like to avoid a case by case decomposition Here the beauty of the grading element comes in because it is easy to prove that in many situations Hl g F is zero in positive degree This is well documented in 47 18 39 among other places 9 FROM THE FUBINI EDS TO FILTERED EDS I now explain how we were led to work with ltered EDS in an effort to use Lie algebra cohomology to determine rigidity of homogeneous varieties 91 Problem 1 Osculating v5 root gradings As mentioned above for CHSS the osculating grading coincides with the root grading but for all other homogeneously embedded homogeneous varieties this fails Thus to have any hope to exploit Lie algebra cohomology we need to work with an EDS that respects the root grading From now on we will work on SLU C GLU which will not change anything regarding our study of rigidity of subvarieties of lPU De ne the 17 JP system on SLU by I wg ph JP Ipvwgiy ln speci c examples after a short calculation the k th order Fubini system can be shown to be strictly stronger than some 17 JP system where of course p depends on At the moment we have no general method of determining this but we do so uniformly for adjoint EDS LIE ALGEBRA COHOMOLOGY AND RIGIDITY 23 varieties in 39 In summary7 this problem is easy to resolve in speci c cases or even classes of cases7 but work remains to resolve the general case Here is the proof in the adjoint case Proposition 91 Every integral manifold of the third order Fvbini system Imb37Jth3 for a given adjoint variety is an integral manifold of the 14714 system for the same adjoint variety Proof Suppose that f C SLU is an integral manifold of third order Fubini system We wish to show that the gilt07valued component of the Maurer Cartan form vanishes when pulled back to f That the giovalued component vanishes is an immediate consequence of the injectivity of the second fundamental form F2 on each homogeneous component Referring to the table aloove7 we see that there remain four blocks of the component of the Maurer Cartan form in gilt0 to consider the three 071 blocks wTiz 8131 wNia N2 and wN74 N3 and the singleton 07 E2 block wN74 N2 The third Fubini form is de ned by 35 of 197 35 The vanishing of the giicomponent of the rst two blocks is a consequence of the 3T1 NE3 component of F3 This is the only nonzero component of F3 The vanishing of the giicomponent of the third and fourth blocks is given by the 3T NE4Ecomponent of F3 D 92 Problem 2 Even the systems de ned by the root grading do not lead to Lie algebra cohomology For simplicity we take p El and k 2 Notice that gsL 5US for all s g 73 Abbreviate wgw 3 ws so that tag ads for all s g 73 Thus 1 wgi wgiz vJE37 wEf The calculations that follow utilize the Maurer Cartan equation see 67 and that 2 g C Q and g gt C gt It is easy to see that dws E 0 modulo 1 when 8 g 73 Next7 computing modulo IE1 Edwgb E wg727wg Edwgii E wgimwgfl wgi wg In order for these two equations to be satis ed7 on an integral element we must have 92 12 01wg71 02wg72 93 wgli 11wg71 t 12wg72 for some AM 6 giL gj Consider the degree two homogeneous component 62 of the Spencer differential 6 A V a W A2V7 where A gfeagoi W g il but we may and will ignore the 24 JM LANDSBERG rst derived system giig V 971 699723 62 i 9 891 69 at 2992 H Bil 891 A 92 69 sz 892 A 92 11 6902 H 1471 A U72 H l11U717U72l U71702U72l 6996727572 H A02z727y72l 96727302y72l Here 72 6 g4 etc This is exactly the Lie algebra cohomology differential 821 Now consider the degree one component 61 61 i at 891 H Bil 891 Nil 69 sz 891 Agiz 01 H 1471 A U71 69 90717 2472 H l01U717U71l U717 01U71l 69 l0195717y72l This fails to be the Lie algebra cohomology differential because we are missing a term 12u1U72l on the right hand side One can try to x this by adding in such a term At rst this appears unnatural but if one takes into account that there is a natural ltration on our manifold it is not unreasonable to weaken the condition Lug 0 to the condition wgillTil O ie Lug A12wg72 where A42 6 gfl gi2 at each point of our manifold We make this x precise and natural with the introduction of ltered EDS 93 The Fix for problem 2 Filtered EDS De nition 94 Let E be a manifold equipped with a ltration of its tangent bundle T 1 C T72 C C T f TE De ne an r ltered Pfa ian EDS on E to be a ltered ideal I C T9 whose integral manifolds are the immersed submanifolds i M gt 2 such that i u iTuir 0 for all u with the conuention that T79 TE when is g 7f Another way to View ltered EDS is to consider the ordinary EDS on the sum of the bundles M X TETu In our case these bundles will be trivial with xed vector spaces as models De ne 1150 to be the p 1 ltered EDS on GLU with ltered ideal I wg and independence condition 9 given by the wedge product of the forms in wgi We maypview this as an ordinary EDS on GLU X 9 972 69 W 699710169 9371 973 69 W 699 69 69 lgiikJrZ Biklgt where giving gL 82 linear coordinates AM we have 95 I wgsiv 8 S 29 k 1 wgiiw H ApimszHL gm AwHMWQa H Apih3k71wgk17 1g 7 P Q JQikl 7 AP2wgi2 However as is explained below it is more natural to work in the category of ltered EDS EDS LIE ALGEBRA COHOMOLOGY AND RIGIDITY 25 Returning to the p El k 2 system the rst derived system is wSE4 Computing similarly to above only now modulo 11 we obtain 96dw73 E wgiszgfll E wgim A712wg72l 7 97dwgi2 E Lug ng wgimwgrl wgiuwgil EL 3 wgiszg l t WQEMAELZQ JQEZH AE12WQE27 1gt2w972gi7 Edltwg 17 12wg72gt E iwgtwmi d LLZMwB 7 1gt2ltlw9727w90l l fwglvwgill l wg iwgiilg E wg72ng wgi wgoT A12wg72wg0 1 A712wg727le d712Wg72 90 ET 7 Ail fwgiszgol wgi wgill 12WQ727A712wgi2lggt Here lg resp lgT denotes the component of the bracket taking values in g resp L g Note that if we were to view the ltered EDS as an ordinary EDS on SLU x gfl g2 the term A71 is part of the torsion whereas here it is simply part of the tableau of the ltered Spencer differential The degree one homogeneous component of 9697918 is as follows 1 g71515 must be in the kernel of the map 61 371g 9271 H Bil 891 Nil 69 gig 891 Agiz de ned as follows Given 741 121 6 g4 99 5131U71 A U71 E l01U717U71l U71701U71l E 712lU717U71l FOF 7171 6 97171172 6 972 5131U71 A U72 E l01U717 Uizl U717 712U72l That is 61 8 where 8 is the Lie algebra cohomology differential described in 8 Moreover g1L 11 gU1 and the Lie algebra cohomology denominator 89gf is the space of admissible normalizations of the prolongation coef cients 1 Thus the vanishing of H11ggi implies that normalized integral manifolds of the Ifl 9 system are in one to one correspondence with integral manifolds of the 5 9 system Punch line by working with ltered EDS and by homogeneous degree we do obtain Lie algebra cohomology The vanishing of the Lie algebra cohomology reduces the system to the 10f 9 system and vanishing of the Lie algebra cohomology group H21ggi moves one to the 11f 9 system etc Moreover there was nothing special about beginning with p El The nal result is 26 JVM LANDSBERG Theorem 910 Let U be a complex vector space and g C gU a represented cornplex semi simple Lie algebra Let Z GP C lP U be the corresponding homogeneous variety the orbit of a highest weight line Denote the induced Z gradings by g gk gk and U U0 69 69 Uf Fix an integerp Z 71 and let 1 9 denote the linear Pfa ian systern given by 95 If H ggi 0 for all at Z p 2 then the hornogenous variety GP is rigid for the 159 systern E E E H H d HH 2 O1 10 OPEN QUESTIONS AND PROBLEMS Does Hlggi nonzero imply exibility If so can one prove this directly and in general without going through the sometimes quite long Cartan algorithm Give a uniform description of the Fk for all GP s to obtain uniform determinations of Fubini rigidity Determine the class of extrinsically realizable non at parabolic geometries modeled on Flagl2C3 as some natural class of parabolic geometries Apply Cap s machinery to study parabolic geometries having families of differential operators Whose kernel is large but not maximal REFERENCES 1 E Berger R Bryant P Grif ths Some isometric embedding and rigidity results for Riemannian 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