Constructing Mironov cycles in complex Grassmannians
aa r X i v : . [ m a t h . S G ] M a y Constructing Mironov cycles in complexGrassmanians
Nikolai Tyurin ∗ BLTPh JINR (Dubna) and NRU HSE (Moscow)
A. Mironov in paper [1] proposed a construction of lagrangian submanifoldsin C n and CP n ; there he was mostly motivated by the fact that these lagrangiansubmanifolds (which can have in general self intersections, therefore below wecall them lagrangian cycles) present new example of minimal or Hamiltonianminimal lagrangian submanifolds. However the Mironov construction of la-grangian cycles itself can be directly extended to much wider class of compactalgrebraic varieties: namely it works in the case when algebraic variety X ofcomplex dimension n admits T k - action and an anti - holomorphic involutionsuch that the real part X R ⊂ X has real dimension n and is transversal to thetorus action. For this case, as we show in [2], one has families of lagrangiansubmanifolds and cycles.In the present small text we show how the construction of Mironov cyclesworks for the complex Grassmannians, resulting in simple examples of smoothlagrangian submanifolds in Gr( k, n + 1), equipped with a standard Kahler formunder the Pl¨ucker embedding. For sure the text is not complete but in thenew reality we would like to fix it, hoping to continue the investigations and topresent in a future complete list of Mironov cycles in Gr( k, n + 1). Acknowledgements.
The author cordially thanks A. Kuznetsov and P.Pushkar’ for valuable discussions and remarks.
General theory.
Let (
X, ω ) be a simply connected compact smooth sym-plectic manifold of real dimension 2 n . Suppose that it admits an incompletetoric action so there are moment maps f , ..., f k which commute with respectto the standard Poisson brackets and generate Hamiltonian T k - torus actionon M . Take a generic set of values c , ..., c k such that the common level set N ( c , ..., c k ) = { f i = c i , i = 1 , ..., k } does not intersect the determinantal locus∆( f , ..., f k ) = { X f ∧ ... ∧ X f k = 0 } and contains an isotropical n − k - dimen-sional submanifold S ⊂ N ( c , ..., c k ) which is transversal at each point to thetorus action which means that T p S and < X f ( p ) , ..., X f k ( p ) > are transversalin T p N ( c , ..., c k ) at each point p ∈ S .Then the toric action applied to S generates a real n - dimensional cycle T k ( S ) which is lagrangian; depending on the situation it can be smooth or haveself intersections, moreover in some cases one can take even special values of f i and nevertheless get smooth lagrangian submanifolds. The proof is based on the ∗ The author is partially supported by Laboratory of Mirror Symmetry NRU HSE, RFGovernment grant, ag. N 14.641.31.0001 < T p ( c , ..., c k ) , X f ( p ) , ..., X f k ( p ) > is lagrangian subspace in T p M for each p ∈ S , and the lagrangian conditionis stable under the Hamiltonian T k - action (we present the details and thesimplest examples in [2]). Algebraic varieties.
Any compact algebraic variety X can be consideredas a real symplectic variety: by the very definition (see f.e. [3]) any X admitsa very ample line bundle L → X such that the corresponding complete linearsystem | L | generates an embedding φ L : X ֒ → CP N , and the lifting ω L = φ ∗ L Ω F S of a standard Kahler form Ω
F S of the Fubini - Study metric gives a symplectic(Kahler) form ω L on X . This form is not unique, but the cohomology class[ ω L ] = c ( L ) ∈ H ( X, Z ) if fixed, and one can expect that the lagrangiangeometry of ( X, ω L ) depends on this cohomology class only. Note howeverthat for different very ample line bundles over a fixed X one can expect ratherdifferent lagrangian geometries.Suppose that ( X, ω L ) admits a Hamiltonian toric action; for our aims weformulate it as follows. Fix a homogenous coordinate system [ z : ... : z N ] on CP N , compatible with fixed Fubini - Study form Ω F S , and consider the standardmoment maps of the form F i = P Nj =0 λ ij | z j | P Nj =0 | z j | , i = 1 , ..., N, λ ij ∈ Z , such that if we add row ( λ j ) = (1 , ...,
1) then integer valued matrix Λ =( λ ij ) , ≤ i, j ≤ N, is non degenerated. Then we suppose that it exists such Λthat k moment maps F i (without loss of generality we can think that they are F , ..., F k ) preserve by the Hamiltonian action the image φ L ( X ) ⊂ CP N , andthese F i ’s generate the corresponding T k - action.For certain algebraic varieties it is possible to find isotropical submanifoldsin the common level sets N ( c , ..., c k ) moreless automatically. Suppose addition-ally that our X admits an appropriate anti holomorphic involution. Again wereformulate it in our simple terms: for the fixed coordinate system [ z : ... : z N ]consider the map σ : [ z : ... : z N ] [¯ z : ... : ¯ z N ];suppose that φ L ( X ) is real with respect to σ which means that σ ( φ L ( X )) = φ L ( X ) and moreover that the real part X R = { x ∈ X | φ ∗ L σ ( x ) = x } ⊂ X issmooth real submanifold of dimension n .Then X R ⊂ X is lagrangian with respect to ω L ; consequently the intersection S R ( c , ..., c k ) = X R ∩ N ( c , ..., c k ) is isotropical, and moreover if the value set( c , ..., c k ) is generic then components of S R ( c , ..., c k ) are smooth and transver-sal to the T k - action. Therefore we can apply General theory which leads tothe construction of lagrangian cycles T n ( S R ( c , ..., c k ) in ( X, ω L ).Indeed, the transversality of the toric action and the real part can be directlychecked in CP N : the image of the real part φ L ( X R ) = φ L ( X ) ∩ RP N , and thetransversality is clear from the coordinate description of T k - action and thereal part RP N ⊂ CP N .Since this construction is a natural extension of the construction for C n and CP n , presented in [1], we call the resulting T k ( S R ( c , ..., c k )) ⊂ ( X, ω L ) Mironovcycles (or Mironov submanifolds in the case when they are smooth).
Grassmannians.
The situation, presented above, takes place for complexGrassmanians Gr( k, n + 1) under the Pl¨ucker embedding to P ( ∧ k C n +1 ) (details2n the geometry of Gr( k, n + 1) can be found in [3]). In the discussion belowwe would like to avoid the algebraic machinery and construct certain Mironovcycles in Gr( k, n + 1) using pure geometrical arguments, however all steps canbe explicitly checked in the Pl¨ucker coordinates.First or all we fix a hermitian structure on the source C n +1 , compatible withcoordinate system ( Z , ..., Z n ); this gives the corresponding K¨ahler structure onthe projective space CP n , equipped with homogenous coordinates [ z : ... : z n ].Then the standard toric action generated by moment maps µ i = | z i | P nj =0 | z j | , i = 1 , ..., n, can be naturally extended to the spaces of all projective subspaces of CP n sinceevidently every k − l ⊂ CP n is moved by theaction to another k − T n - action on the Grassmannian Gr( k, n + 1). For the projectivespace P ( ∧ k C n +1 ) one has the induced K¨ahler structure, the induced Pl¨uckercoordinates w i ,...,i k , and therefore the induced moment maps F ( µ i ) must havethe explicit expressions. Indeed, they read as follows F ( µ i ) = P ( i ,...,i k ) δ ( i, i , ..., i k ) | w i ,...,i k | P ( i ,...,i k ) | w i ,...,i k | , where symbol δ ( i, i , ..., i k ) equals 1 if i = i j for certain j or to zero otherwise.The geometrical meaning of the induced moment maps F ( µ i ) is rather sim-ple: the value of F ( µ i ) at subspace L ⊂ C n +1 equals to the norm of the orthog-onal projection of basis unit vector v i to L ; on the projective level the valueis a derivation from the distance between point [0 : ... : 1 : ...
0] where 1 is onthe i th place and the corresponding projective subspace l = P ( L ). Essentially F ( µ i ) measures the angle of L to v i .From this description it follows that F ( µ i ) has two critical values 0 and 1,and for any other value 0 < c i < X F ( µ i ) does notvanish on the level set N ( c i ) = { F ( µ i ) = c i } ⊂ Gr( k, n + 1). The critical values0 and 1 corresponds to the following ”ends”: recall that Gr( k, n + 1) under thechoice of a vector v ∈ C n +1 can be decomposed into two partsGr( k, n + 1) = Gr( k − , n + 1) ∪ tot( E → Gr( k, n )) (1)where E is a vector bundle of rank k . The first part corresponds to subspaces L which contain v , — and in our case it is equivalent to the fact that the normof the projection of v to L equals to 1; the zero section of E corresponds tothe subspaces L such that the projection equals to 0, hence the zero section of E consists of the subspaces contained by the orthogonal complement < v > ⊥ .Therefore the critical subsets of F ( µ i ) can be described as follows: take decom-position (1) for vector v = v i , then the first part of (1) forms the critical subsetwith critical value 1, and the zero set Gr( k, < v i > ⊥ ) corresponds to criticalsubset with critical value 0.On the other hand we know that the real part Gr R ( k, n + 1) ⊂ Gr( k, n + 1)exists, has right dimension and is transversal to the toric action of each F ( µ i ).Indeed, for any real l ⊂ RP n ⊂ CP n the flow φ tX µi is either trivially acts on l or itmoves l outside of RP n . The trivial action corresponds to the cases F ( µ i )( l ) = 03r 1; otherwise l must contain a real point with non trivial i ’th - coordinate, theflow scales this coordinate by e it and does not change the resting coordinates —hence the point must leave RP n (but it comes there again for a moment when t = π which we will exploit below). Mironov submanifolds of homogeneity 1.
As we have seen above com-plex Grassmanian Gr( k, n + 1) is an algebraic variety which possesses the prop-erties one needs to apply the construction of Mironov cycles. At the same timeinstead of full T n - action, spanned by all the moment maps F ( µ i ), one canreduce the story to any subtorus T k . To distinguish the cases we say that aMironov cycle T k ( S R ( c , ..., c k )) has homogeneity k if it is constructed using k moment maps, derived from the complete set ( F ( µ ) , ..., F ( µ n )). Below wepresent an example of Mironov submanifold constructed using a single momentmap, say, F ( µ n ).Fix a non critical value c n ∈ (0; 1) and study first of all the restricted levelset S R ( c n ) = N ( c n ) ∩ Gr R ( k, n + 1). Since c n = 0 we can exclude the firstcomponent in the decomposition (1). Recall the description of the second part.The bundle E R → Gr R ( k, n ) has as the base (and the zero section) the spaceof k - dimensional subspaces in the orthogonal complement < v n > ⊥ ⊂ R n +1 .For any such subspace L ⊂ < v n > ⊥ the fiber E R | L consists of k dimensionalsubspaces of the direct sum R < v n > ⊕ L which do not contain v n (thereforethey form k - dimensional vector space).Now, since we study N ( c n ), consider k - dimensional subspace L of R < v n > ⊕ L such that the projection of v n to L is fixed; it is equivalent to the conditionthat the angle between v n and L is fixed. Since c n = 0 such L never coincideswith L , and the intersection L ∩ L = M ⊂ L is a proper k − L . Then it is not hard to see that if M ⊂ L is fixed there existexactly two choices of such k - dimensional subspaces in R < v n > ⊕ L withfixed angle, which contains M .These arguments imply that the restricted level set S R ( c n ) ⊂ Gr R ( k, n + 1) isisomorphic to the following manifold. Take Gr R ( k, n ), take the tautological bun-dle τ → Gr R ( k, n ), take the dual bundle τ ∗ and at last take the ”spherization” S k − ( τ ∗ ) of this bundle: then the total spacetot( S k − ( τ ∗ ) → Gr R ( k, n )) (2)is isomorphic to S R ( c n ) = N ( c n ) ∩ Gr R ( k, n + 1).Indeed, as we have seen above, over a point [ L ] ∈ Gr R ( k, n ) two k - dimen-sional subspaces L and L are uniquely defined by the fixed angle to v n andthe intersection M = L ∩ L = L ∩ L . The intersection M is given by a pointof the projectivization of the fiber τ ∗ | [ L ] , therefore one has the double coveringof P ( τ ∗ ) which is the spherization of the fiber; globalization of the local pictureleads to the answer, given in (2).Further, following the strategy, we switch on the Hamiltonian action gen-erated by X F ( µ n ) . Geometrically this means that the unit vector v n variesin the family { e it v n } , and the fiber subspaces L and L vary as well in thespace C n +1 . However under the process L stays stable since it is contained by < v n > ⊥ which is stable for the Hamiltonian action being the critical subset.At the same time it is easy to see that the rotation φ tX F ( µn ) interchanges L and L when t = π .The result of the S - action, generated by the moment map F ( µ n ), on S R ( c n ) has been described by P. Pushkar’ in paper [4]. Take the direct sum4 × S k − ( τ ∗ ) → Gr R ( k, n )) of the trivial S bundle and the spherization of τ ∗ ;there one has the fiberwise diagonal action of Z given by simultaneous actionof the standard antipodal involutions on both the summands (note that boththe summands are spheres). Factorizing with respect to this Z - action onegets the answer: S ( S R ( c n )) = tot( S × S k − ( τ ∗ )) / Z → Gr R ( k, n )) , (3)where the fiber is the Pushkar submanifold L k ⊂ C k = C ⊗ τ ∗ | [ L ] . According to[4] (Proposition 1), we can characterize the topological type of the constructedMironov cycle as follows: it is presented as a fiber bundle over the real Grass-mannian where the fiber is either S × S k − for even k or topologically nontrivial U (1) - bundle over RP k − for odd k (this type was called generalizedKlein bottle in [1]); in both the cases the fiber bundle is topologically non trivialbeing associated with τ ∗ → Gr R ( k, n ).In particular for k = 2 the corresponding Mironov cycle is a T - bundle: forGr(2 ,
3) = CP the construction gives the standard Clifford torus since the baseGr R (2 ,
2) is just a point; for Gr(2 ,
4) it gives a two - torus bundle over RP .Note however that F ( µ n ) is rather simple moment map for the present case:one can study any integer valued linear combination of F ( µ i ) as the momentmap used for the construction of a homogeneity 1 Mironov cycle, and as weknow from [2] the result can have different topological type.We hope to continue the work in the future. References: [1] A. Mironov, “New examples of Hamilton-minimal and minimal Lagrangianmanifolds in C n and CP n ” , Sb. Math., 195:1 (2004) pp. 85–96;[2] N. Tyurin, ”Lagrangian cycles of Mironov in algebraic varieties” , sub-mitted to Sb. Math;[3] P. Griffits, J. Harris, ”Principles of algebraic geometry” , NY, Wiley, 1978;[4] P. Pushkar’, ”Lagrange intersections in a symplectic space””Lagrange intersections in a symplectic space”