Combinatorial identities and Chern numbers of complex flag manifolds
aa r X i v : . [ m a t h . DG ] F e b COMBINATORIAL IDENTITIES AND CHERN NUMBERS OF COMPLEXFLAG MANIFOLDS
PING LI AND WENJING ZHAO
Abstract.
We present in this article a family of new combinatorial identities via purelydifferential/complex geometry methods, which include as a speical case a unified and explicitformula for Chern numbers of all complex flag manifolds. Our strategy is to constructconcrete circle actions with isolated fixed points on these manifolds and explicitly determinetheir weights. Then applying Bott’s residue formula to these models yields the desired results. Introduction
Complex flag manifolds are natural generalizations of complex projective spaces and com-plex Grassmannian manifolds, and forms an important subclass of compact complex mani-folds, which can be described as follows. Arbitrarily fix a positive integer r and r + 1 positiveintegers m , m , . . . , m r , m r +1 and set N := P r +1 j =1 m j . Define the following set F := F ( m , . . . , m r , m r +1 ) := (cid:26) ( L , · · · , L r ) (cid:12)(cid:12)(cid:12)(cid:12) L ⊂ L ⊂ · · · ⊂ L r , L i are linear subspaces in C N , dim C L i = i X j =1 m j (cid:27) (1.1)and call such an ( L , · · · , L r ) ∈ F a flag in C N . When r = 1 or r = 1 and m = 1,it degenerates to the complex Grassmannian consisting of complex m -dimensional linearsubspaces in C m + m or complex m -dimensional projective spaces, which play fundamentalroles in geometry and topology.Note that the unitary group U( N ) acts transitively in a natural manner on F and itsisotropy subgroup is U( m ) × · · · × U( m r +1 ). So the complex flag manifold F can be endowedwith a homogeneous space structure(1.2) U( N )U( m ) × · · · × U( m r +1 ) . It is a classical fact, essentially due to Borel, Koszul, Wang and Matsushima that this ho-mogenous space admits a canonical U( N )-invariant complex structure and can be endowed onthis canonical complex structure with a unique invariant K¨ahler-Einstein metric with positivescalar curvature up to rescaling ([Bor54], [Ko55], [Wa54], [Mat57]). Borel and Hirzebruch([BH58], [BH59]) systematically investigated the characteristic classes of homogeneous space G/U ( G is a compact connected Lie group and U is its closed subgroup) in terms of Lie Mathematics Subject Classification.
Key words and phrases. combinatorial identity, Chern number, complex flag manifold, Bott’s residue for-mula, circle action.Both authors were partially supported by the National Natural Science Foundation of China (Grant No.11471247) and the Fundamental Research Funds for the Central Universities. theoretical information of G and U (root systems, weights, representations and so on). Inparticular, they gave a complete characterization in terms of root systems of when an invariantalmost-complex structure on the complex flag manifolds F is integrable and of the number ofinequivalent invariant structures on F ([BH58, § § F = F ( m , . . . , m r , m r +1 ) endowed with possibly various almost-complex structures. A direct wayto approach this question is to apply the above-mentioned Borel-Hirzebruch theory in [BH58],where the Chern classes of F are described as polynomials in the roots of unitary groups. In-deed this idea has been taken up by Kotschick and Terzi´c in [KT09] where they calculated theChern numbers of the two invariant complex structures of complex flag manifold F ( n, , r = 2 and m = m = 1) and gave some related applications to the geometry of F ( n, , § The main purpose of this paper is to apply Bott’s residue formula to all complex flagmanifolds F ( m , . . . , m r +1 ) to obtain a family of new combinatorial identities, Theorem 2.3,which strictly include a unified and explicit formula for calculating Chern numbers of complexflag manifolds as a special case. The reason why our main results Theorem 2.3 are strongerthan just a formula for Chern numbers is due to the statement of Bott’s residue formula itself.If we take a closer look at the precise statement of Bott’s residue formula, we shall see thatit provides more vanishing-type information for low-degree polynomials than just a methodof calculating Chern numbers (more details can be found in Section 3). Thus accordingly ourmain results contain more information.Note that our inputs (complex flag manifolds and Bott’s residue formula) are purely geo-metric. So it is a little surprising to see that the outputs (Theorem 2.3) are essentiallycombinatorial identities. However, it has been widely known that various aspects of geome-try and topology of complex flag manifolds are deeply related to enumerative combinatorics([Man01], [Fu97]). So our main results in this paper strengthen this point of view and thusin this sense they are natural and expectable. Outline of this paper.
The rest of this paper is organized as follows. we shall state ourmain results and present two examples in Section 2. In Section 3 we briefly review Bott’sresidue formula in the case of holomorphic circle action with isolated fixed points. Section4 is devoted to preliminaries on complex flag manifolds and explicit construction of theirlocal coordinate charts. After these preliminaries, we shall construct in Section 5 concretecircle actions on complex flag manifolds with isolated fixed points and explicitly determine
OMBINATORIAL IDENTITIES AND CHERN NUMBERS 3 the weights around these fixed points with the help of the local coordinate charts establishedin Section 4, from which the proof of our main result easily follows. The last section is anAppendix which contains a detailed calculation on a Chern number discussed in Example 2.8in Section 1.
Acknowledgements
This paper was completed during the first author’s visit to Max-Planck Institute for Math-ematics at Bonn in Fall 2016, to whom the first author would like to express his sincere thanksfor its hospitality and financial support.2.
Main results
We introduce in this section some necessary notation and symbols and then state our mainresults of this paper.We arbitrarily fix as in Section 1 a positive integer r and r +1 positive integers m , . . . , m r +1 and define N := P r +1 j =1 m j and F := F ( m , . . . , m r +1 ) as in (1.1). An ordered sequence I := ( I , . . . , I r +1 ) is called a decomposition of the set { , , . . . , N } if(2.1) I i ⊂ { , , . . . , N } , r +1 [ i =1 I i = { , , . . . , N } , ♯ ( I i ) = m i . Here ♯ ( · ) denotes the cardinality of a set. This means that these I i are mutually disjoint.Note that there are(2.2) (cid:18) Nm (cid:19)(cid:18) N − m m (cid:19) · · · (cid:18) m r + m r +1 m r (cid:19) = N ! m ! m ! · · · m r ! m r +1 !different decompositions for this set { , , . . . , N } , which is precisely the Euler characteristicof the complex flag manifold F ( m , . . . , m r +1 ) (see Remark 2.4).Let x , . . . , x N be N variables. For each decomposition I = ( I , . . . , I r +1 ), we formulate aset W I as follows. W I := [ ≤ i Suppose that m = m = · · · = m r +1 = 1 and then N = r + 1. In this casethe decompositions correspond to S N , the permutation group on N objects, W σ = {− x σ ( i ) + x σ ( j ) | ≤ i < j ≤ N } , ∀ σ ∈ S N , and ♯ ( W σ ) = N ! . Recall that a partition λ is a finite sequence of positive integers ( λ , λ , . . . , λ l ) in non-increasing order: λ ≥ λ ≥ · · · ≥ λ l ≥ . The weight of λ is defined to be P li =1 λ i . PING LI AND WENJING ZHAO We denote by c i ( · , . . . , · ) the i -th elementary symmetric polynomial of d variables. If λ =( λ , λ , . . . , λ l ) is a partition, we define c λ ( · , . . . , · ) := l Y i =1 c λ i ( · , . . . , · )to be the product of these elementary symmetric polynomials c λ i . It will be clear soon thatthe Chern numbers of F corresponding to the partition λ shall be described in terms of thequantity c λ . Notation convention 2.2. If { y , . . . , y n } is a set of n variables, we define e ( { y , . . . , y n } ) := n Y i =1 y i to be the product of the elements in the set. If f ( · , . . . , · ) is a symmetric polynomial of n variables, we define f ( { y , . . . , y n } ) := f ( y , . . . , y n ) . With the above-defined symbols and notation understood, we can now state our mainresults as follows. Theorem 2.3 (Main results) . Suppose that f ( · , . . . , · ) is a homogeneous symmetric polynomialof d variables. We denote by deg ( f ) the degree of f . Then we formulate a rational functionof the variables x , . . . , x N as follows. (2.3) R f ( x , . . . , x N ) := X I =( I ,...,I r +1 ) f ( W I ) e ( W I ) , where the sum of the right hand side is over all the decompostions I of the set { , . . . , N } .Then we have (1) if deg ( f ) < d , then R f ( x , . . . , x N ) ≡ . (2) If deg ( f ) = d , then R f ( x , . . . , x N ) is a constant depending only on f . Moreover, if λ is a partition of weight d , then R c λ ( x , . . . , x N ) is the Chern number of the complexflag manifold F ( m , . . . , m r , m r +1 ) corresponding to the partition λ . Remark 2.4. (1) If deg( f ) < d , it is quite difficult, at least at the first glance, to imagine that theright hand side of (2.3) vanishes. So this situation provides us a family of nontrivialcombinatorial identities.(2) If deg( f ) = d , Theorem 2.3 tells us that R f ( x , . . . , x N ) is a constant dependingonly on f . This means that, for any N mutually distinct numbers (integral, real orcomplex etc) ( α , . . . , α N ), R f ( α , . . . , α N ) is equal to this constant. This providesus an effective method to calculate Chern numbers of the complex flag manifolds inpractice.(3) Put f = c d in (2.3), then c d ( W I ) = e ( W I ) and thus each summand in the right handside of (2.3) is 1 and so R c d ( x , . . . , x N ) = N ! m ! m ! · · · m r ! m r +1 ! , OMBINATORIAL IDENTITIES AND CHERN NUMBERS 5 which is equal to the Chern number of the complex flag manifold F ( m , . . . , m r +1 )corresponding to the partition ( d ). This Chern number is famously known to be equalto the Euler characteristic of F ( m , . . . , m r +1 ) (compare to (2.2)).If deg( f ) > d , the right hand side of (2.3) is still well-defined. But in general the expressions R f ( x , . . . , x N ) depend on x , . . . , x N and thus lack geometrical meanings. Nevertheless, inour situation, for two special homogenous symmetric polynomials of degree d + 1: c d +11 and c d c , we still have the following Proposition 2.5. ( R c d +11 ( x , . . . , x N ) = 0 R c d c ( x , . . . , x N ) = 0(2.4) Remark 2.6. (1) The reason for the first equality in (2.4) is related to the residue formula of the Futakiintegral invariant ([FM85]), which obstructs the existence of K¨ahler-Einstein metricson Fano manifolds (compact complex manifolds with positive first Chern classes). Thewell-known existence of such a metric on F ( m , . . . , m r +1 ) leads to the first equalityin (2.4). We shall explain this in more detail in Section 3.(2) In contrast to the nontriviality of the first one in (2.4), the second one of (2.4) is quiteobvious: R c d c ( x , . . . , x N ) = X I c ( W I ) = X I (the sum of elements in W I ) = 0 , which is indeed a special case of a general result proved by the first author in [Li13-1],which we will mention again in Section 3.Before ending this section, we would like to illustrate Theorem 2.3 by two simple examples. Example 2.7. As in Example 2.1 we assume m = · · · = m r +1 = 1 and r + 1 = N . Then wehave X σ ∈ S N f (cid:0) {− x σ ( i ) + x σ ( j ) | ≤ i < j ≤ n } (cid:1)Q ≤ i Note that F (1 , , C ∞ ==== U (4) U (1) × U (1) × U (2) C ∞ ==== U (4) U (1) × U (2) × U (1) C ∞ ==== F (1 , , . Here “ C ∞ ” means diffeomorphism. This means that as smooth manifolds F (1 , , 2) and F (1 , , 1) are diffeomorphic to each other. However, Borel and Hirzebruch applied the resultsobtained in [BH58] to show that ([BH59, § c of them are(2.5) c [ F (1 , , = 4860 = c [ F (1 , , , which implies that the canonical complex structures on F (1 , , 2) and F (1 , , 1) are different .This gives the first example of two compact complex manifolds which are diffeomorphic toeach other, but have different Chern numbers (hence not biholomorphic to each other). We PING LI AND WENJING ZHAO can also apply Theorem 2.3 to calculate them, whose details are presented in Appendix 6.This was revisited again by Hirzebruch in [Hi05]. Remark 2.9. It can be shown that (cf. [Hi05] or [KT09]) the complex flag manifolds F (1 , , n )can be identified with P ( T C P n +1 ), the projectivizations of the holomorphic tangent of thecomplex projective space C P n +1 . This point of view was taken up in [KT09, § 4] to deduce theChern number c n +11 [ F (1 , , n )] for general n . In principle, this formula can also be obtaineddirectly via our Theorem 2.3. However, practicely for general n the expression we need todeal with is quite complicated and thus it is difficult to obtain the closed formula as in [KT09,p. 604, Theorem 3] without resorting to the characteristic classes of P ( T C P n +1 ).3. Bott’s residue formula In this section we briefly review Bott’s residue formula for circle actions and give somerelated remarks.Suppose that M is a compact complex manifold with complex dimension n and it isequipped with a holomorphic circle action with isolated fixed points. We denote by P suchan isolated fixed point. At each P there are well-defined n integers k , · · · , k n (not necessarilydistinct) induced from the isotropy representation of this circle action on the holomorphictangent space T P M . Namely, the circle acts on T P M ∼ = C n in the following manner: S × C n −→ C n , (cid:0) g, ( v , . . . , v n ) (cid:1) ( g k · v , . . . , g k n · v n ) . (3.1)Note that these k , · · · , k n are nonzero as these fixed points P are isolated and called weights at P with respect to this circle action.Following the notation and symbols in Section 2, let f ( x , · · · , x n ) be a homogeneous sym-metric polynomial in the variables x , · · · , x n . Then f ( x , · · · , x n ) can be written in an es-sentially unique way in terms of the elementary symmetric polynomials e f ( c , · · · , c n ), where c i = c i ( x , · · · , x n ) is the i -th elementary symmetric polynomial of x , · · · , x n . For instance, f ( x , · · · , x n ) = x + · · · x n = c − c = e f ( c , . . . , c n ) . With the above-mentioned notation and symbols understood, we can now state a versionof the Bott’s residue formula [Bo67], which reduces the calculation of Chern numbers of M to the weights k i around the isolated fixed points of this holomorphic circle action. Theorem 3.1 (Bott’s residue formula) . Suppose a compact complex n -dimensional manifold M admits a holomorphic circle action with isolated fixed points { P } and the weights aroundeach P are denoted by k , · · · , k n , which depend on P . Then we have X P f ( k , . . . , k n ) Q ni =1 k i = (cid:26) , deg ( f ) < n, R M e f ( c ( M ) , · · · , c n ( M )) , deg ( f ) = n. (3.2) Here the sum is over all the isolated fixed points P and c i ( M ) denotes the i -th Chern class of M . Remark 3.2. (1) If λ is partition of weight n and e f ( c , . . . , c n ) = c λ , this formula precisely gives amethod to calculate the Chern number of M with respect to the partition λ in termsof the weights k i around the isolated fixed pionts. OMBINATORIAL IDENTITIES AND CHERN NUMBERS 7 (2) The conclusion that the left hand side of (3.2) vanishes when deg( f ) < n is by nomeans trivial. Thus this formula provides us a family of vanishing-type results aboutthe weights k i around the isolated fixed points rather than just how to compute theChern numbers of M . This observation played a dominant role in establishing mainresults in the first author’s previous works ([Li12], [Li13-2], [Li14], [LL11] and [LL13]).Note that (3.2) tells us nothing about those f with deg( f ) > n . For general f withdeg( f ) > n the left hand side of (3.2) may not vanish or may depend on the weights k i , whichcan be easily tested by the data presented in Appendix 6. Nevertheless, for two specail f ofdegree n + 1: c n +11 and c n c , we have the following Theorem 3.3. We make the same assumptions as in Theorem 3.1. (1) If M admits a K¨ahler-Einstein metric, then (3.3) X P c n +11 ( k , . . . , k n ) Q ni =1 k i = X P ( k + · · · + k n ) n +1 Q ni =1 k i = 0(2) The following identity holds ( unconditionally )(3.4) X P c n c ( k , . . . , k n ) Q ni =1 k i = X P ( k + · · · + k n ) = 0 . Remark 3.4. (1) The left hand side of (3.3) was showed by Futaki-Morita ([FM85, Prop. 2.3]) to bethe Futaki integral invariant with respect to the holomorphic vector field generatedby the circle action, which vanishes if M admits a K¨ahler-Einstein metric ([Fut84]).Some of the considerations in [FM84] and [FM85] have recently been improved by thefirst author in [Li13-2].(2) (3.4) is a particular case of [Li13-1, Corollary 1.3] by the first author, which is in turnan application of the rigidity phenomenon of Dolbeault-type operators on compact(almost) complex manifolds.After constructing in the next section concrete circle action on the complex flag manifolds F ( m , . . . , m r +1 ) with isolated fixed points and with the desired weights we shall see thatTheorem 2.3 and Proposition 2.5 follow from Theorems 3.1 and 3.3 respectively.4. Complex flag manifolds and their coordinate charts In this preliminary section we recall the matrix-description and a typical local coordinatechart of complex flag manifolds , which are crucial to explicitly determine the weights of circleactions constructed in Section 5. Although these materials must be well-known to experts,we are unfortunately not able to find a suitable reference to them, except a book written inChinese by Q.-K. Lu ([Lu63]). Therefore for reader’s convenience we present these materialsin detail in this section. As the description of the local coordinate charts of F ( m , . . . , m r +1 )is a little bit complicated for general r and m i , at least at the first glance, to make this sectionmore accessible to readers not familiar with these materials, we would like to first illustratethe idea for complex Grassmannian in detail, which can be found, for example, in [GH78, p.193].First recall the following standard fact for complex projective spaces. PING LI AND WENJING ZHAO Example 4.1. C P n := (cid:8) ( z , . . . , z n +1 ) ∈ C n +1 { } (cid:9) / ∼ =: { [ z , . . . , z n +1 ] } , where ( z , . . . , z n +1 ) ∼ ( w , . . . , w n +1 ) if and only if there exists t ∈ C − { } such that( z , . . . , z n +1 ) = t ( w , . . . , w n +1 ) and [ · ] denotes the coset element in the quotient space. Let U i := { [ z , . . . , z n +1 ] ∈ C P n | z i = 0 } , ≤ i ≤ n + 1 . Then ϕ i : U i ∼ = −→ C n [ z , . . . , z n +1 ] ( z z i , . . . , b z i z i , . . . , z n +1 z i ) , (4.1)where “ b · ” means deletion. These ( U i , ϕ i ) (1 ≤ i ≤ n + 1) form local coordinate charts for C P n .The above construction in Example 4.1 can be extended to complex Grassmannian mani-folds as follows ([GH78, p. 193]). Example 4.2. Let Gr( m, n ) denotes the complex Grassmannian manifold consisting of com-plex m -dimensional sublinear spaces in C m + n . Note that dim C Gr( m, n ) = mn . Denote by M ( r, s ) the set of matrices with r rows and s columns and GL( r, C ) the general linear groupof rank r over C . An element in Gr( m, n ) can be represented by a set of m linearly indepen-dent column vectors in C m + n spanning this element, i.e., by a matrix A = ( α , . . . , α m ) ∈ M ( m + n, m ) of rank m such that the column vectors α , . . . , α m form a basis of this element.Obviously two matrices A, B ∈ M ( m + n, m ) with rank( A ) = rank( B ) = m represent thesame element in Gr( m, n ) if and only if there exists Q ∈ GL( m, C ) such that A = BQ . Thuswe have the following matrix-description for complex Grassmannian manifolds.Gr( m, n ) = { [ A ] } := { A | A ∈ M ( m + n, m ) , rank( A ) = m } / ∼ , where A ∼ B if and only if there exists Q ∈ GL( m, C ) such that A = BQ .With the notation given above, the local coordinate charts of Gr( m, n ) can be describedin the following manner. Let I = ( i , . . . , i m ) be an increasing integer sequence such that1 ≤ i < i < . . . < i m ≤ m + n . For A ∈ M ( m + n, m ), denote by A I ∈ M ( m, m ) the I -minorof A consisting of its i , . . . , i m rows, i.e., if A := ( β , . . . , β m + n ) t , then A I := ( β i , . . . , β i m ) t .Here “ t ” denotes the transpose of a matrix. Then U I := { [ A ] ∈ Gr( m, n ) , A I ∈ GL( m, C ) } . Clearly this definition is independent of the choice of A in the coset as A = BQ implies that A I = B I Q and thus U I is well-defined. Note that inside each coset [ A ] ∈ U I there contains aunique matrix representative such that its I -minor is the identity matrix. Indeed, arbitarilychoose A ∈ [ A ], A · ( A I ) − is the desired representative. In this case, the resulting mn entriesin the matrix A · ( A I ) − can be used to be the local coordinates of [ A ]. More precisely,if we define I c to be the increasing integer sequence complemenatry to I with respect to { , . . . , m + n } , i.e., I c = { , . . . , m + n } − I, then ϕ I : U I ∼ = −→ M ( n, m ) ∼ = C mn , [ A ] A I c · ( A I ) − . (4.2)Note that the roles played by A I c and A I in (4.2) are the same as those of ( z , . . . , b z i , . . . , z n +1 )and z i in (4.1). OMBINATORIAL IDENTITIES AND CHERN NUMBERS 9 With the construction of the local coordinate charts for complex Grassmannians in Example4.2 in mind, we can now proceed to the general complex flag manifolds F ( m , . . . , m r , m r +1 ).We still use the notation and symbols introduced in Section 1. A flag ( L , . . . , L r ) ∈ F ( m , . . . , m r , m r +1 ) can be represented by a matrix A = ( A , A , . . . , A r ) ∈ M ( N, P rj =1 m j )with A j ∈ M ( N, m j ) and rank( A ) = P rj =1 m j such that the column vectors of the matrix( A , . . . , A i ) form a basis of the linear subspace L i . The following lemma tells us that underwhat conditions two matrices represent the same flag. Lemma 4.3. Two matrices A = ( A , A , . . . , A r ) , B = ( B , B , . . . , B r ) ∈ M ( N, r X j =1 m j ) with rank ( A ) = rank ( B ) = P rj =1 m j represent the same flag ( L , . . . , L r ) if and only if thereexists a block upper triangular matrix Q ∈ GL ( P rj =1 m j , C ) of the following form (4.3) Q = Q Q · · · Q r Q · · · Q r ... ... . . . ... · · · Q rr , Q ii ∈ GL ( m i , C ) , Q ij ∈ M ( m i , m j ) , such that AQ = B .Proof. The “only if” part. Suppose the matrices A and B represent the same flag ( L , . . . , L r ),which implies that there exist Q ji , e Q ji ∈ M ( m j , m i ) for 1 ≤ j ≤ i ≤ r such that(4.4) B i = i X j =1 A j Q ji , A i = i X j =1 B j e Q ji . It suffices to show that these Q ii are nonsingular. Indeed, we have from (4.4) that(4.5) B i = i X j =1 (cid:2) ( j X k =1 B k e Q kj ) Q ji (cid:3) = B i · ( e Q ii Q ii ) + i − X j =1 (cid:2) B j ( i X k = j e Q jk Q ki ) (cid:3) , Note that { B , . . . , B i } is a basis of L i . Then (4.5) tells us that e Q ii Q ii = E m i , i X k = j e Q jk Q ki = (0) ∈ M ( m j , m i ) , where E m i is the identity matrix of rank m i . This means that these Q ii are nonsingular.The “if” part. Suppose A and B represent two flags ( L , . . . , L r ) and ( e L , . . . , e L r ) respec-tively and satisfy (4.3). Then B i = P ij =1 A j Q ji . Then means that e L i ⊂ L i . Notice, however,that dim C e L i = dim C L i = i X j =1 m j , which tells us that e L i = L i . (cid:3) With Lemma 4.3 in hand we can now describe general complex flag manioflds as follows. F ( m , . . . , m r , m r +1 ) = { [ A ] } := n A = ( A , A , . . . , A r ) (cid:12)(cid:12) A j ∈ M ( N, m j ) , rank( A ) = r X j =1 m j o(cid:14) ∼ , (4.6)where A ∼ B if and only if B = AQ with Q satisfying (4.3).Our next task is to give a description of local coordinate charts in the spirit of (4.2). Notation convention 4.4. Suppose that I = ( I , . . . , I r +1 ) is a decomposition of { , , . . . , N } (see (2.1) for its definition) and A = ( A , . . . , A r ) with A i ∈ M ( N, m i ). A ( j ) i := I j -submatrix of A i consisting of the rows in I j , i.e., if A i = ( β , . . . , β N ) t and I j = ( k , . . . , k m j ) with 1 ≤ k < · · · < k m j ≤ N , then A ( j ) i := ( β k , . . . , β k mj ) t ∈ M ( m j , m i ) . Define A ( I ) := A (1)1 A (1)2 · · · A (1) r A (2)1 A (2)2 · · · A (2) r ... ... . . . ... A ( r +1)1 A ( r +1)2 · · · A ( r +1) r . Note that A ( I ) is nothing but rearranging the rows of A in terms of the data I =( I , . . . , I r +1 ).For a decomposition I = ( I , . . . , I r +1 ) of { , . . . , N } , we define U I := n [( A , . . . , A r )] ∈ F ( m , . . . , m r +1 ) (cid:12)(cid:12) A (1)1 · · · A (1) i ... . . . ... A ( i )1 · · · A ( i ) i are nonsingular for 1 ≤ i ≤ r o . (4.7)This definition is independent of the choice of the matrix representative in the coset as( AQ )( I ) = A I · Q . Our next proposition shows that these U I can be viewed as local co-ordinate charts. Proposition 4.5. (1)(4.8) [ I U I = F ( m , . . . , m r +1 ) . (2) For each matrix representative A = ( A , . . . , A r ) in [ A = ( A , . . . , A r )] ∈ U I , thereexists a unique Q A of the form as that of (4.3) such that ( A · Q A )( I ) = A ( I ) · Q A is OMBINATORIAL IDENTITIES AND CHERN NUMBERS 11 of the following form(4.9) ( A · Q A )( I ) = A ( I ) · Q A = E m · · · ∗ E m · · · ∗ ∗ · · · ∗ ∗ · · · E m r ∗ ∗ · · · ∗ , where E m denotes the identity matrix of rank m . Proof. (1) Suppose A i ∈ M ( N, m i ) for 1 ≤ i ≤ r and rank( A , . . . , A r ) = P ri =1 m i . Then wecan choose m rows, say 1 ≤ k (1)1 < · · · < k (1) m ≤ N , such that these rows of A are linearly independent as rank( A ) = m . We denote by I := ( k (1)1 , . . . , k (1) m ). Byour choice the k (1)1 , . . . , k (1) m rows of the matrix ( A , A ) are also linearly independent.Since rank( A , A ) = m + m , we are able to supplement these m rows with another m rows, say k (2)1 < · · · < k (2) m , such that these m + m rows of ( A , A ) are linearlyindependent. We denote by I := ( k (2)1 , . . . , k (2) m ). We continue to apply this idea toobtain I i := ( k ( i )1 , . . . , k ( i ) m i ) ( i ≤ r ) such that the rows whose indices are containedin I , . . . , I i of the matrix ( A , . . . , A i ) are linearly independent. Denote by I r +1 := { , . . . , N } − S ≤ i ≤ r I i and I := ( I , . . . , I r +1 ). Then [( A , . . . , A r )] ∈ U I for thischosen I . This completes the proof of (4.8).(2) Note that the submatrices Q i , . . . , Q ii in Q A are characterized by the following equa-tions A (1)1 · · · A (1) i ... . . . ... A ( i )1 · · · A ( i ) i Q i Q i ... Q ii = E m i , ≤ i ≤ r, whose existence and uniqueness are then yielded by the invertibility of the matrices( A ( s ) r ) ≤ r,s ≤ i . (cid:3) Proposition 4.5 tells us that each coset [ A = ( A , . . . , A r )] ∈ U I contains a unique matrixrepresentative such that it is of the form of the right hand side of (4.9) after rearranging itsrows in terms of I . Now we can use the bottom left entries of this unique matrix representativeto be the local coordinates. To be more precise, we have ϕ I : U I ∼ = −→ C d , ( d = X ≤ i In this section we construct a holomorphic circle action on the complex flag manifolds F ( m , . . . , m r , m r +1 ), show that it has isolated fixed points, and explicitly determine theweights on these fixed points. Then Theorems 3.1 and 3.3 will yield our main results inSection 2.We arbitrarily choose N = P r +1 j =1 m j mutually distinct integers k , . . . , k N and use them toconstruct the following circle action. S × F ( m , . . . , m r +1 ) ψ −→ F ( m , . . . , m r +1 ) , (cid:0) g, [ A = ( A , . . . , A r )] (cid:1) h g k . . . g k N A i , (5.1)i.e., the action of g ∈ S is given by multiplying the entries of the i -row of the matrix A with g k i . It is clear that ψ is well-defined and gives rise to a circle action on F ( m , . . . , m r +1 ).Define ψ g ([ A ]) := ψ ( g, [ A ]) . By the definition of U I in (4.7) we know that ψ g ( U I ) = U I for any decomposition I . Recallin (4.10) the local coordinate description of U I under ϕ I and let˜ ψ g : C d ∼ = −→ C d ( d = X ≤ i Lemma 5.1. The map ˜ ψ g in ( ) behaves as follows: ˜ ψ g (cid:16) E m · · · B E m · · · B B · · · ... ... . . . ... B r B r · · · E m r B r +1 , B r +1 , · · · B r +1 ,r (cid:17) = E m · · · g k I ) B ( g − k I ) E m · · · g k I ) B ( g − k I ) ( g k I ) B ( g − k I ) · · · ... ... . . . ... ( g k Ir ) B r ( g − k I ) ( g k Ir ) B r ( g − k I ) · · · E m r ( g k Ir +1 ) B r +1 , ( g − k I ) ( g k Ir +1 ) B r +1 , ( g − k I ) · · · ( g k Ir +1 ) B r +1 ,r ( g − k Ir ) . (5.3) Here, as we have done in ( ) , still denote by the bottom left entries of the matrix thecoordinate components of C d . ( ψ g is holomorphic forany g ∈ S . This means that the circle action ψ constructed in ( ) is holomorphic.Proof. Recall the definitions of ϕ I (cid:0) (4.9) and (4.10) (cid:1) and ψ g . It suffices to show that, for[ A ] ∈ U I , if A ( I ) · Q A is equal to the matrix on the left hand side of (5.3), then (cid:0) ψ g ( A ) (cid:1) ( I ) · Q ψ g ( A ) is equal to the matrix on the right hand side of (5.3).First note that (cid:0) ψ g ( A ) (cid:1) ( I ) = (cid:16) g k . . . g k N A (cid:17) ( I ) = ( g k I ) . . . ( g k Ir +1 ) · A ( I ) . Therefore, (cid:0) ψ g ( A ) (cid:1) ( I ) · Q A = ( g k I ) . . . ( g k Ir +1 ) · A ( I ) · Q A = ( g k I ) . . . ( g k Ir +1 ) E m · · · B E m · · · B B · · · B r B r · · · E m r B r +1 , B r +1 , · · · B r +1 ,r = E m · · · g k I ) B ( g − k I ) E m · · · g k I ) B ( g − k I ) ( g k I ) B ( g − k I ) · · · g k Ir ) B r ( g − k I ) ( g k Ir ) B r ( g − k I ) · · · E m r ( g k Ir +1 ) B r +1 , ( g − k I ) ( g k Ir +1 ) B r +1 , ( g − k I ) · · · ( g k Ir +1 ) B r +1 ,r ( g − k Ir ) ( g k I ) . . . ( g k Ir ) . Thus we have established(5.4) (cid:0) ψ g ( A ) (cid:1) ( I ) · h Q A ( g − k I ) . . . ( g − k Ir ) i = the matrix on the right hand side of (5.3) . Note that Q A is block upper triangular and so is the product matrix inside [ · ] on the lefthand side of (5.4). The uniqueness of Q ψ g ( A ) showed in (cid:0) Prop. 4.5, (2) (cid:1) tells us that thisproduct matrix is precisely Q ψ g ( A ) and thus yields the desired proof. (cid:3) With this key lemma in hand, we are now ready to show the following results and completethe proof of our main results in Section 2. Proposition 5.2. (1) The fixed points of the holomorphic circle action are indexed by the decompositions I of { , . . . , N } , say { P I } . More precisely, P I = ϕ − I (0), where 0 denotes the origin of C d .(2) The weights around P I induced by the circle action on the holomorphic tangent spaceto P I are(5.5) a ≤ i In this appendix we apply Theorem 2.3 to work out (2.5) again. Since the calculations for c [ F (1 , , c [ F (1 , , F (1 , , r = 2 , m = m = 1, m = 2 and N = 4. There are 12 decomposi-tions of the set { , , , } : I (1 , := ( { } , { } , { , } ) , I (1 , := ( { } , { } , { , } ) , I , := ( { } , { } , { , } ) ,I (2 , := ( { } , { } , { , } ) , I (2 , := ( { } , { } , { , } ) , I , := ( { } , { } , { , } ) ,I (3 , := ( { } , { } , { , } ) , I (3 , := ( { } , { } , { , } ) , I , := ( { } , { } , { , } ) ,I (4 , := ( { } , { } , { , } ) , I (4 , := ( { } , { } , { , } ) , I , := ( { } , { } , { , } ) . As we have remarked in Remark 2.4, we may assume that x i = i (1 ≤ i ≤ 4) to calculate theChern number. For simplicity, we denote by W ( i,j ) := W I ( i,j ) , e ( i,j ) := e ( W ( i,j ) ) , c i,j ) := c ( W ( i,j ) ) . Then we have (cid:0) W (1 , , e (1 , , c , (cid:1) = (cid:0) { , , , , } , , (cid:1) , (cid:0) W (1 , , e (1 , , c , (cid:1) = (cid:0) { , , , − , } , − , (cid:1) , (cid:0) W (1 , , e (1 , , c , (cid:1) = (cid:0) { , , , − , − } , , (cid:1) , (cid:0) W (2 , , e (2 , , c , (cid:1) = (cid:0) {− , , , , } , − , (cid:1) , (cid:0) W (2 , , e (2 , , c , (cid:1) = (cid:0) { , − , , − , } , , (cid:1) , (cid:0) W (2 , , e (2 , , c , (cid:1) = (cid:0) { , − , , − , − } , − , − (cid:1) , (cid:0) W (3 , , e (3 , , c , (cid:1) = (cid:0) {− , − , , , } , , (cid:1) , (cid:0) W (3 , , e (3 , , c , (cid:1) = (cid:0) {− , − , , − , } , − , − (cid:1) , (cid:0) W (3 , , e (3 , , c , (cid:1) = (cid:0) { , − , − , − , − } , , − (cid:1) , (cid:0) W (4 , , e (4 , , c , (cid:1) = (cid:0) {− , − , − , , } , , (cid:1) , (cid:0) W (4 , , e (4 , , c , (cid:1) = (cid:0) {− , − , − , − , } , , − (cid:1) , (cid:0) W (4 , , e (4 , , c , (cid:1) = (cid:0) {− , − , − , − , − } , − , (cid:1) . Therefore, Theorem 2.3 tells us that c [ F (1 , , X ≤ i = j ≤ c i,j ) e ( i,j ) = 12 · · · + ( − − . We also note that in this case X ≤ i = j ≤ c i,j ) e ( i,j ) = 12 · · · + ( − − , which is consistent with the first equality in (2.4). References [AS68] M.F. Atiyah, I.M. Singer: The index theory of elliptic operators: III, Ann. Math. (1968), 546-604.[Bor54] A. Borel: K¨ahlerian coset spaces of semisimple Lie groups , Proc. Nat. Acad. Sci. U. S. A. (1954),1147-1151.[BH58] A. Borel, F. Hirzebruch: Characteristic classes and homogeneous spaces. I, Amer. J. Math. (1958),458-538.[BH59] A. Borel, F. Hirzebruch: Characteristic classes and homogeneous spaces. II, Amer. J. Math. (1959),315-382.[Bo67] R. Bott: Vector fields and characteristic numbers , Michigan Math. J. (1967), 231-244.[Fu97] W. Fulton: Young tableaux , Cambridge University Press, (1997).[Fut84] A. Futaki: An obstruction to the existence of Einstein K¨ahler metrics , Invent. Math. (1983),437-443.[FM84] A. Futaki, S. Morita: Invariant polynomials on compact complex manifolds , Proc. Japan Acad. (1984), 369-372.[FM85] A. Futaki, S. Morita: Invariant polynomials of the automorphism group of a compact complex mani-fold , J. Differential Geom. (1985), 135-142.[GH78] P. Griffiths, J. Harris: Principles of algebraic geometry , Pure and Applied Mathematics, Wiley, NewYork, 1978.[Hi05] F. Hirzebruch: The projective tangent bundles of a complex three-fold , Pure Appl. Math. Q. (2005),441-448.[Ko55] J.L. Koszul: Sur la forme hermitienne canonique des espaces homog`enes complexes , Canad. J. Math. (1955), 562-576.[KT09] D. Kotschick, S. Terzi´c: Chern numbers and the geometry of partial flag manifolds , Comment. Math.Helv. (2009), 587-616.[Li12] P. Li: Circle action, lower bound of fixed points and characteristic numbers , J. Fixed Point TheoryAppl. (2012), 245-251.[Li13-1] P. Li: An application of the rigidity of Dolbeault-type operators , Math. Res. Letters, (2013), 81-89. OMBINATORIAL IDENTITIES AND CHERN NUMBERS 17 [Li13-2] P. Li: Remarks on Bott residue formula and Futaki-Morita integral invariants , Topology Appl. (2013), 488-497.[Li14] P. Li: On the vanishing of characteristic numbers , Homology, Homotopy Appl. (2014), 185-204.[LL11] P. Li, K.-F. Liu: Some remarks on circle action on manifolds , Math. Res. Letters, (2011), 437-446.[LL13] P. Li, K.-F. Liu: On an algebraic formula and applications to group action on manifolds , Asian J.Math. (2013), 383-390.[Lu63] Q.-K. Lu: The Classical Manifolds and Classical Domains , Scientific and Technical Publisher, Shang-hai, 1963. (Chinese).[Man01] L. Manivel: Symmetric functions, Schubert polynomials and degeneracy loci , SMF/AMS Texts andMonographs, Volume 6, SMF/AMS, 2001.[Mat57] Y. Matsushima: Sur la structure du groupe d’homomorphismes analytiques d’une certaine varitk¨ahlrienne , Nagoya Math. J. (1957), 145-150.[MS74] J.W. Milnor, J.D. Stasheff: Characteristic classes , Princeton University Press, Princeton, N. J., 1974.Annals of Mathematics Studies, No. 76.[Wa54] H.C. Wang: Closed manifolds with homogeneous complex structure , Amer. J. Math. (1954), 1-32. School of Mathematical Sciences, Tongji University, Shanghai 200092, China E-mail address ::