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Probabilistic Schubert calculus

  • Peter Bürgisser and Antonio Lerario ORCID logo


We initiate the study of average intersection theory in real Grassmannians. We define the expected degreeedegG(k,n) of the real Grassmannian G(k,n) as the average number of real k-planes meeting nontrivially k(n-k) random subspaces of n, all of dimension n-k, where these subspaces are sampled uniformly and independently from G(n-k,n). We express edegG(k,n) in terms of the volume of an invariant convex body in the tangent space to the Grassmannian, and prove that for fixed k2 and n,


where degG(k,n) denotes the degree of the corresponding complex Grassmannian and εk is monotonically decreasing with limkεk=1. In the case of the Grassmannian of lines, we prove the finer asymptotic


The expected degree turns out to be the key quantity governing questions of the random enumerative geometry of flats. We associate with a semialgebraic set XPn-1 of dimension n-k-1 its Chow hypersurface Z(X)G(k,n), consisting of the k-planes A in n whose projectivization intersects X. Denoting N:=k(n-k), we show that


where each Xi is of dimension m=n-k-1, the expectation is taken with respect to independent uniformly distributed g1,,gmO(n) and |Xi| denotes the m-dimensional volume of Xi.

Award Identifier / Grant number: BU 1371/2-2

Funding statement: The first author was partially supported by DFG grant BU 1371/2-2.

A Appendix

A.1 Proof of Theorem 3.2

Since the density of A is invariant under the orthogonal group O(n), we can assume that B is spanned by the first l standard vectors, i.e.,


where 1 is the × identity matrix, A1 is an ×k matrix and A2 is an (n-)×k matrix. (Again, abusing notation, we denote by A and B also matrices whose columns span the corresponding spaces.) If we sample A with i.i.d. normal Gaussians, the corresponding probability distribution for the span of its columns is O(n) invariant, and consequently it coincides with the uniform distribution. In order to compute the principal angles between A and B using (3.1), we need to orthonormalize the columns of A. Defining


we see that the span of the columns of A and A^ is the same, and the columns of A^ are orthonormal. The cosines of the principal angles between A and B are the singular values 1σ1σk0 of the matrix


The σ1,,σk coincide with the square roots of the eigenvalues 1u1uk0 of the positive semidefinite matrix


which are the same as the eigenvalues of N=(A1TA1+A2TA2)-1A1TA1 (eigenvalues are invariant under cyclic permutations). Consider the Cholesky decomposition


Then the eigenvalues of N equal the eigenvalues of


We use now some facts about the multivariate Beta distribution (see [37, Section 3.3]). By its definition, the matrix U has a Betak(12l,12(n-l)) distribution, and [37, Theorem 3.3.4] states that the joint density of the eigenvalues of U is given by


Recall now that ui=(cosθi)2 for i=1,,k, which implies the change of variable


and thus (A.1) becomes the stated density in (3.2).

A.2 Proof of Lemma 4.3

We begin with a general reasoning. Assume that Ae(B). The unit normal vectors of e(B) at A, up to a sign, are uniquely determined by A. Lemma 4.1 provides an explicit description for them as follows. Let a1,a2,,ak and a1,b2,,bn-k be the orthonormal bases given by Lemma 3.1 for A and B, respectively; note that dim(AB)=1 since Ae(B). In particular, ai,bi=cosθi for ik, where θ1θk are the principal angles between A and B. Let f=(A+B) with f=1. According to Lemma 4.1, the unit vector ν:=fa2ak spans the normal space of e(B) at A. (We use here the representation of elements of G(k,n) and its tangent spaces by vectors in Λkn; cf. Section 2.1.)

Fix now BG(n-k,n) and recall that the functions θ1,θ2:G(k,n) give the smallest and second smallest principal angle, respectively, between AG(k,n) and B. Let 0<εδ<π2 and put


We first prove that 𝒯(e(B)δ,ε)T.

For Ae(B) we consider the curve NA(t):=(a1cost+fsint)a2ak for t. We note that NA(t) arises from A by a rotation with the angle t in the (oriented) plane spanned by a1 and f, fixing the vectors in the orthogonal complement spanned by a2,,ak. We have NA(0)=A and N˙A(0)=ν. It is well known (see, e.g., [13]) that NA(t) is the geodesic through A with speed vector ν, that is, NA(t)=expA(tν) and NA(0)=A. From the definition of the ε-tube, we therefore have


Since both a1 and f are orthogonal to b2,,bn-k, the principal angles between NA(t) and B are |t|,θ2,,θk, where aj,bj=cosθj. By our assumption Ae(B)δ, we have δθ2θk. Hence we see that |t| is the smallest principle angle between NA(t) and B if |t|δ. We have thus verified that 𝒯(e(B)δ,ε)T.

For the other inclusion, let CT; assume that Ce(B)δ, otherwise we clearly have C𝒯(e(B)δ,ε). Let (c1,,ck) and (b1,,bn-k) be orthonormal bases of C and B, respectively, as provided by Lemma 3.1. So we have bj,cj=cosθj(C) for all j, where θ1(C)ε and δθ2(C)θk(C) by the assumption CT. In particular, b1 is orthogonal to c2,,ck and b1,c1 are linearly independent. We define AG(k,n) as the space spanned by b1,c2,,ck. By construction, θ1(A)=0 and θj(A)=θj(C) for j2. Hence, Ae(B)δ. Let NA(t)G(k,n) be defined as above as the space resulting from A by a rotation with the angle t in the oriented plane spanned by b1 and c1. By construction, we have C=NA(τ), where τ=θ1(C)ε. Therefore, we indeed have C𝒯(e(B)δ,ε) by (A.2).

A.3 Proof of Lemma 4.5

More generally, we consider a semialgebraic set YPn-1 of dimension dn-k-1. Consider the semialgebraic set


together with the projections on the two factors: π1:C(Y)G(k,n) and π2:B(Y)Y. As Z(Y)=π1(C(Y)), the set Z(Y) is semialgebraic. Note that C(Y) is compact if Y compact, and C(Y) is connected if Y is connected (since π1 is continuous). In order to determine the dimension of C(Y), we note that the fiber π2-1(y) over yY is isomorphic to G(k-1,n-1). As a consequence, we get


The fibers π1-1(A) over AZ(Y) consist of exactly one point, except for the A lying in the exceptional set

Z(2)(Y):={AZ(Y)(A)Y consists of at least two points}.

Note that (A)Y consists of one point only, for all AZ(Y)Z(2)(Y).

In order to show that dimZ(2)(Y)<dimZ(Y), we consider the semialgebraic set


with the corresponding projections π3:C(2)(Y)G(k,n) and π4:C(2)(Y)Y×Y. Note that Z(2)(Y)=π3(C(2)(Y)). For (y1,y2)Y×Y such that y1y2, we have


since the fibers of π4 are isomorphic to G(k-2,n-2). Therefore,


and we see that dim(C(Y)C(2)(Y))=dimC(Y). For the projection f:C(2)(Y)C(Y) defined by f(A,y1,y2):=(A,y1), we have π1-1(Z(2)(Y))=f(C(2)(Y)), hence


Moreover, the projection C(Y)C(2)(Y)Z(Y)Z(2)(Y), (A,y)y is bijective, hence


Using dimZ(Y)dimC(Y), we see that




In the special case where dimX=n-k-1, we conclude that Z(X) is a hypersurface.

We consider now the following set of “bad” AZ(X):


We claim that this semialgebraic set has dimension strictly less than Z(X). This follows for Z(2)(X) from (A.6), for Z(SingX) from (A.5) applied to Y=Sing(X), for π1(Sing(C(X)) from Proposition 2.2 and (A.5), and finally for Sing(Z(X)) by Proposition 2.2.

Thus generic points AZ(X) are not in S(X) and hence satisfy the following:

  1. The intersection (A)X consists of one point only (let us denote this point by p),

  2. The point p is a smooth point of X,

  3. The point A is a regular point of Z(X),

  4. The point (A,p) is a regular point of C(X).

It remains to prove that for every AZ(X)S(X) we have (4.3). To this end, let us take (A,p)C(X) with AS(X). We work in local coordinates (w,y)N×n-1 on a neighborhood U of (A,p) in G(k,n)×Pn-1, where N=k(n-k). For simplicity we center the coordinates on the origin, so that (0,0) are the coordinates of (A,p). In this coordinates, the set C(X)U can be described as


where F(w,x)=0 represents the reduced local equations describing the condition x(W) and G(x)=0 the reduced local equations giving the condition xX. Since (A,p) is a regular point of C(X), the tangent space of C(X) at (A,p) is described by


Note that p is a smooth point of X, hence (D0G)x˙=0 is the equation for the tangent space to X at p. On the other hand, since Z(X)=π1(C(X)), we have


Let now Bn be the linear space corresponding to TpX and let us write the equations for C(B) in the same coordinates as above (by construction we have (A,p)C(B)):


Note that the same equation F(w,x)=0 as in (A.7) appears here (recall that this is the equation describing x(W)), but now G=0 is replaced with its linearization at zero. In particular,


which coincides with (A.8). Since Ω(B)=π1(C(B)), this finally implies


which finishes the proof.

A.4 Proof of Proposition 5.12

We extend the function f to k×m{0} by setting


denoting it by the same symbol. Similarly, we extend g by setting


We assume now that Xk×m has i.i.d. standard Gaussian entries. Then we can write


where pSVD denotes the joint density of the ordered singular values of X. This density can be derived as follows. The joint density of the ordered eigenvalues λ1>>λk>0 of the Wishart distributed matrix XXT is known to be [37, Corollary 3.2.19]




From this, using λi(XXT)=σi(X)2 and the change of variable dλi=2σidσi, we obtain


As a consequence, we obtain


where in the second line we have switched to polar coordinates σ=rθ with θSk-1 and r0. Note that the power of the r-variable arises as




we obtain


It is immediate to verify that


which completes the proof.

A.5 Generalized Poincaré’s formula in homogeneous spaces

The purpose of this subsection is to prove the kinematic formula in homogeneous spaces for multiple intersections and to derive Theorem 3.19. The proofs are similar to [26], to which we refer the reader for more details.

A.5.1 Definitions and statement of the theorem

In the following, G denotes a compact Lie group with a left and right invariant Riemannian metric.[2] See [6] for background on Lie groups. We denote by eG the identity element and by Lg:GG,xgx the left translation by gG. The derivatives of Lg will be denoted by g*:TeGTgG. By assumption, this map is isometric.

In (3.5) we defined a quantity for capturing the relative position of linear subspaces of a Euclidean vector space. We can extend this notion to linear subspaces ViTgiG in tangent spaces of G at any points g1,,gmG, assuming idimVidimG. This is done by left-translating the gi to the identity e: so we define


Let now KG be a closed Lie subgroup and denote by p:GG/K the quotient map. We endow K with the Riemannian structure induced by its inclusion in G, and G/K with the Riemannian structure defined by declaring p to be a Riemannian submersion. For example, when G=O(n) with the invariant metric defined in Section 2.8 and K=O(k)×O(n-k), then G/K with the quotient metric is isometric to the Grassmannian G(k,n) with the metric defined in Section 2.1.

Note that G acts naturally by isometries on G/K; if gG and yG/K, we denote by gy the result of the action. Further, we denote by y0=p(e) the projection of the identity element. The multiplication with an element kK fixes the point y0; as a consequence, the differential of k induces a map denoted k*:Ty0G/KTy0G/K, so that we have an induced action of K on Ty0G/K.

Given a submanifold X of a Riemannian manifold M, we denote by NX its normal bundle in M (i.e., for all xX the vector space NxX is the orthogonal complement to TxX in TxM). Also, the restriction of the Riemannian metric of M to X allows to define a volume density on X; if f:X is an integrable function, we denote its integral with respect to this density by Xf(x)𝑑x.

Definition A.1.

For given submanifolds Y1,,YmG/K, we define the function


as follows. For (y1,,ym)Y1××Ym let ξiG be such that ξiyi=y0 for all i. We define


where the expectation is taken over a uniform (k1,,km)K××K.

The reader should compare this definition with [26, Definition 3.3], which is just a special case. The main result of this section is the following generalization of Poincaré’s kinematic formula for homogeneous spaces, as stated in [26, Theorem 3.8] for the intersection of two manifolds. We provide a proof, since the more general result is crucial for our work and we were unable to find it in the literature.

Theorem A.2.

Let Y1,,Ym be submanifolds of G/K such that


Then, for almost all (g1,,gm)Gm, the manifolds g1Y1,,gmYm intersect transversely, and


where the expectation is taken over a uniform (g1,,gm)G××G.

We note that when G acts transitively on the tangent spaces to each Yi (formally defined as in Definition 3.4), then the function σK:Y1××Ym introduced in Definition A.1 is constant. As a consequence we obtain:

Corollary A.3.

Under the assumptions of Theorem A.2, if moreover G acts transitively on the tangent spaces to Yi for i=1,,m, then we have


where (y1,,ym) is any point of Y1××Ym.

Let us look now at the special case G=O(n) and K=O(k)×O(n-k). If Y1,,Ym are coisotropic hypersurfaces of G(k,n), then G acts transitively on their tangent spaces by Proposition 4.6. Moreover, when m=k(n-k), it is easy to check that the constant value of σK equals the real average scaling factor α(k,n-k) defined in Definition 3.18. Hence in this case, the statement of Corollary A.3 coincides with the statement of Theorem 3.19.

A.5.2 The kinematic formula in G

As before, G denotes a compact Lie group. We derive first Theorem A.2 in the special case K={e}, which is the following result (we can without loss of generality assume gm=e).

Lemma A.4.

Let X1,,Xm be submanifolds of G such that




In the special case of intersecting two submanifolds, this is an immediate consequence of the following “basic integral formula” from [26, Section 2.7] (take h=1).

Proposition A.5.

Let M1,M2 be submanifolds of G such that


For almost all gG, the manifolds M1 and gM2 intersect transversely, and if h is an integrable function on M1×M2, then


where φg:gM1M2M1×M2 is the function given by φg(y):=(g-1y,y).

In order to reduce the general case to that of intersecting two submanifolds, we first establish a linear algebra identity.

Lemma A.6.

For subspaces V1,,V,W,Z of a Euclidean vector space, we have



We may assume that WZ=0, since otherwise both sides of the identity are zero. Let us denote by (vi,1,,vi,di), (w1,,wa) and (z1,,zb) orthonormal bases of Vi, W and Z, respectively. Moreover, we denote by (w1,,wa,z~1,,z~b) an orthonormal basis for Z+W obtained by completing (w1,,wa). By definition we have


On the other hand, since W+Z=span{w1,,wa,z1,,zb}, we have


By substituting the last line into (A.9) and recalling the definition of σ(V1,,V,W,Z) from equation (3.5), the assertion follows. ∎

Corollary A.7.

Let M1,M2 be submanifolds of G and let V1,,V be linear subspaces of tangent spaces of G (possibly at different points) such that


Then we have



We apply Proposition A.5 with the function h:M1×M2 defined by


When gM1 and M2 intersect transversely, we have Ny(gM1M2)=Ny(gM1)+Ny(M2), which implies h(φg(y))=σ(V1,,V,Ny(gM1gM2)) for ygM1M2. Hence we obtain with Proposition A.5,


where the last equality is due to Lemma A.6. ∎

Proof of Lemma A.4.

Recall that we already established Lemma A.4 in the case m=2 as a consequence of Proposition A.5. Let now m3 and abbreviate Y:=g2X2Z, where Z:=g3X3gm-1Xm-1Xm. Then we have


where we first applied Lemma A.4 in the case m=2, then interchanged the order of integration, and after that used Corollary A.7. Proceeding analogously, we see that the above integral indeed equals


which completes the proof. ∎

Proof of Theorem A.2.

We consider Xi:=p-1(Yi), which is a submanifold, since the projection p:GG/K is a submersion, cf. [9, Theorem A.15]. Moreover, g1X1,,gmXm intersect transversally if Y1,,Ym do so. We can rewrite the integral in the statement as


For justifying the last equality, note that the coarea formula [9, Theorem 17.8] yields


for any submanifold Y of G/K, since p is a Riemannian submersion. Applying Lemma A.4 to the integral in the last line, we obtain


The projection P:X1××XmY1××Ym defined by


is a Riemannian submersion with fibers isometric to Km. Using the coarea formula


where the last equality is due to Definition A.1. (Note that here is where we use the right invariance of the metric under the action of the elements in K, as one can verify by a careful inspection of the last steps. The reader can see the proof of [26, Theorem 3.8], which is almost identical and where all the details of the calculation are shown). This completes the proof. ∎


We are very grateful to Frank Sottile who originally suggested this line of research. We thank Paul Breiding, Kathlén Kohn, Chris Peterson, and Bernd Sturmfels for discussions. We also thank the anonymous referees for their comments.


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Received: 2017-04-23
Revised: 2018-01-13
Published Online: 2018-05-03
Published in Print: 2020-03-01

© 2021 Walter de Gruyter GmbH, Berlin/Boston

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