Equidistribution and freeness on Grassmannians
aa r X i v : . [ m a t h . N T ] F e b EQUIDISTRIBUTION AND FREENESS ON GRASSMANNIANS
TIM BROWNING, TAL HORESH, AND FLORIAN WILSCH
Abstract.
We associate a certain tensor product lattice to any primitive integerlattice and ask about its typical shape. These lattices are related to the tangentbundle of Grassmannians and their study is motivated by Peyre’s programme on“freeness” for rational points of bounded height on Fano varieties.
Contents
1. Introduction 12. Basic facts about lattices 53. Tangent lattices for Grassmannians 84. Free rational points dominate 125. Tangent lattices equidistribute 146. Consequences of equidistribution 21References 231.
Introduction
Understanding the density of rational points on smooth Fano varieties lies at theconfluence of algebraic geometry, harmonic analysis and analytic number theory.The guiding conjecture is due to Manin [9], and its refinement by Peyre [12].Given a smooth Fano variety V defined over a number field k such that V ( k ) isZariski dense in V , and given an anticanonical height function H : V ( k ) → R , it isconjectured that there exists a thin set of rational points Z ⊂ V ( k ) and an explicitconstant c V,H > { x ∈ V ( k ) \ Z : H ( x ) B } ∼ c V,H B (log B ) ̺ ( V ) − , as B → ∞ , (1.1)where ̺ ( V ) = rank Pic( V ). (Here, as proposed by Serre [18, § Z ⊂ V ( k ) is said to be thin if it is a finite union of subsets which are eithercontained in a proper closed subvariety of V , or contained in some π ( Y ( k )) where π : Y → V is a generically finite dominant morphism, with deg( π ) > Y irreducible.)Despite major progress by Lehmann, Sengupta and Tanimoto [11] on identifyingproblematic thin sets for Fano varieties, they can be hard to work with in general. Date : February 24, 2021.2010
Mathematics Subject Classification.
TIM BROWNING, TAL HORESH, AND FLORIAN WILSCH
An alternative path based on a notion of “freeness” has surfaced in recent workof Peyre [13]. It posits the idea that the distribution of “sufficiently free” rationalpoints of bounded height on smooth Fano varieties should conform to the asymp-totic behaviour in (1.1), without the need to first identify appropriate thin sets ofrational points. This conjecture has been confirmed for smooth hypersurfaces over Q of low degree by Browning and Sawin [4]. However, at the same time it has alsobeen rendered incapable of neatly dealing with all Fano varieties by Sawin [14].Indeed, Sawin shows that Peyre’s proposal fails for the Hilbert scheme Hilb ( P n ),since the obvious thin subset of rational points (consisting of points that lift to acertain double cover) contains many points with relatively large freeness.The primary aim of this paper is to study the notion of freeness in the classicalsetting of Grassmannian varieties over Q . Not only shall we be able to show thatPeyre’s freeness conjecture holds true for Grassmannians, but we shall even beable to prove a natural equidistribution statement for certain tangent lattices thatemerge in the definition of freeness.For any integers 1 m n −
1, there is a natural Q -scheme Gr( m, n ) whose Q -rational points Gr( m, n )( Q ) coincide with m -dimensional linear subspaces of Q n .The Grassmannian is a smooth Q -variety of dimension m ( n − m ), admitting aPl¨ucker embedding ι : Gr( m, n ) → P ( mn ) − Q . The Picard group Pic(Gr( m, n )) is isomorphic to Z and it is generated by thedivisor L = ι ∗ O (1). The anticanonical bundle is ω − = L ⊗ n . This confirmsthat Gr( m, n ) is Fano and allows one to define an anticanonical height function H = H ω − on Gr( m, n )( Q ). On setting Λ = V ∩ Z n for a linear subspace V , arational point x ∈ Gr( m, n )( Q ) is the same thing as a primitive lattice Λ ⊂ Z n ofrank m . Then H ( x ) = covol(Λ) n , where covol(Λ) is the covolume of the lattice.Schmidt [15] has shown that { x ∈ Gr( m, n )( Q ) : H ( x ) B } = c m,n B + O (cid:16) B − max { mn , n − m ) n } (cid:17) , (1.2)for any 1 m n −
1, where c m,n = 1 n (cid:18) nm (cid:19) V ( n ) V ( n − . . . V ( n − m + 1) V (1) V (2) . . . V ( m ) ζ (2) . . . ζ ( m ) ζ ( n ) . . . ζ ( n − m + 1) (1.3)and V ( N ) = vol { x ∈ R N : k x k } for any N ∈ N . This agrees with theconjectured asymptotic formula (1.1), with Z = ∅ . Moreover, Peyre [12, §
6] hasshown that the leading constant c m,n agrees with his prediction. In fact, Franke,Manin and Tschinkel [9] have a far-reaching generalisation of Schmidt’s result toarbitrary Flag varieties over arbitrary number fields and Peyre has confirmed hisconjectured constant for this more general class.In [13, D´ef. 4.11], Peyre defines a freeness function ℓ ( x ) on points x ∈ V ( Q ),for any smooth Fano Q -variety. This function takes values in [0 ,
1] and its precisedefinition will be recalled in (2.7). When V = Gr( m, n ) is the Grassmannian it QUIDISTRIBUTION AND FREENESS ON GRASSMANNIANS 3 admits a particularly succinct description. Associated to x ∈ Gr( m, n )( Q ) is thetensor product lattice T Λ = Λ ∗ ⊗ Z Λ π , where Λ ⊂ Z n is the primitive rank m lattice associated to x , with Λ ∗ being the dual lattice and Λ π = (Λ ⊥ ) ∗ the factor lattice . (The precise definitions of these willbe recalled in Section 2.) The lattice T Λ has rank m ( n − m ) and will henceforth bereferred to as the tangent lattice . The freeness function ℓ ( x ) measures the extentto which T Λ is lopsided, with a small value of ℓ ( x ) corresponding to the existenceof an unusually large largest successive minimum.Suppose for the moment that m = 1, so that Gr(1 , n ) = P n − Q . In this case itfollows from Peyre [13, Prop. 7.1] that ℓ ( x ) > n − n (1.4)for rational points x ∈ P n − Q ( Q ). The same bound applies to the case m = n − m , it turns out that the freeness functioncan become arbitrarily small as one goes over rational points on the GrassmannianGr( m, n ). In fact we have the following result. Theorem 1.1.
Let m, n be integers such that < m < n − . Then there existinfinitely many x ∈ Gr( m, n )( Q ) such that ℓ ( x ) = 0 . Let 1 k < m . Suppose we fix a k -dimensional linear subspace L of Q n . Thenamong the linear subspaces parameterised by Gr( m, n )( Q ) is the set V L of thosethat contain L . Any point in V L whose height is large compared to that of L willhave low freeness. The example constructed to prove Theorem 1.1 is of exactly thissort. Although all points of low freeness have this kind of special structure, theydo not form a thin subset. Theorem 1.2.
Let m, n be integers such that < m < n − . Then, for all ε > ,the set Ω ε = { x ∈ Gr( m, n )( Q ) : ℓ ( x ) < ε } of non- ε -free points is not thin. For a suitable range of ε >
0, Peyre suggests focusing on the restricted countingfunction which only counts ε -free points , meaning those points x ∈ V ( Q ) for which ℓ ( x ) > ε . For suitable Fano varieties it is expected that such a restriction capturesthe behaviour articulated in the Manin–Peyre prediction. The following resultconfirms this for Grassmannians. Theorem 1.3.
For any integers m n − and any ε < , we have { x ∈ Gr( m, n )( Q ) : H ( x ) B, ℓ ( x ) > ε } = c m,n B + o ( B ) , as B → ∞ , where c m,n is given by (1.3) . TIM BROWNING, TAL HORESH, AND FLORIAN WILSCH
We note that Theorem 1.3 is a direct consequence of (1.2) when m = 1 or m = n −
1, at least if ε nn − . We shall give two distinct proofs of Theorem 1.3.The first is based on adapting the geometry of numbers arguments developed bySchmidt [15] and actually leads to the power saving B − − εm ( n − m ) in the error term.This is the object of Section 4. The second proof occurs in Section 6, but it doesn’tgive a power saving. It is based on a general equidistribution statement that willbe discussed in Section 5.Using this equidistribution statement, we can also address a question raised byPeyre [13, §
9] concerning an alternative counting function, in which one ordersrational points by the maximal slope µ max ( T Λ ) of their tangent lattice, as definedin Section 2, instead of by height. In the same spirit as counting sufficiently freepoints, this approach excludes most points whose tangent lattice is very lopsided,since such points lead to tangent lattices whose maximal slope is large comparedto the height. We shall prove the following result in Section 6. Theorem 1.4.
For any integers m n − , we have { Λ ∈ Gr( m, n )( Q ) : µ max ( T Λ ) log B } = c ′ m,n B m ( n − m ) + o (cid:0) B m ( n − m ) (cid:1) , where < c ′ m,n < c m,n is given in (6.2) . In particular, this confirms the expectation expressed in [13, Rem. 7.8] for pro-jective space. This alternative counting function behaves quite differently from thestandard counting function with respect to an anticanonical height function. Inparticular, it is compatible with products by [13, Rem. 7.21(b)]. Thus it followsfrom Theorem 1.4 that { x ∈ X ( Q ) : µ max ( T x ) log B } = c X B dim X + o (cid:0) B dim X (cid:1) , for a suitable constant c X >
0, when X is a product of Grassmannians. A strikingfeature of this example is that the rank of Pic( X ), which appears in (1.1), canbecome arbitrarily large.Returning to his asymptotic formula (1.2), Schmidt [16, 17] also proved variouscounting statements about lattices drawn from the set L m,n = { Λ ⊂ R n is a lattice of rank m } . (1.5)This is a homogeneous space for the group SL n ( R ) and carries an SL n ( R )-invariantmeasure that is unique up to a multiplication by a positive scalar. The space L m,n has infinite volume with respect to this measure, but it naturally projects onto thefinite volume space of unimodular lattices U L m,n = { Λ ∈ L m,n : covol(Λ) = 1 } , to which the volume form on L m,n restricts. Schmidt focuses on proving equidis-tribution statements about the projections of the lattices in L m,n to the spaceof similarity classes of rank m lattices. However, as worked out by Horesh andKarasik [10, Thm. I (3)], it is also possible to show that primitive lattices in L m,n QUIDISTRIBUTION AND FREENESS ON GRASSMANNIANS 5 equidistribute in
U L m,n , as covol(Λ) → ∞ . Theorem 1.3 tells us that among therational points x ∈ Gr( m, n )( Q ), it is rare to find points whose associated tangentlattice T Λ = Λ ∗ ⊗ Z Λ π is very lopsided. Our final result is an equidistributionstatement about the set of tangent lattices { T Λ : Λ ∈ Gr( m, n ) } , as H (Λ) → ∞ .Tangent lattices are lattices of rank m ( n − m ) in R n ⊗ R n , and so belong tothe SL n ( R )-homgeneous space L m ( n − m ) ,n . Inspired by our discussion above, wemight aim for an equidistribution statement about the projections of elements oftangent lattices to the space of unimodular lattices U L m ( n − m ) ,n , with respect tothe natural probability measure induced by the an SL n ( R )-invariant measure on L m ( n − m ) ,n . However, tangent lattices lie inside the submanifold G m,n = { Λ ⊗ Λ ⊥ : Λ ∈ L m,n } ⊂ L m ( n − m ) ,n , on which there is no obvious SL n ( R )-action. (Accordingly, the projections lieinside the submanifold U G m,n = G m,n ∩ U L m ( n − m ) ,n of U L m ( n − m ) ,n , on whichthere is no obvious SL n ( R )-action.)In Section 5 we shall construct a natural probability measure on U G m,n , whichwe denote by ν , paving the way to a proof of the following equidistribution result. Theorem 1.5.
With respect to ν , the projections of the set { T Λ : Λ ∈ Gr( m, n )( Q ) } to U L m ( n − m ) ,n equidistribute in the manifold U G m,n , as H (Λ) → ∞ , with rateof convergence O ( H (Λ) − n ) . The explicit equidistribution statement is given below in Theorem 5.7 with avariant phrased in terms of probability measures in Theorem 5.8. The proof takesplace in Section 5 and involves a reduction to an equidistribution statement aboutpairs of lattices (Λ , Λ π ), for primitive Λ ∈ L m,n , that appears in recent work ofHoresh and Karasik [10, Thm. 1 (4)]. Acknowledgements.
The authors are very grateful to Will Sawin for useful remarksabout this topic. While working on this paper the first two authors were supportedby EPSRC grant
EP/P026710/1 , and the first and last authors by FWF grantP 32428-N35. 2.
Basic facts about lattices
Recall that a lattice Λ is a discrete subgroup of R N , or more generally, of an N -dimensional real vector space equipped with an inner product. There existsan integer r N and linearly independent vectors b , . . . , b r ∈ R N such thatΛ = span Z { b , . . . b r } . We then say that the rank of Λ is rank(Λ) = r and call Λ a full lattice if r = N . The covolume is covol(Λ) = p det( B t B ) = p det( h b i , b j i ) i,j ,where the basis matrix B is the N × r matrix formed from the column vectors b , . . . , b r . This definition is independent of the choice of basis. We say that alattice Λ is unimodular if covol(Λ) = 1. It will be convenient to denote by Λ R thesubspace span R (Λ) that a lattice Λ generates. TIM BROWNING, TAL HORESH, AND FLORIAN WILSCH
A sublattice Λ of a lattice Γ is said to be primitive in
Γ if there is no othersublattice Λ ′ ⊂ Γ of the same rank which properly contains Λ. A lattice Λ ⊂ R N is said to be integral if it is contained in Z N . We say that an integral lattice isprimitive if it is primitive in Z N . Note that an integral lattice Λ is primitive if andonly if Λ R ∩ Z N = Λ.In this section we collect together some facts about lattices, most of which aretaken from the book by Cassels [6]. A special role in our work will be played bythe dual lattice and the factor lattice, and so we begin by defining these. Dual, orthogonal, and factor lattice.
Let Λ ⊂ R N be a lattice of rank r withbasis matrix B . The dual lattice is defined to beΛ ∗ = { x ∈ Λ R : h x , y i ∈ Z for all y ∈ Λ } . This lattice has basis matrix B ( B t B ) − . It immediately follows that rank(Λ ∗ ) = r and covol(Λ ∗ ) = covol(Λ) − . Moreover, we have (Λ ∗ ) ∗ = Λ.Now suppose that Λ ⊂ Z N is a primitive lattice. The orthogonal lattice Λ ⊥ isthe primitive lattice Λ ⊥ = (cid:8) a ∈ Z N : h a , z i = 0 for all z ∈ Λ (cid:9) . We have (Λ ⊥ ) ⊥ = Λ and covol(Λ) = covol(Λ ⊥ ).If π : R N → Λ ⊥ R is the orthogonal projection, then we define the factor lattice Λ π to be the projection π ( Z N ). We have Λ π = (Λ ⊥ ) ∗ and socovol(Λ π ) = 1covol(Λ) . Successive minima and slopes.
Let Λ ⊂ R N be a lattice of rank r . For each1 k r , let s k (Λ) be the least σ > k linearly in-dependent vectors of Euclidean length bounded by σ . The s k (Λ) are the successiveminima of Λ, and they satisfy0 < s (Λ) s (Λ) · · · s r (Λ) . (Note that when we speak of successive minima, we shall always mean with respectto the Euclidean norm k·k .) It follows from Minkowski’s second convex bodytheorem [6, § VIII.1] thatcovol(Λ) r Y i =1 s i (Λ) ≪ N covol(Λ) , (2.1)where the implied constant depends only on the dimension N of the ambient vectorspace. Appealing to work of Banaszczyk [1, Thm. 2.1], we have1 s k (Λ) s r − k +1 (Λ ∗ ) r, (2.2)for 1 k r . QUIDISTRIBUTION AND FREENESS ON GRASSMANNIANS 7
The slope of a lattice Λ ⊂ R N of rank r is defined to be µ (Λ) = − r log covol(Λ) . The maximal slope µ max (Λ) of Λ is the maximum of the slopes of all non-zerosublattices of Λ. On the other hand, the minimal slope µ min (Λ) of Λ is the minimumof the slopes of all quotients Λ / M, as M ⊂ Λ runs over proper and primitivesublattices. According to Bost and Chen [3, Eq. (2)], the maximal and minimalslopes are related via the formula µ max (Λ ∗ ) = − µ min (Λ) , (2.3)where Λ ∗ is the dual of Λ. Appealing to Theorems 1 and 3 of Borek [2], we canalso deduce that 0 log s r (Λ) + µ min (Λ) c r and0 log s (Λ) + µ max (Λ) c r , (2.4)for a certain explicit constant c r > Tensor products.
Suppose we are given two lattices Λ and Λ in R N , withrank(Λ i ) = r i for i = 1 ,
2. We may form the tensor product lattice Λ ⊗ Z Λ ,which has rank r r . Recall that Λ = (Λ ∗ ) ∗ . Then we see that Λ ⊗ Z Λ =(Λ ∗ ) ∗ ⊗ Z Λ , which is isomorphic to the space Hom Z (Λ ∗ , Λ ) via the map whichtakes an elementary tensor ϕ ⊗ v ∈ (Λ ∗ ) ∗ ⊗ Z Λ to the linear map Λ ∗ → Λ definedby ( ϕ ⊗ v )( v ) = ϕ ( v ) v , for v ∈ Λ ∗ and v ∈ Λ . The covolume of the tensorproduct lattice is covol(Λ ⊗ Z Λ ) = covol(Λ ) r covol(Λ ) r . (2.5)Several authors have investigated the relationship between the maximal slopes oftensor product lattices and the maximal slopes of the factors. Appealing to Chen [7,Thm. 1.1], for example, one finds that µ max (Λ ) + µ max (Λ ) µ max (Λ ⊗ Z Λ ) µ max (Λ ) + µ max (Λ ) + r + r . Applying this to (Λ ⊗ Z Λ ) ∗ = Λ ∗ ⊗ Z Λ ∗ , we deduce that µ min (Λ ) + µ min (Λ ) − r − r µ min (Λ ⊗ Z Λ ) µ min (Λ ) + µ min (Λ ) . Once taken in conjunction with (2.4), this implies that s r (Λ ) s r (Λ ) ≪ s r r (Λ ⊗ Z Λ ) ≪ s r (Λ ) s r (Λ ) , (2.6)where the implied constants depend only on r and r . TIM BROWNING, TAL HORESH, AND FLORIAN WILSCH
Freeness.
Suppose for the moment that V is a smooth Fano Q -variety of di-mension r , extended to a smooth scheme V Z over Spec( Z ). Any rational point x ∈ V ( Q ) extends to a unique integral point x ∈ V Z ( Z ) of this scheme. The pull-back ( T V Z ) x of its tangent bundle T V Z along x is a free Z -module of rank r inside( T V ) x ⊗ R = ( T V Z ) x ⊗ R . Fixing a Riemannian metric on V ( R ) induces an innerproduct on ( T V ) x ⊗ R and makes ( T V Z ) x a lattice. The model V Z induces normson ( T V ) ⊗ K v at all finite places K v , and we get an adelic metric on T V , as in [13,Ex. 3.4]. The adelic metric on the tangent bundle induces an adelic norm on theanticanonical bundle ω ∨ V = V r T V , hence an anticanonical height. The logarithmicanticanonical height is defined in [13, D´ef. 3.11], and it satisfies h ( x ) = − log covol(( T V ) x ) = rµ (( T V ) x ) , by [13, D´ef. 4.1, Rem. 4.2]. In [13, D´ef. 4.11], Peyre defines the freeness of x to be ℓ ( x ) = max { µ min (( T V ) x ) , } µ (( T V ) x ) = max { rµ min (( T V ) x ) , } h ( x ) . (2.7)Since the minimal slope is bounded by the slope, we have0 ℓ ( x ) . Tangent lattices for Grassmannians
We now interpret the above in the case V = Gr( m, n ) of Grassmannians. Arational point x ∈ V ( Q ) is the same thing as a primitive lattice Λ ⊂ Z n rank m .Following [8, § m, n ). Consider the trivial bundle O ⊕ nV , carrying thestandard inner product. It admits the subbundle S ⊂ O ⊕ nV whose fibre at a pointΛ is Λ R ⊂ R n , and the quotient bundle Q = O ⊕ nV / S . Then T V ∼ = H om ( S , Q ) ∼ = S ∨ ⊗ Q . The inner product on O ⊕ nV induces inner products on these other bundles (usingthe canonical isomorphism Λ ∗ R ∼ = Λ R induced by the scalar product). Thus it alsoinduces a Riemannian metric. All of these constructions work over Spec Z , givingrise to a smooth integral model Gr Z ( m, n ) together with bundles S Z (with ( S Z ) Λ =Λ ⊂ Z n ), Q Z (with ( Q Z ) Λ = Z n / Λ), and T V Z = H om ( S Z , Q Z ). Note that Z n / Λ isisometric to Λ π via the orthogonal projection. For a point x = Λ ∈ Gr( m, n )( Q ),we are interested in the tangent lattice T Λ = ( T V Z ) x = H om ( S Z , Q Z ) x = Hom(Λ , Z n / Λ) ∼ = Λ ∗ ⊗ Z Λ π inside Λ R ⊗ Λ ⊥ R ∼ = Λ ∗ R ⊗ R n / Λ R ∼ = ( T V ) x ⊗ R , QUIDISTRIBUTION AND FREENESS ON GRASSMANNIANS 9 where all isomorphisms are isometries. In view of (2.5), this lattice has covolumecovol( T Λ ) = covol(Λ ∗ ) n − m covol(Λ π ) m = covol(Λ) m − n · m = covol(Λ) − n . (3.1)The logarithmic anticanonical height of x = Λ is thus h ( x ) = − log covol(Λ ∗ ⊗ Λ π ) = n log covol(Λ) . The corresponding exponential height is H ( x ) = covol(Λ) n . It follows from (2.4) that there is an explicit constant c n > log s m ( n − m ) ( T Λ ) + µ min ( T Λ ) c n , where s m ( n − m ) ( T Λ ) is the largest successive minimum of the tangent lattice. Hence,the definition (2.7) yields ℓ ( x ) = max n ( m ( n − m ) n ) µ min ( T Λ ) , o log covol(Λ)= max n − ( m ( n − m ) n ) log s m ( n − m ) ( T Λ ) , o log covol(Λ) + O (cid:18) (cid:19) , (3.2)for any primitive rank m lattice Λ ⊂ Z n representing a point x ∈ Gr( m, n )( Q ).One notes that s m ( n − m ) ( T Λ ) > covol( T Λ ) m ( n − m ) = covol(Λ) − nm ( n − m ) , by (2.1) and (3.1). This argument suggests that if x is “typical”, in the sense thatthe successive minima of T Λ all have equal order of magnitude, then ℓ ( x ) = 1+ o (1),as H ( x ) → ∞ . Projective space.
We proceed by discussing the quantities we’ve introduced inthe most familiar case m = 1, corresponding to projective space P n − Q . The lattice T Λ then admits a particularly concrete description, as follows. Lemma 3.1.
Let x ∈ Gr(1 , n )( Q ) = P n − Q ( Q ) be identified with the lattice Λ = Z x ,for a primitive vector x ∈ Z n . Then T Λ is isometric to k x k − ( Z n ∩ x ⊥ ) ∗ .Proof. We begin by noting that covol(Λ) = k x k . Let us put Γ x = Z n ∩ x ⊥ , whichwe note is a primitive lattice of rank n −
1. We begin by showing thatΓ π x = k x k − Γ ⊥ x . (3.3)Clearly, Γ ⊥ x = Z x , which is a primitive lattice of rank one contained in the space R x . We compute Γ π x using the fact that Λ π = (cid:0) Λ ⊥ (cid:1) ∗ when Λ is primitive. Recalling the definition of the dual lattice, we therefore obtainΓ π x = (cid:0) Γ ⊥ x (cid:1) ∗ = (cid:8) z ∈ (cid:0) Γ ⊥ x (cid:1) R : h y , z i ∈ Z for all y ∈ Γ ⊥ x (cid:9) = { t x : t ∈ R such that h t x , m x i ∈ Z for all m ∈ Z } . But h t x , m x i = tm h x , x i = tm k x k . Moreover, tm k x k ∈ Z for every m ∈ Z ifand only if t ∈ k x k − Z . We conclude thatΓ π x = (cid:8) t x : t ∈ k x k − Z (cid:9) = k x k − Z x = k x k − Γ ⊥ x , as required for (3.3).Next we observe that T Λ = ( Z x ) ∗ ⊗ Z ( Z x ) π = Γ π x ⊗ Z Γ ∗ x . We wish to prove that this is isometric to k x k − Γ ∗ x . We know that Γ π x ⊂ (cid:0) Γ ⊥ x (cid:1) R = R x and Γ ∗ x ⊂ Γ x , R = x ⊥ , because the dual latticealways lives in the same space as the original lattice, and the factor lattice alwayslives in the orthogonal space. Thus T Λ ⊂ R x ⊗ x ⊥ . We claim that the map ϕ : R x ⊗ x ⊥ → x ⊥ α x ⊗ w α k x k w is an isometry, where the inner product on the tensor product is the product of h· , ·i in each of the components. In the light of (3.3), this will suffice to completethe proof of the lemma, since then ϕ clearly maps the lattice Γ π x ⊗ Z Γ ∗ x to k x k − Γ ∗ x .To check the claim, it suffices to find an orthornormal basis of R x ⊗ x ⊥ that istaken to an orthonormal basis of x ⊥ . Clearly, the orthonormal basis (cid:8) k x k − x ⊗ w , . . . , k x k − x ⊗ w n − (cid:9) suffices, where { w , . . . , w n − } is an orthonormal basis for x ⊥ . (cid:3) We next consider what the definition (3.2) has to say when m = 1. ApplyingLemma 3.1, we obtain h ( x ) = n log k x k and ℓ ( x ) = n − n · max (cid:26) − log s n − (( Z n ∩ x ⊥ ) ∗ )log k x k , (cid:27) + O (cid:18) h ( x ) (cid:19) , if x ∈ P n − Q ( Q ) is represented by the primitive vector x ∈ Z n . Butlog s n − (( Z n ∩ x ⊥ ) ∗ ) = − log s ( Z n ∩ x ⊥ ) + O (1) , by (2.2). Since s ( Z n ∩ x ⊥ ) >
1, this shows that ℓ ( x ) > n − n + o (1) , as h ( x ) → ∞ , which essentially recovers (1.4). QUIDISTRIBUTION AND FREENESS ON GRASSMANNIANS 11
The general case.
Let m > ∈ Gr( m, n )( Q ). The largest successiveminimum of the tangent lattice T Λ = Λ ∗ ⊗ Z Λ π is closely related to those of Λ ∗ andΛ π . Indeed, it follows from (2.6) that s m (Λ ∗ ) s n − m (Λ π ) ≪ s m ( n − m ) ( T Λ ) ≪ s m (Λ ∗ ) s n − m (Λ π ) , where the implied constants depend only on n .Returning to (3.2), there exists a constant C >
0, which depends only on n , suchthat any given x ∈ Gr( m, n )( Q ) has ℓ ( x ) < ε + o (1) if and only if s m ( n − m ) ( T Λ ) > C (covol(Λ)) − εnm ( n − m ) . On redefining C , this is equivalent to s m (Λ ∗ ) s n − m (Λ π ) > C (covol(Λ)) − εnm ( n − m ) . It now follows from (2.2) that ℓ ( x ) < ε + o (1) if and only if s (Λ ⊥ ) s (Λ) C (covol(Λ)) εnm ( n − m ) , (3.4)after a further modification to C . Proof of Theorem 1.1.
Let 1 < m < n −
1. In particular, it follows that n >
4. Ourtask is to show that there are infinitely many x ∈ Gr( m, n )( Q ) such that ℓ ( x ) =0. For this we work directly with the definition (3.2) of ℓ ( x ) in terms of slopes.Consider the two vectors u = ( q, , , . . . , ∈ Z n and v = (1 , − q, , . . . , ∈ Z n ,and let e , . . . , e n be the standard basis vectors of R n . We takeΛ = Z u ⊕ Z e ⊕ · · · ⊕ Z e m +1 . This is a primitive lattice of rank m , with covol(Λ) = q and s (Λ) = 1. Theorthogonal complement isΛ ⊥ = Z v ⊕ Z e m +2 ⊕ · · · ⊕ Z e n . This is a primitive lattice of rank n − m , with covol(Λ ⊥ ) = q and s (Λ ⊥ ) = 1. Onappealing to (2.3), it now follows that µ min ( T Λ ) = − µ max (Λ ⊗ Z Λ ⊥ ) log covol(span Z ( e m +1 ⊗ e n )) = 0 , whence ℓ ( x ) = 0. (cid:3) Let Ω ε = { Λ ∈ Gr( m, n )( Q ) : ℓ ( x ) ε } be the set of non- ε -free points onGr( m, n ). The remaining task for this section is to prove that this set is not thin,as claimed in Theorem 1.2. Lemma 3.2. If ε > and < m < n − , then the image of Ω ε is dense in Q v Gr( m, n )( Q v ) .Proof. Let S be a finite set of places, and let U v ⊂ Gr( m, n )( Q v ) be open subsetsfor v ∈ S . We want to find a Λ ∈ Ω ε whose image lies in Q v ∈ S U v . By weakapproximation, there exists Λ ∈ Gr( m, n )( Q ) whose image lies in Q v ∈ S U v . Pickany rational line l ⊂ Λ , R = Λ ⊗ R and any rational hyperplane H ⊂ R n with Λ ⊂ H . We have l = h u i and H = h v i ⊥ for primitive u , v ∈ Z n . Consider thesubvariety X = { Λ ∈ Gr( m, n ) : l ⊂ Λ R ⊂ H } ⊂ Gr( m, n ) , which is isomorphic to Gr( m − , n − m ensures thatGr( m − , n −
2) is smooth and of positive dimension. Thus the non-empty opensubsets U v ∩ X ( Q v ) contain infinitely many Q v -points for all v ∈ S . An applicationof weak approximation on Gr( m − , n −
2) therefore shows that there are infinitelymany Λ ′ ∈ X ( Q ) whose images lie in U v ∩ X ( Q v ) for all v ∈ S .Let C > ε -free. Since there are only finitely many points in Gr( m − , n − Q ) ofbounded height, we can find Λ ′ ∈ X ( Q ) whose images are in U v for all v ∈ S , with H (Λ ′ ) > (cid:0) C − k u k k v k (cid:1) m ( n − m ) ε . Since the corresponding lattice Λ satisfies s (Λ) k u k and s (Λ ⊥ ) k v k , wehave P ′ ∈ Ω ε . (cid:3) Proof of Theorem 1.2.
Appealing to work of Serre [18, Corollary 3.5.4], the imageof Ω ε would be nowhere dense if it where thin. (cid:3) Remark . We note that Ω is always a thin set, since it fails to be Zariski dense.Indeed, when 1 < m < n −
1, we find that Ω is the union of the proper subvarieties { P ∈ Gr( m, n )( Q ) : u ∈ P } , where u runs over the finitely many vectors in Z n of norm O (1). If, on the other hand, m is 1 or n −
1, then Ω ε = ∅ for ε < n − n by (1.4). 4. Free rational points dominate
In this section we give our first proof of Theorem 1.3. We must provide an upperbound for the quantity E ε ( B ) = { x ∈ Gr( m, n )( Q ) : H ( x ) B, ℓ ( x ) < ε } , with the aim being to show that E ε ( B ) = o ( B ) for any 0 ε <
1. Switching tothe language of primitive lattices, we write P ( m, n ) for the set of primitive latticesΛ ⊂ Z n which have rank m . It then follows from (3.4) that E ε ( B ) is at most n Λ ∈ P ( m, n ) : covol(Λ) B n , s (Λ ⊥ ) s (Λ) C (covol(Λ)) εnm ( n − m ) o , for a suitable constant C > n . It is convenient to break therange for the covolume and the sizes of s (Λ) and s (Λ ⊥ ) into dyadic intervals.Thus we put E ε ( B ) X R =2 j R B /n X S =2 j , S =2 j S S CR εnm ( n − m ) e E ε ( R, S , S ) , (4.1) QUIDISTRIBUTION AND FREENESS ON GRASSMANNIANS 13 where e E ε ( R, S , S ) = Λ ∈ P ( m, n ) : R/ < covol(Λ) R,S / < s (Λ) S ,S / < s (Λ ⊥ ) S . Let P ( m, n ; R, S ) = { Λ ∈ P ( m, n ) : covol(Λ) R, s (Λ) S } . Then, since the covolume is preserved under taking the orthogonal complement,we see that e E ε ( R, S , S ) min { P ( m, n ; R, S ) , P ( n − m, n ; R, S ) } . (4.2)Our attention now shifts to estimating P ( m, n ; R, S ) for given 1 m < n and R, S > r > s , . . . , s r ∈ R with1 s s · · · s r . We let P r,n ( s , . . . , s r ) be the set of primitive lattices Λ ⊂ Z n of rank r whose i thsuccessive minimum lies in the interval [ s i , s i ), for 1 i r . We shall need thefollowing result. Lemma 4.1.
We have P r,n ( s , . . . , s r ) ≪ s n + r − s n + r − · · · s n +1 − rr , where the implied constant only depends on n .Proof. This is extracted from Lemma 6 of Schmidt [15] in work of Browning, LeBoudec and Sawin [5, Lemma 3.18]. (cid:3)
The following general inequality will facilitate our application of Lemma 4.1. Let r > ξ , . . . , ξ r , α , . . . , α r >
0. Then ξ α · · · ξ α r r ( ξ · · · ξ r ) α ··· + αrr , (4.3)provided that 1 ξ · · · ξ r and α > · · · > α r . This is easily proved byinduction on r . Indeed, on putting β = α + · · · + α r − , the induction hypothesisyields ξ α · · · ξ α r r ( ξ · · · ξ r − ) βr − · ξ α r r ( ξ · · · ξ r ) β + αrr · ( ξ · · · ξ r − ) βr ( r − − αrr · ξ α r − β + αrr r . Noting that β/ ( r − > α r , we may take ξ , . . . , ξ r − ξ r in the second factor andthen immediately arrive at the right hand side of (4.3).We are interested in the case r = m . Breaking into dyadic intervals, we see that P ( m, n ; R, S ) X s ··· s m ≪ Rs ...s m ≪ Rs S P m,n ( s , . . . , s m ) . But Lemma 4.1 yields P m,n ( s , . . . , s m ) ≪ s s n + m − s n + m − · · · s n +1 − mr . Note that( n + m −
3) + m − X k =0 ( n + 1 − m + 2 k ) = ( n + m −
3) + mn − m − n + 1= mn − . Hence it follows from (4.3) that P m,n ( s , . . . , s m ) ≪ s ( s · · · s m ) n − m , whence P ( m, n ; R, S ) ≪ X s ··· s m ≪ Rs ··· s m ≪ Rs S s ( s · · · s m ) n − m ≪ R n − m (log R ) m − X s S s ≪ S R n − m (log R ) m − . Returning to (4.2), we apply the inequality min { α, β } √ αβ for any α, β ∈ R > ,in order to deduce that e E ε ( R, S , S ) ≪ q S R n − m (log R ) m − · S R n − n − m (log R ) n − m − = S S R n − nm ( n − m ) (log R ) n − . But then, on returning to (4.1) and summing over dyadic intervals for
R, S , S ,we obtain E ε ( B ) ≪ X R =2 j R B /n R n +( ε − nm ( n − m ) (log R ) n ≪ B − − εm ( n − m ) (log B ) n . This gives an explicit power saving error term if and only if ε <
1. This thereforecompletes the proof of Theorem 1.3.5.
Tangent lattices equidistribute
The goal of this section is to prove Theorem 1.5, namely to establish equidistri-bution of the projections of the tangent lattices Λ ∗ ⊗ Z Λ π in U G m,n with respectto a certain probability measure ν which will be defined below. QUIDISTRIBUTION AND FREENESS ON GRASSMANNIANS 15
Spaces of lattices.
We begin by describing the spaces L m,n and G m,n of lattices asquotients. Recalling the space L m,n defined in (1.5), we note that it is isomorphicto L m,n ∼ = SL n ( R ) / (cid:16)h GL m ( Z ) Mat m,n − m ( R )0 n − m,m GL n − m ( R ) i ∩ SL n ( R ) (cid:17) . For our needs, however, it will be useful to view it as the non-homogeneous quotient L m,n ∼ = Mat × n × m ( R ) / GL m ( Z ), where Mat × n × m ( R ) is the open subset of Mat n × m ( R )consisting of matrices of full rank.With the latter point of view, the subspace G m,n can be described as a quotient Y / GL m ( n − m ) ( Z ) ⊂ Mat × n × m ( R ) / GL m ( Z ), where Y = (cid:26) A ⊗ B : A ∈ Mat n × m ( R ) , B ∈ Mat n × ( n − m ) ( R )span R ( A ) = span R ( B ) ⊥ (cid:27) , and where we interpret A ⊗ B as ( a i,j B ) i,j for A = ( a i,j ) ∈ Mat n × m ( R ). (Note thatthe conditions in Y imply that A and B must be of full rank.)Next, we extend the notion of a factor lattice to lattices that are not necessarilyintegral: Recall that a rank m lattice Λ is primitive inside a full unimodular lattice Γ if there is no rank m sublattice of Γ that properly contains Λ. (Note that when m < n , every lattice is primitive with respect to infinitely many full unimodularlattices in R n .) For each full unimodular lattice Γ in which Λ is primitive, we definethe factor lattice of Λ with respect to Γ as Λ π, Γ = π (Γ), where π is the orthogonalprojection from R n to (span R (Λ)) ⊥ . We havecovol(Λ π, Γ ) = 1covol(Λ) . In particular, Λ π, Γ is unimodular if and only if Λ is.Consider the space of pairs P m,n = (cid:8) (Λ , Λ π, Γ ) : Λ primitive of rank m in a full unimodular lattice Γ ⊂ R n (cid:9) . We observe that P m,n ∼ = SL n ( R ) / nh GL m ( Z ) Mat m,n − m ( R )0 n − m,m GL n − m ( Z ) i ∩ SL n ( R ) o . To see this, note that the first m columns of any matrix in a given coset span thesame lattice Λ, and the last n − m columns of such a matrix span Λ π, Γ , where Γis the full lattice spanned by the columns of the matrix. The space P m,n is also aPGL n ( R )-homogeneous space since P m,n ∼ = PGL n ( R ) / nh GL m ( Z ) Mat m,n − m ( R )0 n − m,m GL n − m ( Z ) io . Subspaces of unimodular lattices.
For the sake of defining the measure ν appearing in Theorem 1.5 and of proving Proposition 5.1 below, we briefly discussthe subsets of unimodular elements inside L m,n , G m,n , and P m,n . Let U L m,n denote the subset of unimodular lattices in L m,n , and observe that U L m,n ∼ = L m,n /D , where D = { diag( α − m I m , α n − m I n − m ) : α ∈ R > } ∼ = R > . This is a one-parameter subgroup of diagonal matrices in SL n ( R ). Moreover, U L m,n ∼ = Mat × , n × m ( R ) / GL m ( Z ), where Mat × , n × m ( R ) is the set of matrices M offull rank that satisfy det( M t M ) = 1.Recall that U G m,n = G m,n ∩ U L m,n . Defining Y = { A ⊗ B ∈ Y : det( A ) n − m det( B ) m = 1 } , we get U G m,n ∼ = Y / GL m ( n − m ) ( Z ) . Let
U P m,n be the subset of pairs in P m,n for which Λ (and therefore also Λ π, Γ )is unimodular.The space L m,n decomposes as L m,n ∼ = U L m,n × R > via Λ (covol(Λ) − /m Λ , covol(Λ)), with inverse map α /m L ← [ ( L, α ).For G m,n , we use the decomposition G m,n ∼ = U G m,n × R > , given by Λ (cid:16) covol(Λ) − m ( n − m ) Λ , covol(Λ) − (cid:17) , with the inverse map being given by α − / ( m ( n − m )) L ← [ ( L, α ).Lastly, P m,n decomposes as P m,n ∼ = U P m,n × R > via (Λ , Λ π, Γ ) (covol(Λ) − m Λ , covol(Λ) n − m Λ π, Γ , covol(Λ)) , the inverse map being given by( α m L, α − n − m L π, Γ ′ ) ← [ ( L, L π, Γ ′ , α ) . In all three cases, we denote by u the projection to the unimodular component. Inparticular, G m,n → U G m,n × R > maps T Λ ( u ( T Λ ) , H (Λ)), for Λ ∈ Gr( m, n ).The decomposition of P m,n extends to a decomposition of homogeneous measurespaces as follows. Let vol P m,n be an SL n -invariant measure on P m,n . Then theIwasawa decomposition of the Haar measure on SL n yields( P m,n , vol P m,n ) ∼ = ( U P m,n , vol UP m,n ) × ( R > , x n − d x ) , for a finite measure vol UP m,n on U P m,n , and we normalise all measures so thatthis measure is a probability measure.
QUIDISTRIBUTION AND FREENESS ON GRASSMANNIANS 17
From pairs (Λ , Λ π ) to tensors Λ ∗ ⊗ Z Λ π . We now summarise our strategy forproving Theorem 1.5. We construct a diffeomorphism ϕ : U P m,n → U G m,n thatwill enable us to apply an equidistribution statement for the primitive lattices in
U P m,n with respect to the measure vol UP m,n . The measure in our equidistributiontheorem will be the pushforward ν = ϕ ∗ vol UP m,n of vol UP m,n . Proposition 5.1.
The map ϕ : P m,n → G m,n sending (Λ , Λ π, Γ ) Λ ∗ ⊗ Z Λ π, Γ is a diffeomorphism, as is its restriction ϕ : U P m,n → U G m,n to the unimodular elements. Moreover, we have ϕ = ϕ × ( · ) n . Let U = nh I m Mat m,n − m ( R )0 I n − m io , so that P m,n = PGL n ( R ) / (cid:16) U ⋊ h GL m ( Z ) 00 GL n − m ( Z ) i(cid:17) . Clearly, Λ ∗ ⊗ Z Λ π is a lattice of rank rank(Λ ∗ ) · rank(Λ π ) = m ( n − m ) inside thespace R n ⊗ R n ∼ = R n . The diffeomorphism will be constructed in three steps:(i) define a differentiable map ˜ ϕ : GL n ( R ) → Y ;(ii) obtain a diffeomorphism ˆ ϕ : PGL n ( R ) /U → Y ; and(iii) reduce ˆ ϕ modulo the integral subgroups to ϕ : P m,n → G m,n . Step (i).
Let ˜ ϕ : GL n ( R ) → Y be the map which is defined as follows. For g ∈ GL n ( R ), let A g ∈ Mat × n × m ( R ) be the matrix consisting of the first m columns of g .Let ˜ A g = A g ( A t g A g ) − ∈ Mat × n × m ( R ), and let ˜ B g ∈ Mat × n × ( n − m ) ( R ) be the matrixobtained from projecting the columns of g to the orthogonal subspace of ˜ A g (or A g , as they span the same space). Then we define˜ ϕ ( g ) = ˜ A g ⊗ ˜ B g , which clearly lies in Y . We note that ˜ ϕ is differentiable since it is a composition ofdifferentiable maps. Moreover, if g = ( A | B ) generates a lattice Γ and A generatesa lattice Λ, then ˜ ϕ ( g ) generates Λ ∗ ⊗ Z Λ π, Γ . Step (ii).
By the definition of ˜ B g , the map ˜ ϕ is invariant under the unipotentsubgroup U . On the other hand, multiplying g by a constant λ changes ˜ A g by λ − and ˜ B g by λ , so ˜ ϕ is also invariant under the center of GL n . Hence ˜ ϕ descends tothe quotient X = PGL n ( R ) /U , and we denote this map by ˆ ϕ : X → Y . Lemma 5.2. ˆ ϕ : X → Y is a diffeomorphism. Proof.
Let A ⊗ B ∈ Y . For ˜ A = A ( A t A ) − , let ψ : Y → PGL n ( R ) be the map A ⊗ B ( ˜ A | B ); indeed, ( ˜ A | B ) has full rank since A and B do. We claim that themap ˆ ψ : Y → X obtained by reducing ψ mod U is the inverse of ˆ ϕ . Note first that ψ is well defined; indeed, f λA = λ − ˜ A and therefore λ − A ⊗ λB ( λ ˜ A | λB ) = ( ˜ A | B ) . To conclude bijectivity, note that the matrix ψ ◦ ˆ ϕ ( A | B ) differs from ( A | B ) byadding multiples of the first columns to the last columns, i.e., by an element of U ,so ˆ ψ ◦ ˆ ϕ = id X . Conversely, the tensor ˆ ϕ ◦ ˆ ψ ( A ⊗ B ) is again A ⊗ B , since ˜˜ A = A , and A , B are already orthogonal, so the orthogonalisation step in ˆ ϕ changes nothing.Thus ˆ ϕ ◦ ˆ ψ = id Y . Finally, ˆ ϕ is a differentiable since ˜ ϕ is, and ˆ ψ is differentiablebecause ψ is. (cid:3) Step (iii).
We first note that ˆ ϕ descends to the quotient. Indeed, for a matrix( A | B ) ∈ GL n ( R ) and ( γ m , γ n − m ) ∈ GL m ( Z ) × GL n − m ( Z ), we haveˆ ϕ : ( A | B ) ( γ m , γ n − m ) ( Aγ m ⊗ Bγ n − m ) = ( A ⊗ B )( γ m ⊗ γ n − m ) , so ( A | B )(GL m ( Z ) × GL n − m ( Z )) maps into ( A ⊗ B ) GL m ( n − m ) ( Z ). Also note that ϕ is surjective since ˆ ϕ is. The crucial part is then to show injectivity, which we dousing two technical lemmas. Lemma 5.3.
Let A ⊗ B ∈ Y . If g ∈ GL m ( n − m ) ( R ) is such that ( A ⊗ B ) g ∈ Y ,then g is an elementary tensor.Proof. Write ( A ⊗ B ) g = A ′ ⊗ B ′ ∈ Y . Since g ∈ GL m ( n − m ) ( R ), it follows thatspan( A ⊗ B ) = span( A ′ ⊗ B ′ ), which means that span( A ) ⊗ span( B ) = span( A ′ ) ⊗ span( B ′ ). As a result, span( A ) = span( A ′ ) and span( B ) = span( B ′ ), and thereforethere exist g m ∈ GL m ( R ) and g n − m ∈ GL n − m ( R ) such that A ′ = Ag m and B ′ = Bg n − m . Thus ( A ⊗ B )( g − g m ⊗ g n − m ) = 0, which implies that g = g m ⊗ g n − m ,since A ⊗ B has full rank. (cid:3) Lemma 5.4.
Let A ⊗ B ∈ Y , and let γ ∈ GL m ( n − m ) ( Z ) such that ( A ⊗ B ) γ ∈ Y .Then γ = γ m ⊗ γ n − m for γ m ∈ GL m ( Z ) and γ n − m ∈ GL n − m ( Z ) .Proof. Let A ⊗ B and γ be as in the statement. Appealing to Lemma 5.3 and usingthe fact that Mat m ( n − m ) × m ( n − m ) ( Z ) = Mat m × m ( Z ) ⊗ Mat ( n − m ) × ( n − m ) ( Z ), we have γ = γ m ⊗ γ n − m , with γ i ∈ GL i ( R ) ∩ Mat i × i ( Z ) for i ∈ { m, n − m } . Sincedet( γ m ) n − m det( γ n − m ) m = det( γ m ⊗ γ n − m ) = det( γ ) ∈ {± } , it follows that det( γ m ) , det( γ n − m ) ∈ {± } . We conclude that γ m ∈ GL m ( Z ) and γ n − m ∈ GL n − m ( Z ). (cid:3) Proof of Proposition 5.1.
As already noted, ϕ is well-defined and surjective. It isalso injective by Lemma 5.4, since if two elements in Y are equivalent moduloGL m ( n − m ) ( Z ), then their pre-images are equivalent modulo GL m ( Z ) × GL n − m ( Z ). QUIDISTRIBUTION AND FREENESS ON GRASSMANNIANS 19
Thus ϕ is also bijective, and therefore the inverse map of ˆ ϕ descends to an inversemap of ϕ . Since ˆ ϕ and its inverse are differentiable, so are ϕ and its inverse.Composing ϕ with the decompositions of P m,n and G m,n results in U P m,n × R > P m,n G m,n U G m,n × R > ( L, L π, Γ , α ) ( α m L, α − n − m L π, Γ ) α − nm ( n − m ) L ∗ ⊗ Z L π, Γ ( L ∗ ⊗ Z L π, Γ , α n ) , ϕ hence ϕ restricts to a diffeomorphism ϕ : U P m,n → U G m,n that satisfies ϕ = ϕ × ( · ) n , as required. (cid:3) The main equidistribution theorem.
The equidistribution statement in The-orem 1.5 employs counting in subsets of the spaces P m,n and U P m,n satisfyingthe following boundary condition.
Definition . The (topological) boundary of a set inside a manifold M is said tobe controlled if it consists of finitely many subsets of embedded C submanifoldswhose dimension is strictly smaller than dim M .Having proved that the space P m,n is diffeomorphic to G m,n via ϕ , we mayapply the following counting result for pairs in P m,n , in order to deduce our mainequidistribution result about tangent lattices. Theorem 5.6 (Horesh–Karasik [10]) . Let m, n be integers such that m n − .Let E ⊆ U P m,n be a subset whose boundary is controlled. For any B > , let E B = E × (0 , B ] ⊂ P m,n denote the subset of pairs (Λ , Λ π, Γ ) that project to E andfor which covol(Λ) B . Then { (Λ , Λ π ) ∈ E B : Λ primitive } = c m,n vol UP m,n ( E ) B n + O ε,E ( B n (1 − δ E )+ ε ) , for any ε > , where c m,n was defined in (1.3) and δ E = ⌈ n − ⌉ n if E is bounded, ⌈ n − ⌉ n · [2(max( m,n − m ) − n − n ] if E is not bounded. To be precise, this result follows on combining [10, Thm. I(4)] with the discussionin [10, Remark 1.2] around normalising the invariant measures to get probabilitymeasures. We are now ready to prove the following result, which is a more preciseversion of Theorem 1.5.
Theorem 5.7.
Let m, n be integers such that m n − . Let E ⊆ U G m,n bea subset whose boundary is controlled. Define E B = E × (0 , B ] ⊂ G m,n , for B > . Then { Λ ∈ Gr( m, n )( Q ) : T Λ ∈ E B } { Λ ∈ Gr( m, n )( Q ) : H (Λ) B } = ν ( E ) + O E ( B − n ) , as B → ∞ . Proof.
We claim that δ E > n in the statement of Theorem 5.6. This is obvious when E is bounded. Alternatively,when E is not bounded, we note that2(max( m, n − m ) − n −
1) + n n − n −
1) + n < n − n , for any 1 m n − n >
2. But then it follows that δ E > ( n − n · n − n = 116 n , as claimed.The set E B consists of all the lattices L ∈ G whose normalisation lies in E and forwhich covol( L ) − B . It contains precisely those T Λ for primitive Λ that projectto E and have H (Λ) B . Since ϕ defines a bijection between the pairs (Λ , Λ π ),where Λ is primitive, and the elements T Λ = Λ ∗ ⊗ Z Λ π , where Λ is primitive, wededuce that the number of tangent lattices in E B is in fact equal to the numberof “primitive” pairs in ϕ − ( E B ). The latter can be estimated by Theorem 5.6since Proposition 5.1 implies that ϕ − ( E B ) = ( ϕ ) − ( E ) × (0 , B /n ], where ϕ isa diffeomorphism, so that in particular ( ϕ ) − ( E ) has controlled boundary. Ontaking a sufficiently small choice of ε in Theorem 5.6, an application of this resultnow yields { Λ ∈ Gr( m, n )( Q ) : T Λ ∈ E B } = c m,n vol UP m,n (( ϕ ) − ( E )) B + O E (cid:16) B − n (cid:17) = c m,n ν ( E ) B + O E (cid:16) B − n (cid:17) . The statement of the theorem now follows with the asymptotic expression (1.2) forthe denominator. (cid:3)
The following reformulation of Theorem 5.7 will be very useful in applications.Call a Borel set E ⊂ U G m,n a continuity set if ν ( ∂E ) = 0, and denote by C b ( U G m,n ) the space of bounded, continuous functions
U G m,n → R . Let N m,n ( B ) = { Λ ∈ Gr( m, n )( Q ) : H (Λ) B } be the number of rational points on Gr( m, n ) of height at most B . Theorem 5.8.
Let ν B = 1 N m,n ( B ) X H (Λ) B δ u ( T Λ ) be the sequence of probability measures on U G m,n counting tangent lattices of pointsof bounded height. Then ν B converges weakly to ν as B → ∞ , in the sense that(i) lim B →∞ R f d ν B = R f d ν for all f ∈ C b ( U G m,n ) ,(ii) lim B →∞ ν B ( E ) = ν ( E ) for all continuity sets E ⊂ U G m,n , and(iii) lim sup B →∞ ν B ( E ) ν ( E ) for all closed E ⊂ U G m,n . QUIDISTRIBUTION AND FREENESS ON GRASSMANNIANS 21
Moreover,(iv) let f : U G m,n → R > be a bounded, continuous function. Let N m,n ( f ; B ) = { Λ ∈ Gr( m, n )( Q ) : H (Λ) f ( u ( T Λ )) B } . (5.1) Then N m,n ( f ; B ) N m,n ( B ) → Z f d ν, B → ∞ . Proof.
The statements (i), (ii), and (iii) are equivalent. It suffices to check (ii) ona basis of the topology closed under finite intersections. Since the intersection oftwo boundary-controlled open sets is boundary-controlled and open, and since allopen balls along local charts are boundary controlled, the assertion follows.The last statement can be proved analogously to [12, Prop. 3.3 (b)] for continu-ous functions with compact support. It follows for arbitrary bounded, continuousfunctions f using Prokhorov’s theorem, cutting off f outside progressively largecompact subsets of U G m,n . (cid:3) Consequences of equidistribution
Equidistribution and freeness.
As a first application of the equidistributiontheorem, we reprove Theorem 1.3.
Alternative proof of Theorem 1.3.
Recall that E ε ( B ) = ε ( B ), whereΩ ε ( B ) = { Λ ∈ Gr( m, n )( Q ) : ℓ (Λ) ε, H (Λ) B } . Let κ = lim sup B →∞ E ε ( B ) N m,n ( B ) . We want to prove that κ = 0. Note that µ ( αL ) = µ ( L ) − log α for all lattices L . Applying this to quotient lattices yields µ min ( u ( L )) = µ min ( L ) − µ ( L ). More-over, recall that the logarithmic anticanonical height verifies h (Λ) = log H (Λ) = − log covol( T Λ ), whence µ ( T Λ ) = h (Λ) m ( n − m ) . (6.1)Thus ℓ (Λ) ε is equivalent to µ min ( u ( T Λ )) ε − m ( n − m ) h (Λ) . For
R >
0, setΩ (0) ( R ) = { Λ ∈ Gr( m, n )( Q ) : H (Λ) R } andΩ (1) ε ( R, B ) = (cid:26) Λ ∈ Gr( m, n )( Q ) : µ min ( u ( T Λ )) ε − m ( n − m ) log R, H (Λ) B (cid:27) . We note that Ω ε ( B ) ⊂ Ω (0) ( R ) ∪ Ω (1) ε ( R, B ) , for all R >
0. Clearly, lim sup B →∞ (0) ( R ) N m,n ( B ) = 0 . To bound the number of elements of the second set, let Z ε ( R ) = (cid:26) L ∈ U G m,n : µ min ( L ) ε − m ( n − m ) log R (cid:27) , and note thatΩ (1) ε ( R, B ) = { Λ ∈ Gr( m, n )( Q ) : u ( T Λ ) ∈ Z ε ( R ) , H (Λ) B } . Since µ min is clearly continuous, Z ε ( R ) is closed. It follows that κ lim sup B →∞ (1) ε ( R, B ) N m,n ( B ) ν ( Z ε ( R )) , by Theorem 5.8 (iii). The sets Z ε ( R ) form a decreasing sequence whose intersectionis empty, hence ν ( Z ε ( R )) → R → ∞ , completing the proof. (cid:3) Counting by maximal slope.
In [13, § x on a Fano variety V with a smooth integral model V Z by the maximal slope of their tangent lattice, instead of by height, so as to countthe quantity N µ max V ( B ) = { x ∈ V ( Q ) : µ max ( T V Z ,x ) log B } . Since the logarithmic anticanonical height h ( x ) = log H ( x ) verifies h ( x ) = dim( V ) µ ( T V Z ,x ) dim( V ) µ max ( T V Z ,x ) , bounding the maximal slope automatically also bounds the anticanonical height,yielding a trivial upper bound N µ max V ( B ) N V ( B dim V ) , where N V ( B ) = { x ∈ V ( Q ) : H ( x ) B } . For P n − = Gr(1 , n ), this results in theupper bound N µ max P n − ( B ) c ,n B n − (1 + o (1)) , by (1.2). In [13, Rem. 7.8], Peyre provides the lower bound N µ max P n − ( B ) ≫ η B n − − η for any η >
0, and expects that N µ max P n − ( B ) ∼ c ′ ,n B n − for a suitable constant c ′ ,n >
0. Theorem 1.4 confirms this expectation for projective space and providesan analogous asymptotic formula for all other Grassmannians.
QUIDISTRIBUTION AND FREENESS ON GRASSMANNIANS 23
Proof of Theorem 1.4.
Let N µ max m,n ( B ) = { Λ ∈ Gr( m, n ) : µ max ( T Λ ) log B } . Recall that µ ( αL ) = µ ( L ) − log α . Applying this to all sublattices of T Λ , we see that µ max ( T Λ ) = µ max ( u ( T Λ )) + µ ( T Λ ). Using this and (6.1), the condition e µ max ( T Λ ) B can be seen to be equivalent to H (Λ) / ( m ( n − m )) e − µ max ( u ( T Λ )) B. Hence N µ max m,n (cid:16) B m ( n − m ) (cid:17) = N m,n ( f ; B ) , for f = e − m ( n − m ) µ max , in the notation of (5.1). Since µ max is obviously continuous,so is f , and since µ max ( L ) > L , the function 0 f N µ max m,n ( B ) ∼ c ′ m,n B m ( n − m ) , where c ′ m,n = c m,n Z L ∈ U G m,n e − m ( n − m ) µ max ( L ) d ν. (6.2)It remains to prove that 0 < c ′ m,n < c m,n . Note that the non-empty open subsets U = { µ max ( L ) < } and U = { µ max ( L ) > } of U G m,n have positive measure.Now f | U > e − m ( n − m ) , hence R f d ν >
0. This implies that c ′ m,n >
0. From f R f d ν
1, and since f | U e − m ( n − m ) <
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