Degrees of unitary Deligne–Lusztig varieties
Abstract.
We prove an explicit degree formula for certain unitary Deligne–Lusztig varieties. Combining with an alternative degree formula in terms of Schubert calculus, we deduce several algebraic combinatorial identities which may be of independent interest.
1. Introduction
1.1. Degrees of unitary Deligne–Lusztig varieties
Let be a finite field of size . Let be the quadratic extension of . Let be a (nondegenerate) -hermitian space of dimension . Let be the Grassmannian of -dimensional subspaces of , which is a smooth projective variety over of dimension .
We assume that has dimension () over and take . Define to be the closed -subscheme parametrizing -dimensional subspaces such that , where is the orthogonal complement of in (see the more precise Definition 2.1). It is smooth, projective and geometrically irreducible of dimension , and is known as a unitary Deligne–Lusztig variety, as it can be identified as a generalized Deligne–Lusztig variety associated to a parabolic subgroup of the odd unitary group ([VW11, §4.5]). This class of varieties shows up in the study of the supersingular locus of unitary Shimura varieties ([VW11]) and plays an important role in the arithmetic of unitary Shimura varieties, such as the Kudla–Rapoport conjecture ([KR11, KR14]).
Our first main result is a simple degree formula for . Recall that the degree of a projective variety of pure dimension is given by the geometric intersection number of with general hyperplanes in . Denote by the degree of under the Plücker embedding .
Theorem 1.1.
Let be a -hermitian space of dimension . Then
(1.1.1) |
Remark 1.2.
Here (and below) we regard the empty product (when ) as 1. The right hand side of (1.1.1) can also be interpreted as the -analogue of the double factorial
where is the -analogue of .
Theorem 1.1 will be proved in §2.6. Its proof is inspired by the higher local modularity in the recent proof of the Kudla–Rapoport conjecture (cf. [LZ22a, §6.4]). In fact, the key formulas (Propositions 2.11, 2.12) can be extracted from certain vertical intersection formulas ([LZ22a, Lemmas 6.4.5, 6.4.6]) on unitary Rapoport–Zink spaces. To make the ideas more transparent, here we work directly on and introduce the notions of special cycles (Definition 2.3) and a special line bundle (Definition 2.5) on . These notions may be viewed respectively as finite field analogues of special cycles and tautological line bundles on unitary Shimura varieties. From this perspective, the degree formula in Theorem 1.1, or more precisely the formula for in Proposition 2.12, may be viewed as a finite field analogue of the constant term formula in Kudla’s geometric Siegel–Weil formula (relating the geometric volume of unitary Shimura varieties and an abelian -value, cf. [Kud04, (4.4)] for the analogue for orthogonal Shimura varieties).
The proof of Theorem 1.1 ultimately relies on identifying the special cycles as unitary Deligne–Lusztig subvarieties and the (proved) Tate conjecture for , in order to perform induction on the dimension . These inductive structures are available for Deligne–Lusztig varieties beyond those of unitary types (e.g., the type considered in [LZ22b, §7.6] and two other types listed in [HLZ19, §3]), and it would be interesting to extend the method to obtain degree formulas for more general Deligne–Lusztig varieties.
1.2. Schubert calculus and applications to algebraic combinatorics
We also prove a different formula for in terms of Schubert calculus on Grassmannians (see also the related general works [Kim20, HP20]).
Theorem 1.3.
The following identity holds in :
(1.3.1) |
where the sum runs over all integer tuples with . Here (as recalled in §3.1):
-
,
-
is the Schubert class,
-
is the complement of defined by ,
-
is the conjugate of .
Corollary 1.4.
-
(i)
The right hand side of (1.3.1) is equal to .
-
(ii)
The -coefficient of
is equal to .
-
(iii)
Let . The following two sets have the same size:
-
The set of standard Young tableaux of skew shape , where runs over such that and . Here is the dual of (recalled in §3.1).
-
The set of ordered partitions of satisfying and for all .
-
1.3. Acknowledgments
It is the author’s pleasure to dedicate this paper to Steve Kudla on the occasion of his 70th birthday. The influence of his original insights on the geometric and arithmetic Siegel–Weil formula on this paper should be evident to the readers. The author is also grateful to Z. Yun and W. Zhang for helpful conversations, and to M. Rapoport and the anonymous referee for useful comments. The author’s work is partially supported by the NSF grant DMS-2101157.
1.4. Notation
Let be a smooth projective variety over a finite field . Let be the Chow group of codimension algebraic cycles of defined modulo rational equivalence. Fix a prime , denote by
the -adic cycle class map. Denote by the subspace of Tate classes, i.e., the elements fixed by an open subgroup of . Then intertwines the intersection product on the Chow ring and the cup product on the cohomology ring, namely the following diagram commutes,
(1.4.1) |
When , the cycle class map is injective. When , we often identify via the degree isomorphism (cf. [Ful98, Example 1.6.6]). Recall that the Tate conjecture ([Tat65, Conjecture 1], or [Tat94, Conjecture ]) asserts that for any , the image of -adic cycle class map -spans .
For a subvariety of pure codimension , we denote by its class in . When the ambient variety is clear we suppress the subscript and simply write . For any vector bundle on , denote by its -th Chern class. By abusing notation we also denote by the same symbols , and their images in . For any vector bundle on , denote by its dual bundle.
2. Degrees of unitary Deligne–Lusztig varieties
Let be a prime and let be a power of . Let be a finite field of size . Let be the quadratic extension of . Denote by the absolute -Frobenius endomorphism on any scheme of characteristic .
2.1. Hermitian spaces
Let be a (nondegenerate) -hermitian space of dimension , i.e., a -vector space of dimension equipped with a (nondegenerate) paring that is linear in the first variable, -linear in the second variable and satisfies
(2.0.1) |
for any . For any -subspace , denote by
its orthogonal complement.
More generally, for any -scheme , put . Then there is a unique pairing extending that is -linear in the first variable and -linear in the second variable, given by
for any and . For any subbundle , we denote by
its left orthogonal complement. (Notice that unlike the case , in general the symmetry (2.0.1) does not necessarily hold for , and the left orthogonal complement does not necessarily agree with the right orthogonal complement.)
2.2. Unitary Deligne–Lusztig varieties and special cycles
From now on fix a -hermitian space of dimension ().
Definition 2.1.
Define the unitary Deligne–Lusztig variety to be the closed -subscheme parametrizing -dimensional subspaces such that , i.e., for any -scheme ,
The relation between and classical Deligne–Lusztig varieties associated to the unitary group can be found in [Vol10, Corollary 2.17] and the reader can recognize that the Frobenius appearing in the definition of classical Deligne–Lusztig varieties arises from the -linearity in the second variable of the hermitian form .
Definition 2.2.
Define a special subspace to be a -subspace such that . If is a special subspace, then is a (nondegenerate) -hermitian space under the pairing induced from .
Definition 2.3.
Let be a -subspace. Define the special cycle to be the closed subscheme parametrizing subspaces satisfying , i.e., for any -scheme ,
By definition it is clear that
-
is nonempty only if is a special subspace.
-
,
-
for two subspaces ( denotes the scheme-theoretic intersection).
-
if .
We summarize several known results on and which we will need later.
Proposition 2.4.
-
(i)
is smooth, projective and geometrically irreducible of dimension .
-
(ii)
Let be a special subspace of codimension . Then there is a canonical isomorphism of -schemes
(2.4.1) In particular, is smooth, projective and geometrically irreducible, and has codimension in , and we call a codimension special cycle in .
-
(iii)
The Tate conjecture (see §1.4) holds for .
-
(iv)
For any , the space is spanned by the cycle classes of codimension special cycles , where runs over all special subspace of codimension .
-
(v)
For any , the cup product induces a perfect pairing
Proof.
2.3. Natural vector bundles on and
Definition 2.5.
Define
-
the universal bundle on ,
-
and the two universal subbundles on ,
-
the universal quotient bundle on ,
-
which we call the special line bundle on .
Similarly for a special cycle associated to a special subspace , define
-
the universal bundle on ,
-
and the two universal subbundles on ,
-
the universal quotient bundle on ,
-
which we call the special line bundle on .
Denote by
the natural closed embedding. By definition, we have
(2.5.1) |
Also by definition, the canonical isomorphism (2.4.1) induces a canonical isomorphism and thus
(2.5.2) |
Proposition 2.6.
Let be a special subspace of codimension .
-
(i)
We have a canonical isomorphism
(2.6.1) -
(ii)
Let be the normal bundle of in . Then we have an isomorphism
In particular, when has codimension 1, we have an isomorphism of line bundles
(2.6.2)
2.4. Relations with Grassmannians
For , let be the Grassmannian of -dimensional subspaces of , which is a smooth projective variety over of dimension . We have the Plücker embedding
defined by sending an -dimensional subspace with basis to the line generated by (independent of the choice of the basis).
Definition 2.7.
Define (resp. ) to be the universal subbundle (resp. universal quotient subbundle) on . Denote by
the natural closed embedding.
Lemma 2.8.
The following identities holds:
-
(i)
,
-
(ii)
,
-
(iii)
,
-
(iv)
.
Proof.
-
(i)
It follows from the definition of and .
-
(ii)
It follows from Item (i) and the nondegenerate pairing (-linear in the second variable).
-
(iii)
It follows from the defining exact sequence
- (iv)
2.5. Chern classes of the special line bundle
Lemma 2.10.
Let be a 1-dimensional special cycle associated to a special subspace of codimension . Then the following identity holds in :
Proof.
Proposition 2.11.
The following identity holds in :
(2.11.1) |
here the sum runs over all special subspaces of codimension 1.
Proof.
When , the number of special subspaces is equal to the number of isotropic lines in the 3-dimensional -hermitian space , which is (e.g., by [LZ22a, Lemma 1.9.1]). Hence the right hand side of (2.11.1) evaluates to
in , which agrees with the left hand side by (2.9.1).
When , by Proposition 2.4 (iv) (v) and the commutativity of (1.4.1), to prove (2.11.1) it suffices to show that for any 1-dimensional special cycle (associated to any codimension special subspace ), the following identity holds in ,
(2.11.2) |
For the terms of the right hand side of (2.11.2), we have three cases.
- (i)
- (ii)
-
(iii)
When and is not a special subspace, we know that is empty, and thus
Now we count the number of terms on the right hand side of (2.11.2) in first two cases.
-
The association gives a bijection between the set of codimension 1 special subspaces in Case (ii) and the set of isotropic -lines in . Hence the number of terms in Case (ii) is equal to the number of vectors in times the number of isotropic lines in the -hermitian space . This is , as is of dimension and is of dimension 3.
Thus the right hand side of (2.11.2) evaluates to
which is equal to the left hand side by Lemma 2.10 applied to the 1-dimensional special cycle . ∎
Proposition 2.12.
The following identity holds in :
Proof.
When , this is (2.9.1). In general, we induct on . By Proposition 2.11, we obtain
(2.12.1) |
By the projection formula and (2.5.2), we have (under the isomorphism (2.4.1)),
which evaluates to by the induction hypothesis.
The association gives a bijection the set of codimension 1 special subspaces and the set of isotropic -lines in . Thus the total number of terms in (2.12.1) is the number of isotropic -lines in the -dimensional -hermitian space , which is (e.g., by [LZ22a, Lemma 1.9.1]). Thus (2.12.1) evaluates to
This completes the proof. ∎
2.6. Proof of Theorem 1.1
Recall is the Plücker embedding. By definition
(2.12.2) |
is the intersection number of with general hyperplanes in . By the projection formula, we obtain that
in . By the definition of the Plücker embedding, we have
Thus
and hence
The result then follows from Proposition 2.12.
3. Schubert calculus
3.1. Reminder on Schubert calculus (cf. [EH16, Chapter 4],[Ful98, §14.7])
Let . The Schubert classes of are indexed by -tuples of integers satisfying
in other words, indexed by Young diagrams inside the rectangles. For such an -tuple , define a Schubert cycle
where
is a complete flag in . The Schubert cycle is a closed subvariety of codimension . Define the Schubert class
which is independent of the choice of the complete flag . We use the standard notation suppressing trailing zeros in the indices. In particular, by definition is the hyperplane class under the Plücker embedding.
Define the dual . The Schubert classes form a -basis of and the intersection pairing
is perfect and has Schubert classes as dual basis, with and dual to each other if and only if .
Define the conjugate such that is the number of ’s such that . The Young diagrams of and are mutual reflections along the main diagonal. The canonical isomorphism
(3.0.1) |
maps to .
3.2. The class of
Let be the variety parametrizing partial flags
where and . Define a closed embedding
Also consider the closed embedding
Then by definition induces an isomorphism between and .
Proposition 3.1.
The following identity holds in :
where the sum runs over
-
satisfying ,
-
satisfying , for .
Proof.
Let and be two transverse complete flags in ([EH16, Definition 4.4]). Recall that and are transverse means that for any . Let
Then by definition
and
By the transversality it is easy to see that the intersection of with is nonempty if and only if
in which case the intersection is transverse at the unique point given by
Therefore
The desired result then follows from the fact that Schubert classes form dual basis under the intersection pairing. ∎
Corollary 3.2.
The following identity holds in :
where the sum runs over such that , and is the complement of defined by .
Proof.
3.3. Proof of Theorem 1.3
3.4. Proof of Corollary 1.4
- (i)
-
(ii)
Let
() be the Schur polynomial associated to ([Mac92]). It is a symmetric polynomial of degree . Write
where runs over with . By definition, the coefficient is given by the coefficient of in
(3.3.1) Since the class of a point is , we know that
i.e., the coefficient of in (3.3.1). It remains to compute (3.3.1). By the dual Cauchy identity for Schur polynomials ([Mac92, 0.11’]), we have
By definition, we have
Therefore
and thus
(3.3.2) The result then follows from Item (i).
-
(iii)
By applying Pieri’s formula ([EH16, Proposition 4.9]) times, we know that the term is equal to the number of sequences of Young diagrams starting with and ending with such that each has exactly one more box than . Equivalently, it is the number of standard Young tableaux of skew shape . Now Item (i) shows that the number of such standard Young tableaux with is equal to the coefficient of in
which is equal to the number of ordered partitions of satisfying the extra conditions
Example 3.4.
We end with an example illustrating Corollary 1.4 (ii) (iii).
-
When , the coefficient of is equal to
as in Corollary 1.4 (ii). In Table ‣ 3.4 we list all standard Young tableaux of skew shape with as in Corollary 1.4 (iii). Notice the total number of such Young tableaux is 8, which indeed agrees with the coefficient of .
{ytableau} \none \none 1 \none \none \none 2 3 {ytableau} \none \none 2 \none \none \none 1 3 {ytableau} \none \none 3 \none \none \none 1 2 Table 1. standard Young tableaux of skew shape References
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