On the Chow groups of hypersurfaces in symplectic Grassmannians
aa r X i v : . [ m a t h . AG ] D ec ON THE CHOW GROUPS OF HYPERSURFACES IN SYMPLECTICGRASSMANNIANS
ROBERT LATERVEERA
BSTRACT . Let Y be a Pl¨ucker hypersurface in a symplectic Grassmannian I Gr(3 , n ) or abisymplectic Grassmannian I Gr(3 , n ) . We show that many Chow groups of Y inject into coho-mology.
1. I
NTRODUCTION
Given a smooth projective variety Y over C , let A i ( Y ) := CH i ( Y ) Q denote the Chow groups of Y (i.e. the groups of i -dimensional algebraic cycles on Y with Q -coefficients, modulo rationalequivalence). Let A homi ( Y ) ⊂ A i ( Y ) denote the subgroup of homologically trivial cycles.The famous Bloch–Beilinson conjectures [8], [25] predict that the Hodge level of the coho-mology of Y should have an influence on the size of the Chow groups of Y . For surfaces, this isthe notorious Bloch conjecture, which is still an open problem. For hypersurfaces in projectivespace, the precise prediction is as follows: Conjecture 1.1.
Let Y ⊂ P n be a smooth hypersurface of degree d . Then A homi ( Y ) = 0 ∀ i ≤ nd − . Conjecture 1.1 is still open; partial results have been obtained in [16], [23], [19], [5], [6].In [14], I considered a version of Conjecture 1.1 for Pl¨ucker hyperplane sections of Grass-mannians. In this note, we look at the case of Pl¨ucker hyperplane sections of symplectic Grass-mannians . Recall that inside the Grassmannian
Gr(3 , n ) (of -dimensional subspaces of an n -dimensional vector space), the symplectic Grassmannian I Gr(3 , n ) ⊂ Gr(3 , n ) parametrizessubspaces that are isotropic with respect to some fixed skew-symmetric 2-form. The preciseprediction (cf. subsection 3.2 below) is as follows: Conjecture 1.2.
Let Y := I Gr(3 , n ) ∩ H ⊂ P ( n ) − be a smooth hyperplane section (with respect to the Pl¨ucker embedding). Then A homi ( Y ) = 0 ∀ i ≤ n − . The main result of this note is a partial verification of Conjecture 1.2:
Mathematics Subject Classification.
Primary 14C15, 14C25, 14C30.
Key words and phrases.
Algebraic cycles, Chow groups, motive, Bloch-Beilinson conjectures.
Theorem (=Theorem 3.4) . Let Y := I Gr(3 , n ) ∩ H ⊂ P ( n ) − be a smooth hyperplane section (with respect to the Pl¨ucker embedding). Then A homi ( Y ) = 0 ∀ i ≤ n − . Moreover, in case n ≤ or n = 12 we have A homi ( Y ) = 0 ∀ i ≤ n − . To prove Theorem 3.4, we rely on the recent notion of projections among (symplectic) Grass-mannians [2]. Combined with the Chow-theoretic Cayley trick [9], this reduces Theorem 3.4 tounderstanding the Chow groups of a hyperplane section in an ordinary Grassmannian
Gr(3 , n +1) . This last problem was handled in [14].As a consequence of Theorem 3.4, some instances of the generalized Hodge conjecture areverified:
Corollary (=Corollary 4.1) . Let Y be as in Theorem 3.4, and assume n ≤ or n = 12 . Then H dim Y ( Y, Q ) is supported on a subvariety of codimension n − . Other consequences are as follows:
Corollary (=Corollary 4.2) . Let Y := I Gr(3 , n ) ∩ H ⊂ P ( n ) − be a smooth hyperplane section (with respect to the Pl¨ucker embedding).(i) If n ≤ , then Y has finite-dimensional motive (in the sense of [11] ).(ii) If n ≤ , then Y has trivial Griffiths groups (and so Voevodsky’s smash conjecture is true for Y ).(iii) If n ≤ , the Hodge conjecture is true for Y . Applying the same method, we can also say something about hypersurfaces in bisymplecticGrassmannians . (Recall that the bisymplectic Grassmannian I Gr(3 , n ) ⊂ Gr(3 , n ) is the locusof 3-spaces that are isotropic with respect to two fixed generic skew-forms.) Theorem (=Theorem 3.5) . Assume n is even, and let Y := I Gr(3 , n ) ∩ H ⊂ P ( n ) − be a smooth hyperplane section (with respect to the Pl¨ucker embedding). Then A homi ( Y ) = 0 ∀ i ≤ n − . Moreover, in case n ≤ we have A homi ( Y ) = 0 ∀ i ≤ n − . The hyperplane sections I Gr(3 , ∩ H and I Gr(3 , ∩ H are of particular interest: they areFano varieties of K3 type, and they are related to hyperplane sections Gr(3 , ∩ H and henceto Debarre–Voisin hyperk¨ahler fourfolds, cf. [2, Section 4]. N THE CHOW GROUPS OF HYPERSURFACES IN SYMPLECTIC GRASSMANNIANS 3
Conventions.
In this note, the word variety will refer to a reduced irreducible scheme of finitetype over C . A subvariety is a (possibly reducible) reduced subscheme which is equidimensional. All Chow groups will be with rational coefficients : we denote by A j ( Y ) := CH j ( Y ) Q theChow group of j -dimensional cycles on Y with Q -coefficients; for Y smooth of dimension n thenotations A j ( Y ) and A n − j ( Y ) are used interchangeably. The notations A jhom ( Y ) and A jAJ ( X ) will be used to indicate the subgroup of homologically trivial (resp. Abel–Jacobi trivial) cycles.The contravariant category of Chow motives (i.e., pure motives with respect to rational equiv-alence as in [20] , [17] ) will be denoted M rat .
2. P
RELIMINARIES
Cayley’s trick and Chow groups.Theorem 2.1 (Jiang [9]) . Let E → U be a vector bundle of rank r ≥ over a projective variety U , and let S := s − (0) ⊂ U be the zero locus of a regular section s ∈ H ( U, E ) such that S is smooth of dimension dim U − rank E . Let X := w − (0) ⊂ P ( E ) be the zero locus of theregular section w ∈ H ( P ( E ) , O P ( E ) (1)) that corresponds to s under the natural isomorphism H ( U, E ) ∼ = H ( P ( E ) , O P ( E ) (1)) . There is an isomorphism of integral Chow motives h ( X ) ∼ = h ( S )(1 − r ) ⊕ r − M i =0 h ( U )( − i ) in M Z rat . Proof.
This is [9, Theorem 3.1]. Both the isomorphism and its inverse are explicitly described. (cid:3)
Remark 2.2.
In the set-up of Theorem 2.1, a cohomological relation between X , S and U wasestablished in [12, Prop. 4.3] (cf. also [7, section 3.7], as well as [2, Proposition 46] for ageneralization). A relation on the level of derived categories was established in [18, Theorem2.10] (cf. also [10, Theorem 2.4] and [2, Proposition 47]).2.2. Linear sections of
Gr(2 , n ) .Proposition 2.3. Let Y := Gr(2 , n ) ∩ H ∩ · · · ∩ H s ⊂ P ( n ) − be a smooth dimensionally transverse intersection with s hyperplanes (with respect to the Pl¨uckerembedding). Assume s ≤ . Then A homi ( Y ) = 0 ∀ i . Proof.
This uses a geometric construction that can be found in [4].Let P ⊂ P ( V n ) be a fixed hyperplane, and consider (as in [4, Section 2.3]) the rational map Gr(2 , V n ) P ROBERT LATERVEER sending a line in P ( V n ) to its intersection with P . This map is resolved by blowing up a subvariety σ ( P ) ∼ = Gr(2 , n − , resulting in a morphism Γ : f Gr → P (where f Gr → Gr(2 , V n ) denotes the blow-up with center σ ( P ) ).Let e Y → Y be the blow-up of Y with center σ ( P ) ∩ Y , and let us consider the morphism Γ Y : e Y → P , obtained by restricting Γ .In case s = 1 and P is generic with respect to Y , the morphism Γ Y is a P n − -fibration over P .It follows that e Y , and hence Y , has trivial Chow groups.In case s = 2 , and P chosen generically with respect to Y , the morphism Γ Y is generically a P n − -fibration over P , and there are finitely many points in P where the fiber is P n − . ApplyingTheorem 2.1, this implies that e Y , and hence also Y , has trivial Chow groups. (cid:3) Hyperplane sections of
Gr(3 , n ) .Theorem 2.4. Let Y := Gr(3 , n ) ∩ H ⊂ P ( n ) − be a smooth hyperplane section (with respect to the Pl¨ucker embedding). Then A homi ( Y ) = 0 ∀ i ≤ n − . Moreover, in case n ≤ or n = 13 we have A homi ( Y ) = 0 ∀ i ≤ n − . Proof.
This is [14, Theorems 3.1 and 3.2], which uses the notion of jumps between Grassmanni-ans as developed in [2]. (cid:3)
3. M
AIN RESULTS
Projections.
As in [2], let I r Gr( k, n ) ⊂ Gr( k, n ) parametrize linear subspaces that areisotropic with respect to r fixed generic skew-forms. One has dim I r Gr( k, n ) = k ( n − k ) − r (cid:18) k (cid:19) . For example, I r Gr(2 , n ) is just the intersection of Gr(2 , n ) with r Pl¨ucker hyperplanes. The case I Gr( k, n ) is studied in detail in [1].To relate hyperplane sections of different symplectic Grassmannians, Bernardara–Fatighenti–Manivel [2] have developed a theory of projections . The starting point is a rational map π : Gr( k, n + 1) Gr( k, n ) , determined by the choice of a line in the n + 1 -dimensional vector space. If Y ′ is a hyperplanesection of I r Gr( k, n + 1) , one can restrict π to Y ′ . A detailed analysis of the case k = 3 yieldsthe following: N THE CHOW GROUPS OF HYPERSURFACES IN SYMPLECTIC GRASSMANNIANS 5
Theorem 3.1 ([2]) . Assume n is even and r ≤ , or n is odd and r = 0 . Let Y ′ := I r Gr(3 , n + 1) ∩ H be a smooth hyperplane section. There exists a commutative diagram E ֒ → e Y ′ ← ֓ F ↓ ↓ σ ց τ ց q Z ′ ֒ → Y ′ π ′ I r Gr(3 , n ) ← ֓ Y where Y := I r +1 Gr(3 , n ) ∩ H is a smooth hyperplane section. The morphism σ is the blow-upwith center Z ′ ∼ = I r +1 Gr(2 , n ) . The morphism q is a P -fibration, while τ is a P -fibration overthe complement of Y .Proof. This is contained in [2, Section 3.2] (NB: note that our n is n − in loc. cit.). As explainedin loc. cit., the assumptions on n and r guarantee that the target I r Gr(3 , n ) and the hyperplanesection Y are generic and hence smooth. (cid:3) Motivating the conjecture.
As a consequence of Theorem 3.1, one can compute the Hodgelevel of hyperplane sections Y of symplectic Grassmannians: surprisingly, it turns out that (atleast for n > ) Y is of “Calabi–Yau type”: Theorem 3.2 ([2]) . Let Y := I Gr(3 , n ) ∩ H ⊂ P ( nk ) − be a smooth hyperplane section (with respect to the Pl¨ucker embedding). Assume n > . Then Y has Hodge coniveau n − . More precisely, the Hodge numbers verify h p, dim Y − p ( Y ) = ( if p = n − , if p < n − . Proof.
This is implicit in [2], as we now explain. Let the set-up be as in Theorem 3.1. Then [2,Proposition 6] relates Y and Y ′ on the level of cohomology: one has an isomorphism of Hodgestructures H j − ( Y, Q )( − ⊕ M i =0 H j − i ( I r Gr(3 , n ) , Q )( − i ) ∼ = −→ H j ( Y ′ , Q ) ⊕ c − M i =1 H j − i ( I r +1 Gr(2 , n ) , Q )( − i ) (1)(where c denotes the codimension of Z ′ in Y ′ ). Setting r = 0 and combining with [2, Theorem 3](which gives the Hodge numbers of Y ′ ), plus the fact that Gr(3 , n ) and I Gr(2 , n ) have algebraiccohomology, this gives the required Hodge numbers of Y . (cid:3) ROBERT LATERVEER
Theorem 3.2 motivates Conjecture 1.2. Indeed, the generalized Bloch conjecture [25, Conjec-ture 1.10] predicts that any variety Y with Hodge coniveau ≥ c has A homi ( Y ) = 0 ∀ i < c . Note that at least for n > , the bound of Conjecture 1.2 is optimal: assuming A homi ( Y ) = 0 for j ≤ n − and applying the Bloch–Srinivas argument [3], one would get the vanishing h n − , dim Y − n +3 ( Y ) = 0 , contradicting Theorem 3.2.3.3. A relation of motives.
The cohomological relation (1) between Y and Y ′ also exists as (andactually is implied by) a relation on the level of the Grothendieck ring of varieties [2, Proposition4], and on the level of derived categories [2, Proposition 5]. To complete the picture, we now liftthe relation (1) to the level of Chow motives: Proposition 3.3.
Let notation and assumptions be as in Theorem 3.1. There is an isomorphismof integral Chow motives h ( Y )( − ⊕ M i =0 h ( I r Gr(3 , n ))( − i ) ∼ = −→ h ( Y ′ ) ⊕ c − M i =1 h ( I r +1 Gr(2 , n ))( − i ) in M Z rat . (Here c denotes the codimension of Z ′ in Y ′ .)Proof. The idea is to express the motive of e Y ′ in two different ways:The blow-up formula expresses h ( e Y ′ ) in terms of h ( Y ′ ) ; this gives the right-hand side of therelation.Looking at [2, Section 3.2], one finds that e Y ′ is the total space of a projectivization P ( E ) where E is the vector bundle E := O ⊕ U ∗ on I r Gr(3 , n ) (in the notation of loc. cit.), and Y is given by a section of E . That is, we are in the setting of Cayley’s trick, and so Theorem 2.1expresses h ( e Y ′ ) in terms of h ( Y ) ; this gives the left-hand side of the relation. (cid:3) Hyperplane sections of I Gr(3 , n ) .Theorem 3.4. Let Y := I Gr(3 , n ) ∩ H ⊂ P ( n ) − be a smooth hyperplane section (with respect to the Pl¨ucker embedding). Then A homi ( Y ) = 0 ∀ i ≤ n − . Moreover, in case n ≤ or n = 12 we have A homi ( Y ) = 0 ∀ i ≤ n − . Proof.
A generic hyperplane section Y is attained by the construction of Theorem 3.1 with r = 0 ,i.e. there is a smooth hyperplane section Y ′ := Gr(3 , n + 1) ∩ H ,
N THE CHOW GROUPS OF HYPERSURFACES IN SYMPLECTIC GRASSMANNIANS 7 related to Y via the projection of Theorem 3.1. In this case, Proposition 3.3 implies that there isan injection of Chow groups A homi ( Y ) ֒ → A homi +3 ( Y ′ ) ⊕ M A hom ∗ ( I Gr(2 , n )) . The symplectic Grassmannian I Gr(2 , n ) is nothing but a Pl¨ucker hyperplane section of Gr(2 , n ) ,and so Proposition 2.3 gives the vanishing A hom ∗ ( I Gr(2 , n )) = 0 . The variety Y ′ is a hyperplane section of Gr(3 , n + 1) , and so Theorem 2.4 gives the vanishing A homi +3 ( Y ′ ) = 0 ∀ i ≤ n − , with the additional vanishing for i = n − for small n . This proves the theorem for genericsections Y .A standard spread argument allows to extend to all smooth hyperplane sections: Let Y → B denote the universal family of all smooth hyperplane sections of Gr( k, n ) , and let B ◦ ⊂ B denotethe Zariski open subset parametrizing smooth Y verifying the set-up of Theorem 3.1. Doing theBloch–Srinivas argument [3] (cf. also [13]), the above implies that for each b ∈ B ◦ one has adecomposition of the diagonal(2) ∆ Y b = γ b + δ b in A dim Y b ( Y b × Y b ) where γ b is completely decomposed (i.e. γ b ∈ A ∗ ( Y b ) ⊗ A ∗ ( Y b ) ) and δ b is supported on Y b × W b with codim W b = n − (and codim W b = n − for small n ). Using the Hilbert schemesargument of [24, Proposition 3.7] (cf. also [15, Proposition A.1] for the precise form used here),the γ b , δ b , W b exist relatively, i.e. one can find a cycle γ ∈ ( p ) ∗ A ∗ ( Y ) · ( p ) ∗ A ∗ ( Y ) , a subvariety W ⊂ Y of codimension n − , and a cycle δ supported on Y × B ◦ W such that ∆ Y | b = γ | b + δ | b in A dim Y b ( Y b × Y b ) ∀ b ∈ B ◦ . Let ¯ γ, ¯ δ ∈ A dim Y b ( Y × B Y ) be cycles that restrict to γ resp. δ . The spread lemma [25, Lemma3.2] implies that ∆ Y | b = ¯ γ | b + ¯ δ | b in A dim Y b ( Y b × Y b ) ∀ b ∈ B .
Given any b ∈ B \ B ◦ , using the moving lemma, one can find representatives for ¯ γ and ¯ δ ingeneral position with respect to the fiber Y b × Y b . Restricting to the fiber, this implies that thediagonal of Y b has a decomposition as in (2). Letting the decomposition (2) act on Chow groups,this shows that A homi ( Y b ) = 0 ∀ i ≤ n − , ∀ b ∈ B (with the additional vanishing for i = n − for small n ). (cid:3) Hyperplane sections of I Gr(3 , n ) .Theorem 3.5. Assume n is even, and let Y := I Gr(3 , n ) ∩ H ⊂ P ( n ) − be a smooth hyperplane section (with respect to the Pl¨ucker embedding). Then A homi ( Y ) = 0 ∀ i ≤ n − . ROBERT LATERVEER
Moreover, in case n ≤ we have A homi ( Y ) = 0 ∀ i ≤ n − . Proof.
For n even, a generic hyperplane section Y is attained by the construction of Theorem 3.1with r = 1 , i.e. there is a smooth hyperplane section Y ′ := I Gr(3 , n + 1) ∩ H , related to Y via the projection of Theorem 3.1. In this case, Proposition 3.3 implies that there isan injection of Chow groups A homi ( Y ) ֒ → A homi +3 ( Y ′ ) ⊕ M A hom ∗ ( I Gr(2 , n )) . The bisymplectic Grassmannian I Gr(2 , n ) is nothing but an intersection Gr(2 , n ) ∩ H ∩ H (where the H j are Pl¨ucker hyperplanes), and so Proposition 2.3 gives the vanishing A hom ∗ ( I Gr(2 , n )) = 0 . The variety Y ′ is a hyperplane section of I Gr(3 , n + 1) , and so Theorem 3.4 gives the vanishing A homi +3 ( Y ′ ) = 0 ∀ i ≤ n − , with the additional vanishing for i = n − for small n . This proves the theorem for genericsections Y .The extension to all smooth hyperplane sections Y is done just as in the proof of Theorem3.4. (cid:3)
4. S
OME CONSEQUENCES
Corollary 4.1. (i) Let Y be as in Theorem 3.4 and n ≤ or n = 12 , or as in Theorem 3.5 and n ≤ . Then H dim Y ( Y, Q ) is supported on a subvariety of codimension n − .(ii) Let Y be as in Theorem 3.5 and n ≤ . Then H dim Y ( Y, Q ) is supported on a subvariety ofcodimension n − .Proof. This follows in standard fashion from the Bloch–Srinivas argument [3]. Let us treat (i)(the argument for (ii) is the same). The vanishing A homi ( Y ) = 0 ∀ i ≤ n − (Theorem 3.4) is equivalent to the decomposition ∆ Y = γ + δ in A dim Y ( Y × Y ) , where γ is a completely decomposed cycle (i.e. γ ∈ A ∗ ( Y ) ⊗ A ∗ ( Y ) ), and δ has support on Y × W with W ⊂ Y of codimension n − (to see this equivalence, one can look for instance at [13,Theorem 1.7]). Let H dim Ytr ( Y, Q ) denote the transcendental cohomology (i.e. the complement ofthe algebraic part under the cup product pairing). The cycle γ does not act on H dim Ytr ( Y, Q ) . Theaction of δ on H dim Ytr ( Y, Q ) factors over W , and so H dim Ytr ( Y, Q ) ⊂ H dim YW ( Y, Q ) . Since the algebraic part of H dim Y ( Y, Q ) is (by definition) supported in codimension dim Y / ,this settles the corollary. (cid:3) N THE CHOW GROUPS OF HYPERSURFACES IN SYMPLECTIC GRASSMANNIANS 9
Corollary 4.2.
Let Y := I Gr(3 , n ) ∩ H ⊂ P ( n ) − be a smooth hyperplane section (with respect to the Pl¨ucker embedding).(i) If n ≤ , then Y has finite-dimensional motive (in the sense of [11] ).(ii) If n ≤ , then Y has trivial Griffiths groups (and so Voevodsky’s smash conjecture [22] istrue for Y , i.e. numerical equivalence and smash-equivalence coincide on Y ).(iii) If n ≤ , the Hodge conjecture is true for Y .Proof. This is similar to the argument of Corollary 4.1.(i) The vanishing A homi ( Y ) = 0 ∀ i ≤ n − (Theorem 3.4) is equivalent to the decomposition of the diagonal ∆ Y = γ + δ in A dim Y ( Y × Y ) , where γ is a completely decomposed cycle, and δ has support on Y × W with W ⊂ Y ofcodimension n − (cf. [3] or [13]). The dimension of Y is n − , and so (looking at the actionof the diagonal) one finds that A ∗ AJ ( Y ) = 0 as long as n ≤ . This implies Kimura finite-dimensionality of Y [21, Theorem 4].(ii) The vanishing of Theorem 3.4 implies thatNiveau ( A ∗ ( Y )) ≤ (in the sense of [13]), i.e. the motive of Y factors over a surface. Since surfaces have trivialGriffiths groups, the conclusion follows.(iii) The vanishing of Theorem 3.4 implies thatNiveau ( A ∗ ( Y )) ≤ (in the sense of [13]), i.e. the motive of Y factors over a threefold. Since threefolds verify theHodge conjecture, the conclusion follows. (cid:3) We leave it to the zealous reader to formulate and prove a version of Corollary 4.2 for bisym-plectic Grassmannians.
Acknowledgments.
Thanks to Kai and Len for enjoying Kuifje movies. Thanks to the refereefor constructive comments that helped to improve the presentation.R
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