algebraic cycles and equivalences
TRANSCRIPT
Algebraic cycles and equivalences
B. Wang ( 汪 镔)
October 24, 2021
AbstractThe paper has two related topics: 1) A method of construction of
algebraic cycles, called cone construction; 2) Equivalences on the coneconstruction. The study of them results in a proof of the Lefschetz stan-dard conjecture.
Contents
1 Algebraic cycles 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Statement on algebraic cycles . . . . . . . . . . . . . . . . 11.1.2 Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Cycle-intersection . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.3 Cone family . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3.1 Cone family of cycles . . . . . . . . . . . . . . . . . . . . . 101.3.2 End cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.4 Cone operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2 Equivalences-A proof of the Lefschetz standard conjecture 272.1 Introduction to the Lefschetz standard conjecture . . . . . . . . . 272.2 The proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2.1 The main idea . . . . . . . . . . . . . . . . . . . . . . . . 292.2.2 Extension of rational equivalence . . . . . . . . . . . . . . 302.2.3 Cohomologicity . . . . . . . . . . . . . . . . . . . . . . . . 35
1 Algebraic cycles
1.1 Introduction
1.1.1 Statement on algebraic cycles
We present a construction of algebraic cycles in the first section, followed by thecohomological descend in the second.
Key words: Intersection theory, algebraic cycles, descend, Lefschetz standard conjecture,2020 Mathematics subject classification : 14C17, 14C25, 14F45, 14C15, 14D06
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We work with projective varieties over an algebraically closed field k. Through-out the paper, we let X be a smooth projective variety of dimension n ≥ 2 witha polarization. Let Z(X) be the Q vector space freely generated by irreduciblesubvarieties, CH(X) the total Chow group with Q coefficients. Let the subscriptof them denote the dimension of the cycles, and superscript the codimension.For another smooth projective variety Y , the dash homomorphism
Z(X) 99K Z(Y ) (1.1)
denotes a pair of a subspace Z(X) ⊂ Z(X) and a well-defined homomorphism
Z(X) → Z(Y ). (1.2)
We call Z(X) the domain of the dash homomorphism. The composition andthe equality follow the convention for one variable functions in calculus. For anatural number h ≤ n, let V h be a smooth h-codimensional plane section of X.For any algebraic cycle σ meeting V h properly, through the Serre’s Tor formula,there is an intersection cycle between σ and V h. In this way, we obtain the dashhomomorphism of the intersection with V h
vh : Z(X) 99K ZV h(X) = Z(V h) (1.3)
where the domain consists of cycles meeting V h properly. There are two maintheorems in this paper – Theorem 1.1 and Theorem 2.3.
Theorem 1.1.There exist maps, mostly dash homomorphisms,
fhp : Zp(X) 99K Zp(Vh), 0 ≤ p < n− h (1.4)
ghp : Zp(X) 99K Zp(X), all p (1.5)
Conhp : Zp(V
h) 99K Zp+h(X), 0 ≤ p ≤ n− h (1.6)
L : Zp(Vh) → Zp+h(X), 0 ≤ p ≤ n− h (1.7)
such that for the rational equivalence ∼ on X, there are decomposing formulasas follows:
a) for 0 ≤ p ≤ n− h on Zp(Vh) = Zn−p
V h (X),
vh (Conhp + L)∼ mid (1.8)
where m = deg(X) for the polarization of X, id is the identity map onalgebraic cycles, and L is reduced to a homomorphism
CHp(X) → CHp(X)
whose images are multiples of the plane section class.b) for 0 < p < n− h on Zp(X),
fhp − ghp ∼ mid (1.9)
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c) There is a subgroup Zs(X) ⊂ Z(X) contained in the domains of aboveformulas such that
Zs(X)/∼rat. = CH(X). (1.10)
Remark Theorem 1.1 is a statement resembling the diagonal decomposition.However it is rather primitive in structure, because neither of fhp , g
hp and Conh
p
respect any adequate equivalence relations. However the formula (1.8) has noneed of that, and it directly implies a corollary.
Corollary 1.2. It is known that for h ≤ q ≤ n, vh descends to a homomorphism
vh : CHq−h(X) → CHqV h(X)∩
CHq(X)(1.11)
on the Chow group, where CHqV h(X) is the subgroup
ZqV h(X)
/∼rat.. (1.12)
Then vh is surjective.
1.1.2 Ideas
There is the dictionary of Beilinson-Bloch based on their conjectural filtration:
Algebraic cycles with Q coefficients⇓
Classes of the rational equivalence⇓
Classes of the homological equivalence⇓
Classes of the weaker equivalence⇓...⇓
Classes of the total equivalence
where the down-arrow is the reduction by an equivalence relation, the homo-logical equivalence is conjectural, and the total equivalence is the equivalenceon all algebraic cycles. Theorem 1.1 addresses a primitive structure of algebraiccycles on the top row of the dictionary where the open range between algebraiccycles and Chow classes is not included in the Beilinson-Bloch conjecture but
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blocks primitive structures to descend. For instance if S ⊂ X is a subvariety,then for any algebraic cycle δ0 there is a canonical decomposition in Z(X),
δ0 = δ1 + δ2 (1.13)
where δ1 is supported on S, but none of components of δ2 are. Such a decom-position which will be called a cycle-decomposition does not descend to classes.The study in this open range requires a modification on the fundamentals of in-tersection theory. So to have a clean description of the idea, we use the distinctnotation “ • ” referred to as the cycle-intersection. But at the mean time weretain the standard notations (according to [1]) for the class-intersection in theusual intersection theory denoted by “ · ”. The following is the list of new andold.
Notation 1.3.Notation addresses the category of smooth projective varieties over the alge-
braically closed field k.
(1) Z(∗)=the Q module freely generated by irreducible subvarieties of alldimensions; Each irreducible subvariety is called a component ofthe cycle;
CH(∗)=the total Chow group with Q coefficients;Cycle=an element of Z(∗);Cycle class= an element of CH(∗) or cohomology represented by a cycle;On all these groups and their subgroups, the superscript denotes the codi-mension of cycles or classes, and subscript denotes the dimension ofthem.
(2) The fundamental cycle of a scheme B is still denoted by B.(3) The operations in bold font are in Z(∗) groups, the same alphabets
in blackboard font are their descends to the Chow groups, and thesame alphabets in normal font are their descends to cohomologygroups.
(4) ∆X denotes the diagonal scheme in the product X ×X.(5) A plane section is a complete intersection by hyperplane sections with
the same cohomology class.(6) A plane section class is a class represented by a plane section.(7) The Cartesian product and fibre product for schemes can be linearly ex-
tended to cycles. We’ll use the same notations for the operations.(8) Let [∗] denote rational equivalence classes in a Chow group. Fulton’s
various refined intersection classes are all denoted by [∗] ·F [∗] and [∗ ·F ∗]where F denotes the location of the intersection and the Chow groupscontaining them will be specified in the context. They will be referred toas the class-intersection.
++++++++++++++++++++++++++++++++++++++++++
(9) (cycle-intersection) We’ll use a new notion in intersection, called cycle-
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intersection • which is equivalent to Fulton’s basic construction in caseof proper intersection. Precisely the intersection • is an element in Z(∗)defined as the sum ∑
j
mjWj
where Wj are all irreducible subvarieties, mi are intersection multiplici-ties at Wj if Wj are components with the proper dimension in the inter-section scheme, and zero otherwise. To indicate the ambient spaces wherethe intersection occurs, we adapt the notations from Fulton’s as follows.
(a) Let σ1 ∈ Zi(X), σ2 ∈ Zj(X). Then σ1 •X σ2 ∈ Zi+j−n(X) denotesthe cycle-intersection in a smooth projective X over k.
(b) If f : X → Y is a morphism between two smooth projectivevarieties over k, and x ∈ Zi(X), y ∈ Zj(Y ), then the cycle-intersection
(PX)∗
((x× y) •(X×Y ) graph(f)
)∈ Zi+j−dim(Y )(X) (1.14)
is denoted by x •f y, where PX : X ×Y → X is the projection and(PX)∗ is the pushforward of cycles. In particular X •f y is denotedby f c∗(y).
(10) (Open localization) Continuing from the intersection • in part (9). LetS ⊊ X be a subvariety. In the cycle-intersection σ1 •X σ2, the partialsum supported on S is denoted by b. We define the open localization,
σ1 ⊙S σ2 ∈ Z(X)
with respect to S to be the other part,
σ1 •X σ2 − b.
11) (closure intersection) Let σ1 , σ
2 be two cycles in an open subvariety X
of a smooth projective variety X. Assume σi =
∑j m
ija
ij where aij are
all irreducible subvarieties of X, and mij are integers with finitely many
non-zero. The closure intersection is defined to be
σ1 •X σ
2 :=∑i,j
m1im
2ja
1i •X a2j (1.15)
where ∗ is the cycle closure obtained by taking the closure of eachcomponent in X. Let S = X\X, and σi, i = 1, 2 be the cycle closuresσi . Then the closure intersection has another expression
σ1 ⊙S σ2.
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12) (cycle-correspondence) Let T,X be smooth projective varieties,
Γ ∈ Z(T ×X).
The cycle-correspondence
Γc∗ : Z(T ) → Z(X)
is (PX)∗
(Γ •(T×X) (σ ×X)
)where σ ∈ Z(T ), PX : T ×X → X is the
projection. The usual correspondences on Chow groups or cohomologygroups are referred to as the class-correspondences, denoted by Γ∗.
(13) (family of cycles) Let T,X be smooth projective varieties, Γ ∈ Z(T×X)a cycle decomposed as Γ0+Γ1 such that Γ0 consists of all the componentsof the minimum dimension among those surjective to T . The family ofcycles
Γt ∈ Zp(X), t ∈ T
deduced from Γ is defined by the cycle-correspondence of Γ0 as
Γ0c∗(t)
where t ∈ T .
(14) (fibration of a cycle) Let T,X be smooth varieties. Let E ∈ Z(T ×X)be a cycle, (PX)∗(E) its projection to X. We define the fibres of bothcycles E , (PX)∗(E) to be the same family of cycles
Ec∗(t), t ∈ T.
The fibration is denoted by the symbols
E ⇒ T,(PX)∗(E) ⇒ T.
Remark All dash homomorphims in Theorem 1.1 have trivial extensionsto the cycle-intersection. However the extended homomorphisms do not satisfythe formulas (1.8) and (1.9).
After the basic notion, we come back to the main content of this paper. The-orem 1.1 is proved by using a particular family of cycles, called the cone familyin the cycle-intersection. It starts with the basic idea of diagonal-deformationin a projective space. Let h be a natural number less than n. Let
An+2 = An+2−h ⊕ Ah (1.16)
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be the affine space over k with the decomposition which defines the family oflinear transformations
An+2 = An+2−h ⊕ Ah → An+2
x1 + x2 → x1 + tx2(1.17)
parametrized by t ∈ k. This gives a rise to a family of linear transformationson the projective space over k,
gt : Pn+1 99K Pn+1
[x1 + x2] 99K [x1 + tx2](1.18)
parametrized by t ∈ k ⊂ P1. Then any algebraic cycle σ in Pn+1 has the corre-sponding diagonal-deformation gt(σ). The deformation gives two fundamentallyimportant cycles:
a) The projection
Pn+1 99K Pn+1−h
∥ ∥P(An+2−h) 99K P(Ah)
through the cycle-pushfoward gives a dash homomorphism
Prz : Zi(Pn+1) 99K Zi(P
n+1−h) (1.19)
where z denotes the particular decomposition (1.16). Then
g0(σ) = Prz(σ)
for σ in the domain.b) Let σ be a prime cycle in Zi(P
n+1−h). Then there is a prime cycleσ#P(Ah), the join between two subvarieties. So the map
σ → σ#P(Ah)
can be linearly extended to all cycles to obtain the homomorphism,
Cz : Zi(Pn+1−h) → Zi+h(P
n+1), (1.20)
where z denotes the decomposition (1.16). Then
g∞(σ) = Cz(σ •Pn+1 Pn+1−h) (1.21)
provided the intersection is proper.
For an arbitrary projective variety, the diagonal-deformation still exists buthas lower dimension. So we construct a fibred version to increase the dimension.The fibre structure comes from the variation of Ah as follows. Let the subspaceAh vary in the Grassmannian
G(h, n+ 2),
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expressed as Ahz where the 1-parameter z ∈ Υ ≃ P1 such that z = ∞ corre-
sponds to the original subspace Ah and z = 0 is the only subspace Ah0 that fails
the decompositionAn+2 = An+2−h ⊕ Ah
z . (1.22)
Following the steps of diagonal-deformation, we obtain a 2-parameter family oflinear transformations gzt for generic parameters t, z,
Pn+1 → Pn+1−h ⊕Ph−1z
[x1(z) + x2(z)] → [x1(z) + tx2(z)].(1.23)
Let Ω be the graph of the rational map,
P1 ×Υ×Pn+1 99K Pn+1−h ⊕Ph−1z
(t, z, [x1(z) + x2(z)]) 99K [x1(z) + tx2(z)].(1.24)
LetΘ = τ−1(Ω) ⊂ P1 ×Υ×X ×X, (1.25)
where τ be the regular map (id, id, µ, µ). For each cycle σ ∈ Zp(X), we have acycle family ψt(σ) ∈ Zp(X) deduced from the cycle-correspondence,
Θc∗(Υ× σ) ∈ Zp+1(P1 ×X) (1.26)
whereΘ ⊂ P1 ×Υ×X ×X
(1st× 2nd× 3rd× 4th)
is regarded as a cycle-correspondence
Υ×X −→ P1 ×X(2nd× 3nd) (1st× 4th)
1. Theorem 1.1 is the collection of equivalence-free descriptions for ψ0(σ), ψ1(σ).The family will deform a given cycle in a rational equivalence class to a cycle ofmultiple pieces. The cycle structure of the family is fibred with the diagonal-deformation in a single projective space over the open set Υ − 0. Howeverthe structural complexity enters when the fibre is moved to the unstable pointz = 0 (which is a necessary point for the classes in a complete variety). In thefirst section we pause at this complexity to analyze its set-theoretical structureto obtain the decomposition on cycles of smaller dimensions:
ψ1(∗) = m(∗) + ghp (∗)∼
ψ0(∗) = fhp (∗).(1.27)
where
∼
is the rational equivalence of X. This yields (1.9) which is the particulartype of the diagonal’s cycle-decomposition with all the complexity collected inghp (∗), fhp (∗). Then we prove the formula (1.8) which is leaning toward classes.
We’ll prove it in the usual intersection theory.
1There could be another similar construction where Θc∗ is replaced by the class-correspondence Θ∗. The resulting family is standard in classes. But they are not representedby the cone family. So cone family is more subtle.
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1.2 Cycle-intersection
Parts (9)-(14) in Notation 3 list the terms for the cycle-intersection which isbased on the basic construction in [1]. But it is developed into the set-theoreticalaspects of the theory, and eventually lands in a different category. Even thoughthe difference in statements is clear, the purpose is not. The following areexamples.
Example 1.4. (cycle-correspondence) Let P2 × P2 be a product of projectivespaces over k, with homogeneous coordinates [x0, x1, x2] and [y0, y1, y2] respec-tively. For t ∈ k∗, we defined a non-degenerate linear transformation
gt : [x0, x1, x2] → [x0, tx1, tx2]. (1.28)
Let κt be the graph in P2×P2 that is specialized at t = 0. Then the specializationκ0 denoted by Γ, is a subvariety of P2×P2, whose fundamental cycle is rationallyequivalent to the diagonal ∆P2 . Precisely
Γ = P2 × [1, 0, 0]+ ϑ (1.29)
where ϑ is the surface
(x0, x1y2 − x2y1) ⊂ P2 ×P2. (1.30)
Letσ = (x0)
be the 1-cycle in P2. Then the cycle-correspondence yields
Γc∗(σ) = 0 (1.31)
However the class-correspondence yields
Γ∗([σ]) = [σ] = 0. (1.32)
Example 1.5. (Open localization) Assume the ground field is k. Let [x0, x1] bethe homogeneous coordinates for P1. Let [y0, y1, y2] be the homogeneous coordi-nates for P2. Let S = [0 : 1]×P2. Let Γ be the hypersuface in P1×P2 definedby x1y1 − x0y0 = 0, σi ⊂ P2 for i = 1, 2 be the hyperplane yi = 0 respectively.Then
Γ⊙S (P1 × σ1) = P1 × [0 : 0 : 1]. (1.33)
Γ⊙S (P1 × σ2) = ∆1 (1.34)
where ∆1 the rational curve
x1y1 − x0y0 = 0. (1.35)
Hence Γ⊙S (P1×σ2) and Γ⊙S (P1×σ1) are not rationally equivalent, eventhough σ1, σ2 are.
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Remark Examples show that the cycle-intersection theory does not respectthe rationally equivalence relation. Thus the reinstatement of the results in theusual intersection theory must be verified for the equivalences.
Proposition 1.6.The cycle-intersection satisfies the associativity, commutativity and projec-
tion formula.
Proof. Set-theoretically 3 rules are true. Then properties follow from the sameformulas for the intersection multiplicity in Examples 7.1.7-7.1.9, [1].
Remark The rule of thumb is that all formulas for intersection multiplicitiesin usual intersection theory should still hold in cycle-intersection because thecomponents have the proper dimensions.
1.3 Cone family
1.3.1 Cone family of cycles
The cone family will be parametrized by the projective space P1. Let An+2 bethe affine space over k with the standard basis
e0, · · · , en+1.
Let h be a natural number ≤ n. Consider two subspaces
An+2−h = span(e0, · · · , en+1−h),Ah = span(en+2−h, · · · , en+1).
(1.36)
ThenAn+2−h ⊕ Ah = An+2. (1.37)
Next we define a specific variation of Ah. Let k∪∞ ≃ P1 be the parameterspace of the variation, denoted by Υ, where ∞ is the infinity point of P1. Thefamily of linear spaces in G(h, n+ 2) is defined by
Ahz = span(zen+2−h − e0, en+3−h, · · · , en+1), for z ∈ k (1.38)
and Ah∞ is the original Ah which is the limit of subspaces Ah
z in Grassmannianas z → ∞. Let U = k∗ ∪∞ be the affine open set that parametrizes those Ah
z
satisfying the decomposition
An+2 = An+2−h ⊕ Ahz . (1.39)
The only point z = 0 not in U corresponds to the plane Ah0 that fails the
decomposition (1.39). We call z = 0 the unstable point, others stable points.
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Let x be a vector in An+2. Therefore for each stable point z ∈ U , we have theunique decomposition
x = x1(z) + x2(z). (1.40)
The decomposition gives a family of linear transformations,
A× U × (An+2−h ⊕ Ahz ) → An+2−h ⊕ Ah
z = An+2
(t, z,x1(z) + x2(z)) → x1(z) + tx2(z).(1.41)
which yields a rational map of the projective variety
κ : P1 ×Υ×Pn+1 99K Pn+1
(t, z, [x1(z) + x2(z)]) 99K [x1(z) + tx2(z)].
So we just set up a family of linear transformations gzt on Pn+1 parametrizedby t ∈ k∗, z ∈ U :
gzt : Pn+1 → Pn+1
e0 → e0... ⇒
...en+1−h → en+1−h
en+2−h → t(zen+2−h − e0)en+3−h → ten+3−h
... ⇒...
en+1−h → ten+1
(1.42)
LetΩ = graph(κ) ⊂ P1 ×Υ×Pn+1 ×Pn+1 (1.43)
where the graph of a rational map is defined to be the closure of the graph atthe regular locus ( the same for images of rational maps).
Now we consider the smooth projective variety X of dimension n, equippedwith the polarization u. Let
µ : X → Pn+1
be a birational morphism to a hypersurface of Pn+1 in a general position inthe following sense: µ(X) is in a general position as a subvariety (with respectto the polarization), in particular its first order deformation in gzt (µ(X)) alongt varies with z ∈ U ; X has a very ample line bundle µ∗(OPn+1(1)) such thatu = c1(µ
∗(OPn+1(1))) is the original polarization. The collection of An+2,Ahz ,
gzt , and µ is called cone data. The cone data is extrinsic and it can be obtainedthrough any embedding X ⊂ PN , by taking a projection to a generic subspace:X → Pn+1.
Letµ2 = (µ, µ) : X ×X → Pn+1 ×Pn+1.
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Also let τ = (id, id, µ2) be the morphism
τ : P1 ×Υ×X ×X → P1 ×Υ×Pn+1 ×Pn+1.
LetΘ = τ−1(Ω). (1.44)
Proposition 1.7. For any class in the Chow group there is a representativeσ ∈ Zp(X), such that
Θ ∩ (P1 × |σ| ×X ×X)
has a component surjective to P1, and of the minimum dimension p+ 1.
Proof. Let σ be a prime cycle that does not entirely lie in the plane section
V h = µ−1(P(span(e0, · · · , en+1−h))) (1.45)
and the exceptional locus of µ : X → µ(X). Let U = P1\0, 1. Let
Θ(σ) = Θ ∩ (U × U × σ ×X). (1.46)
Then over each (t, z) ∈ U ×U , the linear transformation gzt is invertible. Hencethe fibre of
Θ(σ) → U × U
is just the hypersurface of the graph of an invertible linear transformation
(µ2)−1
(graph(gzt (σ)) ∩
(Pn+1 × µ(X)
)).
Due to the assumption on the position of σ, the graph will have dimension p−1.Hence dim(Θ(σ)) = p + 1. Then its closure also has dimension p + 1. Thisshows
Θ ∩ (P1 × σ ×X ×X)
has a component of exact dimension p + 1, and dominating P1. Then theargument is extended to cycles σ. By the moving lemma, any class in the Chowgroup can have such a representative σ. This completes the proof.
Remark Θ is reducible with 2 distinct components. The one in the aboveproof is surjective to P1. In the following arguments, the other one will bedismissed due to its non-serjectivity to P1.
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Definition 1.8. Let Zs(X) be the collection of all cycles, each of whose compo-nents has proper intersections with determined finitely many subvarieties fromcone data. In particular they include those from the polarization u, exceptionallocus of µ, first order deformation of gzt (µ(X)) along t for each z. Such a cyclein Zs(X) is referred to as a generally positioned cycle.
Remark So Zs(X) is the group’s version of “genericity”. The determinedsubvarieties in the definition will be introduced in the context where technicaldescriptions are necessary.
Proposition 1.7 gives us a meaningful definition,
Definition 1.9. (Cone family) For σ ∈ Zsp(X), the family of cycles ψt(σ) is
defined by the cycle-correspondence
Θc∗(Υ× σ) in Zp+1(P1 ×X) (1.47)
whereΘc∗ : Z(Υ×X) → Z(P1 ×X)
is induced from the scheme Θ.
Remark The main purpose of cycle-intersection is to create an object inintersection theory that preserves the support, but loses the equivalences. Thestandard intersection theory, on the other hand, goes in opposite direction:preserves the equivalences, but loses the support.
1.3.2 End cycles
By the definition of the family of cycles in Notation 1.3, all end cycles areobtained from components surjective to P1, and of the minimal dimension. Sofor σ ∈ Zs(X), ψ0(σ), ψ1(σ) are rationally equivalent. In this section we analyzetheir set-theoretical structures.
We first introduce homogeneous coordinates in the cone construction. Letx0, · · · , xn+1 be the coefficients of the basis e0, · · · , en+1 for An+2. Then
x0, · · · , xn+1
are homogeneous coordinates for Pn+1. Recall z = 0 are k-numbers parametriz-ing the affine neighborhood of Υ. Then the homogeneous coordinates for
Pn+1−h = P(An+2−h),Ph−1
z = P(Ahz )
13
are
vector x1(z) : x0 +xn+2−h
z, x1, · · · , xn+1−h, 0, · · · , 0
vector x2(z) : 0, · · · , 0,xn+2−h
z, · · · , xn+1.
(1.48)
(in the basis ei as in the decomposition (1.40) ). In the product space
P1 ×Υ×Pn+1 ×Pn+1,
the coordinates for the third factor Pn+1 denoted by x-coordinates as above,the last factor Pn+1 by y-coordinates,
vector y1(z) : y0 +yn+2−h
z, y1, · · · , yn+1−h, 0, · · · , 0
vector y2(z) : 0, · · · , 0,yn+2−h
z, · · · , yn+1.
Applying the coordinates to the graph in (1.43), we obtain the scheme
Ω
is explicitly defined by
xiyj − xjyi, for i, j ∈ [1, n+ 1− h]
xiyj − xjyi, for i, j ∈ [n+ 2− h, n+ 1]
xiyjt1 − yixjt0, for j ∈ [1, n+ 1− h], i ∈ [n+ 2− h, n+ 1]
xyj − yxj , for j ∈ [1, n+ 1− h]
xiyt1 − yixt0, for i ∈ [n+ 2− h, n+ 1]
x = zx0 + xn+2−h,
y = zy0 + yn+2−h.
(1.49)
For any subscheme Λ ofP1 ×Υ×X ×X
orP1 ×Υ×Pn+1 ×Pn+1
we denote its fibres over the parameters t ∈ P1, z ∈ Υ by Λz,Λt,Λzt . Also their
projections to their bases will be denoted by adding the · to the parameters t, z.For instance Θz
tis a subscheme of X ×X, etc.
We define an irreducible subvariety of Pn+1 ×Pn+1.
Definition 1.10. Let
P((span(e0, · · · , en+1−h)) 99K P((span(e1, · · · , en+1−h))∪ ∪
An+1−h → An−h(1.50)
14
be the rational projection that is regular on the affine open set. Define E1 to bethe closure of the fibre product of the affine open sets,
An+1−h ×An−h An+1−h (1.51)
inPn+1−h ×Pn+1−h. (1.52)
• 1-end cycle.
Proposition 1.11. There is a dash homomorphism
ghp : Zp(X) 99K Zp(X),
such that when ψ1 is regarded as a map
Zsp(X) → Zp(X)
it satisfiesψ1 = mid+ gh
p (1.53)
Proof. Notice that Θ has an obvious component
τ−1(Ω1) (1.54)
where Ω1 is the fibre over t = 1. This is the 2nd component of Θ, but notsurjective to P1. Let
Σ = Θ\τ−1(Ω1).
Let σ ∈ Zsp(X). We should note the end cycle is obtained by a selection of
components in the intersection. This selection is included in the definition ofcycle-correspondence. For instance in the case of 1-end cycle, we select thecomponents of
Σ •Y (P1 ×Υ× σ ×X)
surjective to P1, whereY = P1 ×Υ×X ×X.
Hence ψ1(σ) is the projection of the triple intersection(Σ •Y (P1 ×Υ× σ ×X)
)•Y(1 ×Υ×X ×X
).
By the commutativity and associativity, it is the projection of(Σ •Y (1 ×Υ×X ×X)
)•Y(P1 ×Υ× σ ×X
). (1.55)
15
Let’s divide the triple into two parts according to its support: one lies overthe unstable point z = 0, the other does not (primitive decomposition (1.13)).
1) Over z = 0. We denote this part by
L0(σ) ∈ Zp(1 × 0 ×X ×X)
andghp (σ) := (η4)∗(L0(σ)) (1.56)
where η4 : P1 ×Υ×X ×X → X(4th factor) is the projection.2) The part not supported in the fibre over z = 0. We denote this part by
L1(σ). LetS ′ := 1 × 0 ×X ×X.
Then L1(σ) is the open localization(Σ⊙S′ (1 ×Υ×X ×X)
)⊙S′
(P1 ×Υ× σ ×X
)which starts in the first open localization
Σ⊙S′ (1 ×Υ×X ×X).
Notice that no component of the scheme intersection
τ(Σ) ∩ τ(1 ×Υ×X ×X)
lies in the exceptional locus of τ . Thus it suffices to work with the cycle-intersection in the Pn+1 ×Pn+1. The key ingredient in this intersection is thefollowing specialization at t = 1 ( Σ is obtained by the specialization). Let µ(X)be the hypersurface of Pn+1. Assume µ(X) is defined by a polynomial f . Thenµ2(X ×X) is a complete intersection defined by two polynomials f(x), f(y) in
Pn+1 ×Pn+1
for (x,y) ∈ Pn+1 ×Pn+1. Then
µ2(Θzt ), t = 1
is the subvariety ofΩz
t
defined by two hypersurfaces f(x), f(y). As in the setting, we let [x1(z)], [x2(z)]be the points in the decomposition
Pn+1 = Pn+1−h ⊕Ph−1z , dependent of z,
where Ph−1z = P(Ah
z ). At z = ∞, we denote [xi] = [xi(∞)]. Since t is near 1,it can not be 0 or ∞. Then Ωz
tis a graph isomorphic to Pn+1 expressed as the
graph([x1(z) + x2(z)]× [x1(z) + tx2(z)]) ⊂ Pn+1 ×Pn+1.
16
Then µ2(Θzt) is a complete intersection explicitly defined by
f
(x1(z) + x2(z)
), f
(x1(z) + tx2(z)
)(1.57)
inside of Ωzt≃ Pn+1. Two polynomials are equal at t = 1. Thus for the
specialization we consider the expansion along t− 1,
f
(x1(z) + x2(z)
)− f
(x1(z) + tx2(z)
)= (t− 1)rF z
r (x1 + x2) + (t− 1)r+1F zr+2(x1 + x2) + · · ·
where F zr (x1 + x2) is a hypersurface in
[x1 + x2] = Pn+1 ≃ ∆Pn+1
dependent of z. By the assumption on the cone data, r = 1 and
F zr (x1 + x2) = 0
is a varied hypersurface with z ∈ U , of degreem. Then the specialization µ2(Θz1)
at t = 1 in ∆Pn+1 is defined by two polynomials
f(x1 + x2), Fz1 (x1 + x2).
Therefore the specialization Σz1is birational to the hypersurface
F z1 (x1 + x2)
in µ2(∆X) ⊂ ∆Pn+1 , of degree m. This shows the part not supported over z = 0is the closure of the cycle
Σ1 = Σ1 ∩ (1 × U ×X ×X)
whose fibre over each z ∈ U is a degree m hypersurface of the diagonal ∆X . Wedenote this part, which is an m-covering of ∆X , by D1. Then
L1(σ) = D1 •Y (P1 ×Υ× σ ×X).
Soψ1(σ) = (η4)∗(L0(σ) + L1(σ)). (1.58)
For each t around 1 and z = 0, since dim(σ) > 0 and σ is in general position,the hypersurface µ(X) meets (gzt )∗(σ). Then each component of L1(D) doesnot lie over z = 0. Hence the triple intersection (1.55) has a decomposition,
L0(σ) + L1(σ).
The projection formula (Proposition 1.6) asserts (η4)∗(L1(σ)) is deg(X)σ. Wecomplete the proof.
17
• 0-end cycle.
LetPn+1−h = P(span(e0, · · · , en+1−h))
Ph = P(span(e0, en+2−h, · · · , en+1))
be subspaces of dimensions n+ 1− h, h respectively. Let
V h = X ∩Pn+1−h, and V n+1−h = X ∩Ph.
be the n − h and h − 1 dimensional, smooth, irreducible plane sections of Xrespectively. Equivalently,
V h = div(µ∗(xn+2−h)) ∩ · · · ∩ div(µ∗(xn+1))
andV n+1−h = div(µ∗(x1)) ∩ · · · ∩ div(µ∗(xn+1−h))
Proposition 1.12.Let σ ∈ Zs
p(X). If p < n−h, ψ0(σ) denoted by fhp (σ) lies in Vh. Furthermore
there is a fibration fhp (σ) ⇒ Υ whose fibre over generic z ∈ Υ is
V h •Pn+1−h
(Prz µ∗(σ)
). (1.59)
Proof. The following argument is on the intersection schemes containing theintersection cycles. Notice through the generators (1.49), Ω0 is defined by
xiyj − xjyi, for i, j ∈ [1, n+ 1− h]
xiyj − xjyi, for i, j ∈ [n+ 2− h, n+ 1]
yixj , for j ∈ [1, n+ 1− h], i ∈ [n+ 2− h, n+ 1]
xyj − yxj , for j ∈ [1, n+ 1− h]
yix, for i ∈ [n+ 2− h, n+ 1]
x = zx0 + xn+2−h,
y = zy0 + yn+2−h.
(1.60)
By observing the third set of generators
yixj , for j ∈ [1, n+ 1− h], i ∈ [n+ 2− h, n+ 1]
we can see that there are two types of components for cycles. One lies in thescheme
0 ×Υ×Ph ×Pn+1;
18
the other lies in another scheme
0 ×Υ×Pn+1 ×Pn+1−h.
Using µ2, we pull them back to 0 × Υ × X × X to have two types ofcomponents. One lies in the scheme
Υ× V n+1−h ×X
denoted by γ1 (which will be dismissed. See (1.62) below); the other lying in
Υ×X × V h
surjective to P1, denoted byγ2, (1.61)
are the fibres over t = 0, of components of Θ. Now we consider the intersection
(0 × γ1) ∩ (0 ×Υ× |σ| ×X). (1.62)
Since dim(σ) < n−h and σ is in a general position ( i.e. cycle in Zsp(X)) inside
of X, σ must be disjoint with V n+1−h ⊂ X. So (1.62) is empty. Thus thescheme
Θ0 ∩ (0 ×Υ×X ×X) = (0 × γ2) ∩ (0 ×Υ× σ ×X)
lies in0 ×Υ×X × V h.
The projection to the last factor lies in V h.
Furthermore we notice
fhp (σ) = (P3)∗
(γ2 •(Υ×X×X) (Υ× σ ×X)
)where P3 : Υ×X ×X → X(3rd factor) is the projection. Then the fibre of
fhp (σ) ⇒ Υ
is the cycle correspondence(γ2)c∗(z × σ) (1.63)
for z ∈ U , where (γ2)c∗ is the cycle-correspondence
Z(Υ×X) → Z(V h).
By the definition of γ2, (γ2)c∗(z × σ) is the cycle
V h •Pn+1
(Prz µ∗(σ)
). (1.64)
19
1.4 Cone operator
The cycle-factorization of ∞-end cycle results in the cone operator Conhp . How-
ever in this subsection we’ll skip the cone family and study the operator in Chowgroups.
We start with a general lemma following [1] addressing the associativity.
Lemma 1.13. LetX,Y,W (1.65)
be projective varieties and Y,W are smooth.Let
X ′ Y ′ W ′
↓ ↓ ↓X
f1→ Yf2→ W
(1.66)
be a commutative diagram of morphisms. Let
σ1 ∈ Z(X ′), σ2 ∈ Z(Y ′), σ3 ∈ Z(W ′)
Then ([σ] ·f1 [σ2]
)·f2f1 [σ3] = [σ1] ·f1
([σ2] ·f2 [σ3]
)∈
CH(X ′ ×Y Y ′ ×W W ′).
(1.67)
If f1, f2 are embeddings and σ2 = Y , we call Y the transitional variety.
Proof. See the proof in 8.1.1, [1].
The decomposition (1.39), has a natural projection,
An+2−h ⊕ Ahz → An+2−h, z ∈ U (1.68)
which yields the graph B of the rational map,
U ×Pn+1 99K Pn+1−h
(z, [x]) → [x1(z)]
where x = x1(z) + x2(z) is the unique decomposition (1.40). Let B⊺ be itstransposed graph in U ×Pn+1−h ×Pn+1.
Let Ih be the intersection cycle
(Υ× V h ×X) •(id,µ,µ) B⊺ (1.69)
where µ = µ|V h , ¯(∗) is the closure in Υ× V h ×X. We should note Ih is prime,i.e it is an irreducible variety. We denote the hypersurface
0 × V h ×X (1.70)
of Υ× V h ×X by S.
20
Definition 1.14. We define the cone operator Conhp to be the map in open
localization,Zp(V
h) → Zp+h(X)
a → (η3)∗
(Ih ⊙S
(Υ× a×X
)) (1.71)
where η3 : Υ× V h ×X → X is the projection.
In Notation 1.3, there is an equivalent definition that asserts
Conhp(a) = (η3)∗
(Ih •(
U×V h×X) (U × a×X
)),
is a closure intersection where the closure is in Υ× V h ×X.
In the following we have another expression of the cone operator.
Proposition 1.15. For σ ∈ Zsp(V
h), the intersection
Ih ∩ (Υ× |σ| ×X)
is proper in Υ× V h ×X. Furthermore
Conhp(σ) = (η3)∗
(Ih •(Υ×V h×X) (Υ× σ ×X)
). (1.72)
Proof. Let’s calculate
Ih •(Υ×V h×X) (Υ× σ ×X).
Let Y = Υ× V h ×X. By the commutativity and associativity, we obtain that
Ih •(Υ×V h×X) (Υ× σ ×X)
= (Υ× V h ×X) •(id,µ,µ) B⊺ •Y (Υ× σ ×X)
= (Υ× V h ×X) •Y (Υ× σ ×X) •(id,µ,µ) B⊺
= (Υ× σ ×X) •(id,µ,µ) B⊺
(1.73)
Next we consider the fiberation of the cycle,
(Υ× σ ×X) •(id,µ,µ) B⊺ ⇒ Υ. (1.74)
Let B⊺z ⊂ Pn+1−h ×Pn+1 be the fibre of
B⊺ → U
21
over z ∈ U . Since µ is birational and σ is in a general position, the fibre of(1.74) which is
z ×((σ ×X) •µ2 B⊺
z
)is equal to
z × (µ2)c∗((µ(σ)× µ(X)
)•(Pn+1×Pn+1) B
⊺z
). (1.75)
Since the dimension satisfies
dim
(((µ(σ)× µ(X)
)•(Pn+1×Pn+1) B
⊺z
)= p+ h− 1.
the dimension of
dim
(Ih •Y (U × σ ×X)
)= h+ p. (1.76)
Now we consider over the unstable point z = 0. Using the coordinatesexpression (1.49), where xn+1 = · · · = xn+2−h = 0, Gt
z at z = 0 is the schemez
xn+1, · · · , xn+2−h
xiyj − xjyi i, j ∈ [1, n+ 1− h]
yn+2−h
(1.77)
Then for any σ ∈ Zsp(V
h), fibre of
Ih •Y (Υ× σ ×X) ⇒ Υ (1.78)
over 0 has dimension p+ h− 1. So the lower dimension indicates that thereis no component of Conh
p(σ) in that fibre. Hence the
Ih •(Υ×V h×X) (Υ× σ ×X)
is the same as the closure intersection(Ih •(
U×V h×X) (U × σ ×X
)).
So
Conhp(σ) = (η3)∗
(Ih •(Υ×V h×X) (Υ× σ ×X)
). (1.79)
This completes the proof.
Lemma 1.16. For a whole number r and σ ∈ Zr(Vh), the class-correspondence
w∗([σ]) (1.80)
is a multiple of the plane section class in CHr(Vh), where
w = (V h × V h) •µ2 E1.
22
Proof. We recallE1 ⊂ Pn+1−h ×Pn+1−h (1.81)
is the subvariety of dimension n+ 2− h. There is a commutative diagram
Pn+1−h × V h V h ×Pn+1−h Pn+1−h ×Pn+1−h
↓(id,µ) ↓(µ,id) ↓(id,id)Pn+1−h ×Pn+1−h (id,id)→ Pn+1−h ×Pn+1−h (id,id)→ Pn+1−h ×Pn+1−h
and three cycle classes in the top row,
[Pn+1−h × V h] ∈ CH2n−2h+1(Pn+1−h × V h)
[σ ×Pn+1−h] ∈ CHn+1−h+p(Vh ×Pn+1−h)
[E1] ∈ CHn+2−h(Pn+1−h ×Pn+1−h)
They form a triple intersection with the associativity (Lemma 1.13),
[Pn+1−h × V h] ·(id,id)([σ ×Pn+1−h] ·(id,id) [E1]
)=
([Pn+1−h × V h] ·(id,id) [σ ×Pn+1−h]
)·(id,id) [E1]
∈
CHp(Vh × V h)
(1.82)
So the 2nd row of (1.82) is
[σ × V h] ·(id,id) [E1]
= [σ × V h] ·(V h×V h)
([V h × V h] ·(id,id) [E1]
)= [σ × V h] ·(V h×V h) w
whose projection to CHp(Vh) is w∗([σ]).
Let P2 : V h × Pn+1−h → Pn+1−h be the projection. For the 1st row of
23
(1.82), we use the associativity, Lemma 1.14
(P2)∗
([Pn+1−h × V h] ·(id,id)
([σ ×Pn+1−h] ·(id,id) [E1]
))(Use Lemma 1.13)
= (P2)∗
([Pn+1−h × V h] ·(id,id)
([V h ×Pn+1−h] ·(id,id) ([σ ×Pn+1−h] ·(id,id) [E1])
))
= (P2)∗
([V h × V h] ·(id,id)
(([σ ×Pn+1−h] ·(id,id) [E1])
))( Use projection formula for P2 : V h ×Pn+1−h → Pn+1−h)
= [V h] ·µ (P2)∗
([σ ×Pn+1−h] ·(id,id) [E1]
)(1.83)
where (P2)∗
([σ×Pn+1−h] ·(id,id) [E1]
)lies in CHr(P
n+1−h) ≃ Q generated by
the hyperplane section class. Thus
w∗([σ]) = [V h] ·µ (P2)∗
([σ ×Pn+1−h] ·(id,id) [E1]
)(1.84)
is a multiple of a plane section class ∈ CHr(Vh).
For each σ ∈ Zsp(V
h), we’ll study the rational equivalence class [Conhp(σ)]
in X. In the following we state a proposition in Chow groups.
Theorem 1.17. For σ ∈ Zsp(V
h), there is an intersection formula in the Chowgroup of X,
vh
([Conh
p(σ)]
)= m[σ] + h∗([σ]) (1.85)
where the class-correspondence h∗([σ]) is a multiple of the plane section class.Furthermore the formula (1.8) follows from (1.85).
Proof. In the following we’ll apply the rules in the usual intersection theory. Bythe projection formula for the projection
π : Υ× V h ×X → X,
vh
([Conh
p(σ)]
)= π∗
([Υ× V h × V h] ·Y
(Ih ·Y [Υ× σ ×X]
)),
24
where Y = Υ × V h × X. By the associativity of the intersection product inCH(Y), we have
[Υ× V h × V h] ·Y(Ih ·Y [Υ× σ×X]
)=
([Υ× V h × V h] ·Y Ih
)·Y [Υ× σ×X].
We continue
[Υ× V h × V h] ·Y Ih (1.86)
= [Υ× V h × V h] ·Y([Υ× V h ×X] ·(id,µ,µ) [B⊺]
)(1.87)
( By Lemma 1.13 for the transitional [Υ× V h ×X] ) (1.88)
= [Υ× V h × V h] ·(id,µ,µ) [B⊺] (1.89)
( By Lemma 1.13 for the transitional V h = µ−1(Pn+1−h)) (1.90)
= [Υ× V h × V h] ·(id,µ,µ)([Υ×Pn+1−h ×Pn+1−h] ·B B⊺
). (1.91)
where B = Υ×Pn+1−h ×Pn+1. Next to focus on the intersection
[Υ×Pn+1−h ×Pn+1−h] ·B B⊺,
we use coordinates (1.49) to express the intersection scheme
𭟋 = (Υ×Pn+1−h ×Pn+1−h) ∩G (1.92)
where x0, · · · , xn+1−h are homogeneous coordinates for Pn+1−h, y0, · · · , yn+1
for the Pn+1. Then the scheme
𭟋 ⊂ Υ×Pn+1−h ×Pn+1−h
is defined by z(y0xi − x0yi), 0 ≤ i ≤ n+ 1− hyn+2−h, yn+3−h, · · · , yn+1
xiyj − yixj , i, j ∈ [1, n+ 1− h].
The first set shows 𭟋 has two reduced components of dimension n+ 2− h: 𭟋1
defined y0xi − x0yi, 0 ≤ i ≤ n+ 1− hyn+2−h, yn+3−h, · · · , yn+1
xiyj − yixj , i, j ∈ [1, n+ 1− h].(1.93)
and 𭟋2 defined by
z = 0,
yn+2−h, yn+3−h, · · · , yn+1
xiyj − yixj , i, j ∈ [1, n+ 1− h].
(1.94)
25
So𭟋1 = Υ×∆Pn+1−h , 𭟋2 = 0 × E1.
Hence the intersection in Υ×Pn+1 ×Pn+1,
[Υ×Pn+1−h ×Pn+1−h] ·B [G]
is[𭟋1] + [𭟋2] ∈ CHn+2−h(P
n+1−h ×Pn+1−h) (1.95)
where [𭟋1] is onto Υ, but [𭟋2] is not. Hence the intersection with
[Υ× V h × V h]
which is [Υ× V h × V h] ·Y Ih has two parts classified by their support
[Υ× V h × V h] ·(id,µ2) [𭟋1] + [Υ× V h × V h] ·(id,µ2) [𭟋2]. (1.96)
where the first one is an excess intersection and the second one is proper. Noticethe projection of the first part to V h×X is d[∆V h ] (d is the multiplicity) and theprojection of the other is [V h×V h] ·µ2E1, denoted by h. Then after intersectingwith σ×X, followed by the projection formula for the projection X ×X → X,we obtain
[V h] ·X [Conhp(σ)] = d[σ] + h∗([σ]) (1.97)
where h∗ is regarded the correspondence CH(X) → CH(X), and d is an integer.Applying Lemma 1.16, we obtain that as a class h∗([σ]) is a multiple of the planesection class of V h. Since V h is a plane section of X, h∗([σ]) is also a multipleof the plane section class of X.
At last we need to determine the multiplicity d. First we notice the excessintersection in the formula (1.96) is
[Υ× V h ×X] ·(id,µ2) [𭟋1] =
[[Υ× V h ×X] ·(id,µ2) [Υ×∆Pn+1−h ]
]. (1.98)
For this excess intersection, we can use the same type of the linear deformationas gzt , but inside of Pn+1−h and applied to V h. As in the argument part (2) ofProposition 1.11 (for 1-end cycle), we obtain the multiplicity d is the deg(V h)which is deg(X). We complete the proof of (1.85).
Let σ ∈ Zsp(X). Then as in the analysis above, the cycle-intersection
V h •X Conhp(σ)
is proper. Hence the Chow class vh
([Conh
p(σ)]
)is represented by the cy-
cle V h •X Conhp(σ). On the other hand, the class h∗([σ]) can be written as
−[vh(ζ(σ))] for some cycle ζ(σ) ∈ Zq(X), Let L(σ) = ζ(σ). The formula (1.8)follows from (1.85)
26
2 Equivalences-A proof of the Lefschetzstandard conjecture
2.1 Introduction to the Lefschetz standard conjecture
In this section, we introduce equivalences to Theorem 1.1. It leads to a proof ofthe Lefschetz standard conjecture proposed by Grothendieck ([2]). The conjec-ture addresses a cohomological problem over an algebraically closed field k. Theparticular cohomology used by Grothendieck is the l-adic cohomology denotedby Hi
l (X) on a smooth projective variety X of dimension n ≥ 2. Let u be thehyperplane section class in the cohomology H2
l (X). For a whole number h ≤ n,let Lh denote the homomorphism,
Lh :
2n−2h∑i=0
Hil (X) →
2n−2h∑i=0
Hi+2hl (X)
α → α · uh.(2.1)
The hard Lefschetz theorem asserts the Lh is restricted to an isomorphismbetween Hn−h
l (X) and Hn+hl (X). Grothendieck proposed
Conjecture 2.1. (Lefschetz) Let Ai(X) ⊂ H2il (X) be the image of the cycle
map for the cycles of codimension i. If n + h is even, then Lh is restricted toan isomorphism Lh
a,
Lha : A
n−h2 (X) → A
n+h2 (X)
α → α · uh.(2.2)
Conjecture 2.1 is known as one of his two standard conjectures. Grothendieckhas envisioned its root related to his conjectured notion, “motives” whose originis the “universal cohomology”. To address this origin, and also to formulate thedescending of Theorem 1.1, we state an axiomatic “good” cohomology theorymostly based on Kleiman’s approach ([3]).
Definition 2.2. (Good Weil cohomology).A good Weil cohomology theory H(X) is a contravariant functor from the cat-
egory of irreducible, smooth, projective varieties X over an algebraically closedfield to finitely dimensional linear spaces over the coefficient field of K of char-acteristic 0, with the following properties:
1) The linear space H(X) is graded as∑2dim(X)
i=0 Hi(X), where i is calledthe degree and 2dim(X)− i the dimension.
2) There is the cup product
∪ : Hi(X)×Hj(X) → Hi+j(X)(x, y) → x ∪ y (2.3)
27
which has associativity and graded commutativity.3) (Poincare duality) For each X, there is a canonical “orientation” iso-
morphism (∗) : H2n ≃ K such that the compositional bilinear map
Hi(X)×H2n−i(X) → K by (x, y) → (x ∪ y). (2.4)
is non-degenerate and is called the intersection number.4) (projection formula) Let Y be another variety in the category and
f : X → Y a morphism. Denote the contravariant functor map by f∗.The cup product and Poincare duality induce the Gysin homomorphism
Hi(X) → H2dim(Y )−2dim(X)+i(Y ) (2.5)
denoted by f∗. Then cup product is functorial in the following sense: forany (x, y) ∈ H(X)×H(Y ),
f∗(x ∪ f∗(y)) = f∗(x) ∪ y. (2.6)
5) (Kunneth formula) For each X,Y in the category,
H(X × Y ) ≃ H(X)⊗H(Y ).
6) (cycle map) For each X, there is a group homomorphism
cl : Zi(X) → H2i(X), (2.7)
factoring through CH(X) called “cycle map”, satisfying(a) (functoriality) for a morphism f : X → Y ,
f∗ cl = cl f∗, and f∗ cl = cl f∗. (2.8)
( The interchange between cycles and cycle cohomology classes)(b) (multiplicity) For any x ∈ Z(X), y ∈ Z(Y ),
cl(x× y) ≃ cl(x)⊗ cl(y), (2.9)
where ≃ is the Kunneth isomorphism.(c) (calibration) if P is a point, then cl : Z0(P ) → H0(P ) is equal to
the canonical inclusion of integers into the coefficient field K.7) (base model) In a projective space, the homological equivalence is the same
as the rational equivalence.8) (Lefschetz hyperplane theorem) Let h : V → X be the embedding of a
smooth hyperplane section. Then the functor map h∗ : Hi(X) → Hi(V )is an isomorphism for i ≤ dim(X)− 2 and monomorphism fori = dim(X)− 1.
9) (hard Lefschetz theorem) Let H be a hypersurface of X with dimensionn. We define the Lefschetz operator
L : Hi(X) → Hi+2(X) by L(x) := x ∪ cl(H). (2.10)
28
Then for i ≤ n, the (n− i)th iteration of L
Ln−i : Hi(X) → H2n−i(X) (2.11)
is an isomorphism.
In this paper, we use the commutative diagram following from the good Weilcohomology:
Zq−h(X)vh
−→ ZqV h(X)
I−→ Zq(X)cly cl
y cly
Aq−h(X)vh
−→ AqV h(X)
i−→ Aq(X)∩ ∩ ∩
H2(q−h)(X) H2qV h(X) H2q(X)
(2.12)
where vh is the descend of vh, A∗(X) (resp. A∗V h(X)) denotes the image of cl
(resp. Z∗V h(X)) inside of the good Weil cohomology H∗(X), and I and i are the
inclusion maps.
It is known through a collective work of many people that l-adic cohomologyis a good Weil cohomology.
Remark The axioms of Weil cohomology listed in [3] are not sufficientfor our proof which goes through Chow groups. So we add two more axiomsto address the rational equivalence, and call the new theory the good Weilcohomology.
Theorem 1.1 implies that
Theorem 2.3. The Lefschetz standard conjecture is correct for any good Weilcohomology. In particular it is correct for l-adic cohomology.
Remark The implication shows that the Lefschetz standard conjecture is“motivated” by algebraic cycles.
2.2 The proof
2.2.1 The main idea
The cohomology below is a fixed good Weil cohomology.
Definition 2.4. For smooth projective varieties X,Y in the category of goodWeil cohomology, let
ν : Z(X) 99K Z(Y ) (2.13)
be a dash homomorphism. The map ν is cohomological if and only if for any σin a domain Z(X) homologous to 0, ν(σ) is also homologous to 0.
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Notation 2.5. The angle bracket ⟨∗⟩ denotes the cohomology class of the alge-braic cycle ∗.
In this section, we assume n+ h is even and let
q =n+ h
2, p =
n− h
2, p ≥ 2. (2.14)
Note: the case p = 1 is implied by the Lafschetz (1, 1) theorem.
The main idea of the proof is to show in case of (2.14) there is a method deal-ing with equivalences. It resolves the complexity unsolved in the first section.We outline the steps. In the sequence of the good Weil cohomology,
Aq−h(X)vh
−→ AqV h(X)
i−→ Aq(X). (2.15)
vh is surjective due to the Corollary 1.2, and the composition is injective dueto the hard Lefschetz theorem. Hence it is sufficient to prove the surjectivity ofthe inclusion map i.
1). Applying the hard Lefschetz theorem to formula (1.8), we’ll obtain thecohomologicity of Conh
p , and furthermore the cohomological descend
Conhp is injective.2). A non-trivial observation indicates the composition
Conhp fhp (2.16)
preserves the rational equivalence. Then the cohomologicicity of Conhp
leads to the cohomologicity of fhp .3). When the part 2) is applied to the formula (1.9)
mid+ ghp ∼ fhp , (2.17)
we obtain the cohomologicity of ghp . Then the finiteness of cohomology
will imply the class ghp (⟨σ⟩) as a limit lies in V h for σ ∈ Zsp(X). Since
fhp (σ) lies in Vh, the cohomology of σ also lies in V h. So i is surjective.
The structural complexity mentioned in section 1 lies in the dash homomor-phism gh
p that is not in the category of Chow motives due to its set-theoreticalposition over the unstable point.
2.2.2 Extension of rational equivalence
In this subsection, our aim is the calculation of
Conhp fhp (2.18)
30
which is the relative version of
Cz2 Prz1 . (2.19)
There is a rational projection map F
Υ×Pn+1 99K Υ×Pn+1−h
(z,x) 99K (z,x1(z))(2.20)
where x1(z) is from the decomposition (1.40). It induces a correspondence Efrom Pn+1 to Υ×Pn+1, where E is the Zariski closure of the algebraic set(
[x], (z,x1(z))
): z ∈ U
where z lies in the target space2. Hence there are two types of correspondences:Ec∗,E∗.
Lemma 2.6. Let σ ∈ Zp(X) with p ≥ 2. The homomorphism
Ec∗ : Zp(Pn+1) → Zp+h+1(Υ×Pn+1) (2.21)
preserves the rational equivalence, i.e. Ec∗ represents E∗ in the Chow group.
Proof. We show Ec∗ represents the class-correspondence E∗.Let σ ∈ Zp(X) be prime. Recall subspaces
Pn+1−p = P(span(e0, · · · , en+1−p))
Ph = P(span(e0, en+2−p, · · · , en+1)).
Then Ec∗(σ) is the projection of the cycle-intersection
E •(Υ×Pn+1×Pn+1) (Υ× σ ×Pn+1).
So it suffices to show an excess component of
E ∩ (Υ× σ ×Pn+1) (2.22)
has a positive dimension in the fibre of the projection
E ∩ (Υ× σ ×Pn+1) → Υ×Pn+1 (3rd). (2.23)
2The same 2 z′s in U diverge to different points in Υ in the closure.
31
Case 1: σ ⊂ Ph. Since dim(σ) ≥ 2, all components of the intersection (2.22)have positive dimensions in the fibre of the projection
E ∩ (Υ× σ ×Pn+1) → Υ×Pn+1.
In particular the excess components have positive dimensional fibre.Case 2: σ ⊂ Ph. Then σ = σ\(σ ∩ Ph) is irreducible of dimension p. Thenvery components of
F(U × σ)
have dimension p+h+1. Since the proper dimension of the intersection (2.22) isalso p+h+1, then the only excess component of (2.22) are those over z = 0 withone dimensional fibre in the projection (2.23). Therefore Lemma 2.6 is provedfor the prime cycle σ. Then the proof is linearly extended to all cycles.
The individual operators Conhp , f
hp do not respect equivalences. But we’ll
show the composition Conhp fhp (σ) does.
Theorem 2.7. If σ ∈ Zsp(X) is homologous to zero, so is
Conhp fhp (σ) ∈ Zp(X).
Proof. The idea is to have a different expression which shows the cohomologylost in individual operators Conh
p and fhp (σ) is combined to zero in the compo-
sition Conhp fhp (σ).
We define a composition of series of homomorphisms1) ϕ1. Let
ϕ1 : Zi(Pn+1) → Zi+h+1(Υ×Pn+1)
σ → Ec∗(µ∗σ).(2.24)
2) ϕ2. There is a hypersurface Q2 in Υ×Pn+1 whose fibre over each z ∈ Uis
Cz(µ∗Vh)
where z ∈ U , and Cz is the cone in (1.20). Notice Q2 is an irreduciblevariety. Let
ϕ2 : Zsi (Υ×Pn+1) → Zi−1(Υ×Pn+1)σ → Q2 •(Υ×Pn+1) σ
(2.25)
3) ϕ3. There is a rational map Q3,
Υ×Υ×Pn+1 99K Υ×Υ×Pn+1
(z2, z1, [x]) 99K (z1, z2, [x1(z1) + x2(z2)])(2.26)
We let ϕ3 = (Q3)c∗.
32
4) ϕ4. Let
ϕ4 : Zsi (Υ×Υ×Pn+1) → Zi−1(Υ×Υ×X)
σ → (Υ×Υ×X) •(id,id,µ) σ(2.27)
5) ϕ5. Let π4 : Υ×Υ×X → X be the projection. Let ϕ5 = (π4)∗.
Notice besides ϕ1 all these maps are obtained through a proper intersection.Hence by Lemma 2.6 for ϕ1, all homomorphisms can be reduced to well-definedclass homomorphisms on Chow groups. If σ is homologous to zero, by the basemodel axiom
µ∗(σ) ∈ Zp(Pn+1) (2.28)
is rationally equivalent to zero. Thus
ϕ5 ϕ4 ϕ3 ϕ2 ϕ1(σ)
is rationally equivalent to zero. In particular it is homologous to zero. Thereforeit suffices to prove the claim
Claim 2.8. Let σ ∈ Zsp(X). Then
Conhp fhp (σ) = ϕ5 ϕ4 ϕ3 ϕ2 ϕ1(σ) (2.29)
Proof of the claim: We’ll label the same space in a different position of theCartesian product as follows. Let
Ψ = Υ1 ×X1 ×X2 ×Υ2 ×X3, (2.30)
whereX1 = X2 = X3 = X,Υ1 = Υ2 = Υ.
LetS1 = X2 × 0 ×X3 ⊂ X2 ×Υ2 ×X3.
Recall that the correspondence cycle for Conhp is Ih ( see (1.69)). We identify
it as the transposed cycle in X2 ×Υ2 ×X3. Similarly the correspondence cyclefor fhp is γ2 ( see (1.61)). We identify it as a cycle in Υ1 ×X1 ×X2. There isthe commutative diagram There is the diagram of projections
Ψπ1→ X2 ×Υ2 ×X3
π2→ X3. (2.31)
In Ψ, we consider the intersection(Υ1 ×X1 × Ih
)⊙Υ1×X1×S1
((γ2 •(Υ1×X1×X2)
(Υ1 × σ ×X2
))×Υ2 ×X3
)
denoted by T (σ). To remove S1, we restrict the graph to the open set
Ψ := Υ1 ×X1 ×X2 × U2 ×X3), (2.32)
33
for the open set U2 = U . The (2.31) becomes
Ψ π1→ X2 × U2 ×X3π2→ X3. (2.33)
Since π1 in (2.33) is a proper projection, we apply the projection formula inProposition 1.6 to the projection π1. Then we obtain that
(π1)∗(T (σ)) = Ih •(U2×X2×X3) (U2 × fhp (σ)×X3).
Thus(π2)∗ (π1)∗(T (σ)) = Conh
p fhp (σ). (2.34)
On the other hand there is a commutative diagram,
Ψπ1
xx
π3
&&Υ2 ×X2 ×X3
π2
&&
Υ1 ×Υ2 ×X3
π4
xxX3
(2.35)
HenceConh
p fhp (σ) = (π4)∗ (π3)∗(T (σ)).
Comparing both sides of (2.29), it suffices to show
(π3)∗(T (σ)) = ϕ4 ϕ3 ϕ2 ϕ1(σ) (2.36)
as an algebraic cycle in the projective variety Υ1 × Υ2 × X. Considering thefibre projection
Υ1 ×Υ2 ×X → X
if σ is prime, the fibres of both sides of (2.36) are prime cycles(see part (14)in Notation 3 for the fibres of cycles). Thus it suffices to prove the equalityof generic fibres. We consider their fibres over generic points for a cycle σin a general position which in particular satisfies that the projection of eachcomponent of µ∗(σ) to Pn+1−h from infinity Pn+1
z1 is finite to one. Then thefibre of
(π3)∗(T (σ)) ⇒ Υ×Υ
over a generic point (z1, z2) ∈ U × U is expressed in the diagonal-deformationcycles in a single projective space Pn+1. Precisely it is equal to
X •µ Cz2
(µ∗
(V h •µ Prz1(µ∗σ)
))(2.37)
where the subscripts z1, z2 are referred to the decomposition in (1.39). The fibreof
ϕ4 ϕ3 ϕ2 ϕ1(σ) ⇒ Υ×Υ
34
is also expressed in a diagonal-deformation. Precisely it is equal to
X •µ Cz2
(µ∗
(V h •µ Prz1(µ∗σ)
))(2.38)
which is the same as (2.37). So Claim 2.8 is proved.
2.2.3 Cohomologicity
Theorem 2.9. Conhp is cohomological on generally positioned p-cycles.
Proof. We need to consider the operator Conhp which acts on p cycles. Let
σ ∈ Zsp(V
h). First we consider the formula (1.8),
vh (Conhp + L)(σ)∼ mσ (2.39)
If
σhomologically equi.∼ 0,
then
vh Conhp(σ)
homologically equi.∼ 0. (2.40)
Notice the indexes are matched for the hard Lefschetz theorem on (2.40). So
Conhp(σ)
homologically equi.∼ 0.
Hence Conhp on p-cycles is cohomological for the homological equivalence of X.
Theorem 2.10. The operator fhp is cohomological on generally positioned p-cycles.
Proof. Due to Theorem 2.9, we let Conhp be the cohomological descend of Conhp .
There is the Gysin homomorphism
µq∗ : Aq(X) → Aq+1(Pn+1)
where µq∗ = µ∗|Aq(X). Then we have a decomposition
Aq(X) ≃ im(µq∗)⊕ ker(µq
∗). (2.41)
35
For σ ∈ ker(µq∗), the formula (1.8) reduces to
vh Conhp(⟨σ⟩) = m⟨σ⟩.
Thus Conhp |ker(µq∗) is injective. Suppose Con
hp is not injective on the entire group
Aq(X). Then there is σ ∈ Zsp(X) such that µ∗(⟨σ⟩) = 0 and Conhp(⟨σ⟩) = 0.
By the formula (1.85),0 = ⟨σ⟩+ ⟨h∗[σ]⟩. (2.42)
Hence the cohomology of σ is represented by a non-zero multiple of the planesection. Thus it suffices to assume σ is a plane section. Then by Definition 1.14,Conhp(⟨σ⟩) is represented by the cycle through a closure intersection,
(η3)∗(Ih •(U×V h×X) (U × σ ×X)). (2.43)
where η3 : Υ×V h×X → X is the projection. The cycle is a variety fibred overΥ and each fibre over z ∈ U is birational to an irreducible subvariety µ−1(H )where H is the join
P(span(en+2−h, · · · , en+1))#σ.
Hence each fibre is irreducible and has the same dimension. It implies that
(η3)∗(Ih •(U×V h×X) (U × σ ×X)). (2.44)
is a prime cycle. Hence its non-zero multiple does not represent the 0 class inthe CH(X). Therefore Conhp(⟨σ⟩) = 0. This contradiction indicates Conhp is
injective. If σ is homologous to 0, by Theorem 2.7, Conhpfhp (σ) is homologous to
0. By the injectivity of Connp , fhp (σ) is homologous to 0. So it is cohomological.
Proof. of Theorem 2.3: Let σ ∈ Zsp(X). In the following we calculate multiplic-
ities. Let ϱ : µ(X) → Pn be the projection, where Pn is the hyperplane definedby x0 = 0. Notice ϱ is a multiple covering of multiplicity m. Let Ci, i ≤ l be thefinitely many components of gh
p (σ), and |Ci| be its algebraic set. Then the themultiplicity l ≤ m− 1 due to the removing of the component (1.54). Similarlydue the support of cycles, we can also have
ghp (|Ci|) =
∑j
mij |Cj |. (2.45)
Furthermore by the definition of the operator ghp (see (1.56)), the multiplicity
mij is the same intersection multiplicity of
Σ •Y (1 × 0 × |Ci| ×X). (2.46)
36
which is 1 due to the general position of σ. Then the formula (1.9) can berewritten as
σrationally equi.∼
fhpm
(σ)−ghp
m(σ) (2.47)
=fhp (σ)
m−
∑i≤m−1
|Ci|
m(2.48)
We apply the operatorfhpm
−ghp
m
on both sides of (2.48) repeatedly N times to obtain
σhomologically equi.∼ gh
p (σ′) + fhp (σ
′′)−∑i
1
m(l
m)N |Ci| (2.49)
where for some p-cycles σ′, σ′′ with σ′ supported on V h. Notice that the 3rdrow of the expression (1.49) shows gh
p (σ′) is also supported on V h. Since fhp
is cohomological, so is ghp . Then due to the cohomologicity of gh
p , we obtain aformula in the cohomology⟨
σ −∑i
1
m(l
m)N |Ci|
⟩∈ AV h(X). (2.50)
Since l ≤ m− 1, the rational numbers∑i
1
m(l
m)N ≤ 1
m(m− 1
m)N
which converges to 0 as N → ∞. Thus the cohomology class
⟨∑i
1
m(l
m)N |Ci|⟩
=1
m(l
m)N∑i
⟨|Ci|⟩(2.51)
also converges to 0 in the cohomology. Then in the finitely dimensional spaceA(X), σ is the limit of ⟨
σ −∑i
1
m(l
m)N |Ci|
⟩in Zariski topology. Thus ⟨σ⟩ also lies in the closed set AV h(X). Hence theinclusion map
AqV h(X)
i−→ Aq(X)
37
is surjective. Now we observe the sequence in the good Weil cohomology,
Aq−h(X)vh
−→ AqV h(X)
i−→ Aq(X). (2.52)
By Corollary 1.2, the cohomological descend vh of vh is surjective. So thecomposition i vh is also surjective. The hard Lefschetz theorem asserts thecomposition is injective, thus an isomorhpism.
References
[1] W. Fulton, Intersection theory, Springer-Verlag (1980).
[2] A. Grothendieck, Standard conjectures on algebraic cycles, Algebraicgeometry, Bombay Colloqium, 1968, pp 193-199.
[3] S. Kleiman, The standard conjectures, Proceeding of symposis in puremathematics , 1994, pp 3-20.
4 Chessman drive, Sharon, MA02067, USA
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