Consequences of the Lefschetz Standard Conjecture

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In the remaining statements, p is a ﬁxed prime number and F is an algebraic closure of the ﬁeld Fp of p elements. Theorem 2. The Lefschetz standard conjecture for algebraic varieties over F implies (a) the full Tate conjecture for abelian varieties over F; (b) the standard conjecture of Hodge type for abelian varieties in characteristic p.

See §3 for the Tate conjecture and Kleiman 1994, p. 16, for the Hodge standard conjecture.

Theorem 3. The full Tate conjecture for algebraic varieties over F implies Grothendieck’s standard conjectures over F.

We prove these theorems in the ﬁrst four sections of the paper. In Section5, we use a constructionof Schäppi to give unconditional variantsof our theorems, and in Section 6, we list some statements that imply the Lefschetz standard conjecture. We refer to Kleiman 1994 for the various forms, A,B,C,D, of the Lefschetz standard conjecture. We assume that the reader is familiarwith the expository article Milne 2020, cited as HAV. Throughout, ℚ is the algebraic closure of ℚ in . al ℂ 1 ProofofTheorem1

In this section, X is an algebraic variety over a ﬁeld k of characteristic zero.

1.1. Suppose first that k is algebraically closed. We let Br(X) denote the space of absolute Hodge classes of codimension r on X. Thus Br(X) is a finite-dimensional - subspace of the adèlic cohomology group H r(X)(r) (Deligne 1982, §3). Let k → k beℚ 2 ¨ a homomorphism from k into a second algebraicallyA closed field k ; then the canon- ¨ r r r r ical map H (X)(r) → H (Xk )(r) induces an isomorphism B (X) → B (Xk ) (ibid., 2 2 ¨ ¨ 2.9). For anA abelian varietyA over , every Hodge class is absolutely Hodge (ibid., Main Theorem 2.11). ℂ

1.2. Now allow k to be arbitrary of characteristic 0, and let k be an algebraic clo- al sure of k. Then k k acts on Br X through a ﬁnite quotient, and Br X def al k r k k al B X al . Gal( ∕ ) ( ) ( ) = k Gal( ∕ ) ( al ) 1.3. We deﬁne an almost-algebraic class of codimension r on X to be an absolute

Hodge class of codimension r such that there exists a cartesian square → ← X → ← ← X→

f S → ← k Spec( ) and a global section ̃ of R satisfying the following conditions, 2r ⋄ S is the spectrum of af regular∗A(r) integral domain of ﬁnite type over ⋄ is smooth and projective; ℤ; ⋄ thef ﬁbre of over is , and the reductionof at is algebraic for all closed points in ã denseSpec(k) open subset of S. ̃ s s 2 ALMOST-ALGEBRAIC CLASSES ON ABELIAN VARIETIES 3

Cf. Serre 1974, 5.2, and Tate 1994, p.76. Usually, almost-algebraic classes are not required to be absolutely Hodge, but since we have a robust theory of absolute Hodge classes, it is natural to include it.

Theorem 1.4. Assume that the Lefschetz standard conjecture holds for algebraic varieties over finite fields. Then all absolute Hodge classes on abelian varieties over fields of characteristic zero are almost-algebraic.

Proof. Itsuﬃcestoprovethiswith k , where it becomes a question of showing that Hodge classes on abelian varieties are almost-algebraic.=ℂ Let X be an algebraic variety of dimension d over , and let L H X, → H X, be the Lefschetz operator ∗ ∗+2 on Betti cohomologyℂ deﬁned by∶ a hyperplane( ℚ) section.( Accordingℚ(1)) to the strong Lefschetz theorem, the map L H X, → H X, d i is an isomorphism. For i≤d, d−i i 2d−i let H X, d ∶i →( Hℚ)X, denote( theℚ)( inverse− ) isomorphism. i 2d−i i The∶ isomorphism( ℚ)( − ⊗) H( ℚ)X d i → H X is absolutely Hodge (i.e., its i 2d−i i graph is an absolute Hodge1∶ class).A Consider( )( − a) diagramA( as) in 1.3. For a closed point s of S such that X and L have good reduction, ⊗ specializes to the inverse of the iso- i morphism L s H X s → H X s d 1i . As we are assuming the standard d−i i 2d−i conjecture over( ) ,∶ thisA( inverse( )) is algebraicA ( ( ))( (Kleiman− ) 1994, 4-1, ⇔ B). Hence is i almost-algebraic.F Since this holds for all X and i, the Lefschetz standard conjecture holds for almost- algebraic classes on algebraic varieties over . As for algebraic classes, this implies that all Hodge classes on abelian varieties are almost-algebraicℂ (HAV, Theorem 4). ✷

Note that the theorem does not say that an absolute Hodge class becomes algebraic modulo p for any speciﬁc p, even when the abelian variety has good reduction at p. In the next section, we prove this.

2 Almost-algebraic classes on abelian varieties

Fix a prime number p, and let be an algebraic closure of p. In the following, l is a prime number ≠ p. F F

Variation of algebraic classes over Proposition 2.1. Let S be a complete smoothF curve over and X → S an abelian scheme over S. Assume that the Lefschetz standard conjectureF holdsf∶ for X and l-adic étale cohomology. Let t be a global section of the sheaf R r ; if is algebraic for one 2 l s S , then it is algebraic for all s. f∗ℚ (r) t s∈ (F) Proof. For a positive integer n prime to p, let n denote the endomorphism of X S acting as multiplication by n on the ﬁbres. By a standard argument (Kleiman 1968∕ , p. 374), acts as nj on Rj . As commutes with the diﬀerentials of the Leray n∗ l n∗ spectral sequence i S, Rjf∗ℚ i j X, , we see that it degeneratesd2 at the l + l E -term and H ( f∗ℚ ) ⟹ H ( ℚ ) 2 r i j H X, l Ñ H S, R l , ( ℚ )≃ i j r ( f∗ℚ ) + = 2 ALMOST-ALGEBRAIC CLASSES ON ABELIAN VARIETIES 4

i j r j where S, R l is the direct summand of X, l on which n acts as n . We let aH denoteH ( the f∗-subspaceℚ ) of a cohomology groupH (H spannedℚ ) by the algebraic classes. Let s S ℚ and let S, s . The inclusion js Xs ↪ X induces an isomorphism j ∈H (F)S, R r =→ 1(r X), preserving algebraic∶ classes, and so s∗ 0 2 l 2 s l ∶ ( f∗ℚ ) H ( ℚ ) dim aH (S, R r ) ≤ dim aH r(X , ). (1) 0 2 l 2 s l f∗ℚ ℚ Similarly, theGysinmap j H d r X , → H d r X, , where dim(X∕S), s 2 −2 s l 2 −2 +2 l induces a map H d r(X , ∗ ∶) → H( (S,ℚ R d) rf ) preserving( ℚ ) algebraicd = classes, and 2 −2 s l 2 2 −2 l so ℚ ∗ℚ dim aH d r(X , ) ≤ dim aH (S, R d rf ). (2) 2 −2 s l 2 2 −2 l ℚ ∗ℚ Because the Lefschetz standard conjecture holds for Xs (Kleiman 1968, 2A11), dim aH r(X , ) = dim aH d r(X , ). (3) 2 s l 2 −2 s l ℚ ℚ Hence,

1 3) dim aH (S, R rf ) (≤) dim aH r(X , ) = dim aH d r(X , ) 0 2 l 2 s l ( 2 −2 s l ∗ℚ 2 ℚ ℚ (≤) dim aH (S, R d rf ). 2 2 −2 l ∗ℚ The Lefschetz standard conjecture for X implies that

dim aH (S, R rf ) = dim aH (S, R d rf ), 0 2 l 2 2 −2 l ∗ℚ ∗ℚ and so the inequalities are equalities. Thus

aH r(X , ) = aH (S, R rf ), 2 s l 0 2 l ℚ ∗ℚ which is independent of s. ✷

Remark 2.2. The proof shows that t, when regarded as an element of H r(X, (r)), is 2 l algebraic. ℚ

Weil classes

Fix a prime w of dividing p. The residue ﬁeld at w is an algebraic closure of . al p We refer to Deligneℚ 1982 or HAV for facts on abelian varieties of Weil type. F F Proposition 2.3. Assume that the Lefschetz standard conjecture holds for algebraic varieties over and l-adic étale cohomology, some l ≠ p. Let (A, ) be an abelian variety over of splitF Weil type relative to a CM ﬁeld E, and let t∈W (A) ⊂ H r(A) be a Weil al E 2 classℚ on A. If A has good reduction at w to an abelian variety A over , thenA the element (t ) of H r(A , ) is algebraic. 0 F l 2 l 0 0 ℚ The proof will occupy the remainder of this subsection. In outline, it follows the proof of Deligne 1982, Theorem 4.8, but requires a delicate reduction argument of An- dré.

Lemma 2.4. Let (A, ) be an abelian variety over al of split Weil type relative to E. Then there exists a connected smooth variety S over , anℚ abelian scheme f X → S over S, and an action of E on X S such that ℂ ∶ ∕ 2 ALMOST-ALGEBRAIC CLASSES ON ABELIAN VARIETIES 5

(a) for some s S , Xs , s A, 1 1 1 ℂ (b) for all s S∈ (ℂ), Xs(, s is of) split ≈ ( Weil) type; relative to E; (c) for some∈s (ℂ)S ( , Xs )is of the form B ⊗ E with e E acting as id⊗e. 2 ∈ (ℂ) 2 ℚ ∈ Proof. See the proof of Deligne 1982, 4.8. ✷

We shall need to use additional properties of the family X → S constructed by Deligne. For example, there is a local subsystem W (X∕S) of R rf suchthat W (X∕S) = E 2 E s ∗ WE(Xs) for all s ∈ S( ). Also, the variety B in (c) can be chosenℚ to be a power of CM elliptic curve (so Xs isℂ isogenous to a power of a CM elliptic curve). The variety S has2 a unique model over with the property that every CM-point al s ∈ S( ) lies in S( ). This follows from theℚ general theory of Shimura varieties; or al from theℂ general theoryℚ of locally symmetric varieties (Faltings, Peters); or (best) from descent theory (Milne 1999a, 2.3) using that S is a moduli variety over and that the moduli problem is defined over . The morphism f is also defined overℂ , and we al al will now simply write f X → Sℚfor the family over . There is a -localℚ subsystem al W X S of R rf such∶ that W X S W X ℚfor all s S ℚ . The points s E 2 l E s E s al and(s ∕lie) in S ∗ℚ. ( ∕ ) = ( ) ∈ (ℚ ) 1 al We2 now assume(ℚ ) that E contains an imaginary quadratic field in which the prime p splits — this is the only case we shall need, and it implies the general case. The family X → S (without the actionof E) defines a morphism from S intoamoduli variety M over for polarized abelian varieties with certain level structures. Let ℳ al be the correspondingℚ moduli scheme over O and ℳ its minimal compactification w ∗ (Chai and Faltings 1990). Let S be the closure of S in ℳ . ∗ ∗ Lemma 2.5. The complement of S ℳ in S has codimension at least two. ∗ ∗ F ∩ F F Proof. See André 2.4.2. ✷

Recall that s and s are points in S such that X A and X is a power of a al s s CM-elliptic curve.1 As A2 and the elliptic(ℚ curve) have good1 reduction,= the2 points extend to points s and s of S ℳ. Let S̄ denote the blow-up of S centred at the closed sub- ∗ ∗ scheme defined1 by2 the image∩ of s and s , and let S be the open subscheme obtained by removing the strict transform of1 the boundary2 S S ℳ . It follows from 2.5 that ∗ ∗ S is connected, and that any sufficiently general linear∖ ( ∩ section) of relative dimension F N dim(S) − 1 in a projective embedding S̄ ↪ is a projective flat Ow-curve C con- Ow tained in S with smooth geometrically connectedℙ generic fibre (André 2.5.1). Consider (X C) → C . After replacing C by its normalization and pulling back (X C) , we are F F in theð F situation of Proposition 2.1. The class ts is algebraic because the Hodgeð F conjecture holds for powers of elliptic curves (the -algebra2 of Hodge classes is generated by divisor classes). Hence (ts l) is algebraic, andℚ 2.1 shows that (ts l) is algebraic. This completes the proof of Proposition2 0 2.3. 1 0

Absolute Hodge classes on abelian varieties

Again, w is a prime of lying over p and l is a prime number ≠ p. al ℚ Theorem 2.6. Assume that the Lefschetz standard conjecture holds for algebraic varieties over . Let A be an abelian variety over with good reduction at w to an abelian variety al A overF , and let t be an absolute Hodgeℚ class on A. The class (tl) on A is algebraic. 0 F 0 0 3 PROOF OF THEOREM 2. 6

Proof. We first assume that A is CM, say, of type (E, Φ). Let F be a CM-subfield of , finite and Galois over , that splits E. We may suppose that F contains an imaginaryℂ quadratic field in whichℚp splits. For each subset ∆ of Hom(E, F) such that t∆ ∩ Φ = r = t∆ ∩ Φ̄ for all t ∈ F , we let A A ⊗ F. There isð an obviousð homomorphismð ð f A → ∏s E,s Gal(A . The∕ℚ) abelian variety∆ = A ∈∆is of split Weil type, and every absolute Hodge class∆ ∶ t on A∆can be written as a sum ∆t f t with t a Weil class on A (André 1992; HAV, ∑ ∗ Theorem 1). Thus the theorem= in this∆( ∆ case) follows∆ from Proposition∆ 2.3. We now consider the general case. There exists an abelian scheme f X → S over with S a connected Shimura variety, and a section of R rf such that∶ X, A,tℂ 2 s (Deligne 1982, 6.1). As before, we may suppose that f is∗A defined over( ) and= ( that) al s S . There exists a point s S such that s s in S andℚX isaCM al ¨ al ¨ s 0 0 0 ¨ abelian∈ (ℚ variety) (Kisin, Vasiu). Now∈ the(ℚ theorem) for X(s implies) = that (F)tsl is algebraic.✷ ¨ ( )0 3 ProofofTheorem2.

Fix an algebraic closure of p, and let q be the subfield of with q elements. F F F F 3.1. Let X be an algebraic variety over q. For l≠ p, the Tate conjecture T X, l states that the -vector space H X F q is spanned by algebraic classes, and( the) con- l l2∗ Gal(F∕F ) jecture SℚX, l states that the( obvious)(∗) map H X q → H X is l2∗ Gal(F∕F ) l2∗ q an isomorphism.( ) The full Tate conjecture T X states( )(∗) that, for all r, the( pole)(∗) ofGal(F∕F the zeta) function Z X,t at t q r is equal to the rank( ) of the group of numerical equivalence − classes of algebraic( ) cycles= on X of codimension r. It is known (folklore) that, if T X, l and S X, l hold for a single l, then the full Tate conjecture T X holds, in which( case) T X, l( and) S X, l hold for all l. See Tate 1994. ( ) ( We) say thatone( ) of these conjecturesholds for analgebraicvariety X over if it holds for all models of X over finite subfields of (it suffices to check that it holdsF for some model over a sufficiently large subfield). F

Theorem 3.2. Assume that the Lefschetz standard conjecture holds for algebraic varieties over and l-adic étale cohomology (some l≠ p). Then the full Tate conjecture holds for abelianF varieties over ﬁnite ﬁelds of characteristic p.

Proof. In Milne 1999b, the Tate conjecture for abelian varieties over is shown to follow from the Hodge conjecture for CM abelian varieties over . However,F the proof does not use that the Hodge classes are algebraic, but only that thℂ ey become algebraic modulo p. Hence we can deduce from Proposition 2.6 that the Tate conjecture holds for abelian varieties over and some l. As the Frobenius map acts semisimply on the cohomology of abelian varietiesF (Weil 1948), this implies that the full Tate conjecture holds for abelian varieties over . ✷ F Theorem 3.3. Assume that the Lefschetz standard conjecture holds for algebraic varieties over and l-adic étale cohomology (some l ≠ p). Then Grothendieck’s standard conjectureF of Hodge type holds for abelian varieties over ﬁelds of characteristic p and the classical Weil cohomology theories. 4 PROOF OF THEOREM 3. 7

Proof. In Milne 2002 the Hodge standard conjecture for abelian varieties in characteristic p is shown to follow from the Hodge conjecture for CM abelian varieties over . Again, the proof uses only that the Hodge classes become algebraic modulo p, and soℂ the theorem follows from Proposition 2.6. ✷

Corollary 3.4. Assume that the Lefschetz standard conjecture holds for algebraic varieties over and l-adic étale cohomology (some l≠ p). Then the conjecture of Langlands and RapoportF (1987, 5.e) is true for simple Shimura varieties of PEL-types A and C.

Proof. Langlands and Rapoport (ibid., §6) prove this under the assumption of the Hodge conjecture for CM abelian varieties and the Tate and Hodge standard conjectures for abelian varieties over . However, their argument does not use that Hodge classes on CM abelian varieties areF algebraic, but only that they become algebraic modulo p. As this, together with the Tate and Hodge standard conjectures, are implied by the Lefschetz standard conjecture, so also is their conjecture. ✷

4 ProofofTheorem3.

Brieﬂy, the Tate conjecture over implies the Lefschetz standard conjecture over , and hence the Hodge standard conjectureF for abelian varieties (Theorem 2). Now formF the category of abelian motives over : Grothendieck’s standard conjectues hold for it. The full Tate conjecture implies that theF category of abelian motives contains the motives of all algebraic varieties over , and so the Hodge standard conjecture holds for them also. We now prove more preciseF statements.

Proposition 4.1. Let X be an algebraic variety over . If the Tate conjecture holds for X and some l, then the Lefschetz standard conjecture holdsF for X and the same l.

Proof. To prove the Lefschetz standard conjecture for X and a prime l, it suﬃces to show that, for each ≤ def dim(X), there exists an algebraic correspondence inducing an isomorphism H id i(dX)=→ Hi (X) (Kleiman 1994, 4-1, (X) ⇔ B(X)). The inverse i l2 − l of the Lefschetz map Ld i ∶ Hi (X) → H d i(X)( − ) is an isomorphism d i(X)( − − l l2 − l2 − i d i H d i ) → l(X) commuting with the action of the Galois group. Any algebraic class i suﬃcientlyi H close to the graph of will induce the required isomorphism. ✷

Proposition 4.2. Let H be a Weil cohomology theory on algebraic varieties over an algebraically closed ﬁeld k, and let X and Y be algebraic varieties over k. Assume that there exists an algebraic correspondence on X × Y such that

∶ H (X) → H (Y) ∗ ∗ ∗ is injective. If the Hodge standard conjecture holds for Y, then it holds for X.

Proof. Apply Kleiman 1968, 3.11, and Saavedra Rivano 1972, VI, 4.4.2. ✷

Lemma 4.3. Let X be an algebraic variety over q. If S(X × X, l) holds for some l, then the Frobenius endomorphism acts semisimply onF the l-adic étale cohomology of X. 4 PROOF OF THEOREM 3. 8

Proof. The statement S(X×X, l) says that 1, if an eigenvalue of the Frobenius element acting on the l-adic cohomology of X × X, is semisimple. From the Künneth formula

Hr (X × X)≃ Hi (X) ⊗ Hj (X) l Ñi j r l l + = and linear algebra, we see that this implies that all eigenvalues on H (X) are semisim- l∗ ple. ✷

It is conjectured that the Frobenius element always acts semisimply (Semisimplicity Conjecture). Fix a power q of p and a prime l ≠ p. Define a Tate structure to be a finite- dimensional l-vector space with a linear (Frobenius) map $ whose characteristic polynomial liesℚ in [T] and whose eigenvalues are Weil q-numbers, i.e., algebraic numbers such that, forℚ some integer (called the weight of ), ( ) = m for every ó ó ∕2 homomorphism ∶ [ ] → , and,m for some integer , n óis anó algebraicq integer. When the eigenvalues ℚ are all ofℂ weight (resp. algebraicn integers,q resp. semisimple), we say that is of weight (resp. effectivem , resp. semisimple). For example, for any V m i smooth complete variety X over k, Hl(X) is an effective Tate structure of weight ∕2 (Deligne 1974), which is semisimple if X is an abelian variety (Weil 1948, no. 70). i

Proposition 4.4. Every effective semisimple Tate structure is isomorphic to a Tate substructure of H (A) for some abelian variety A over . l∗ q F Proof. We may assume that the Tate structure V is simple. Then V has weight for some ≥ 0, and the characteristic polynomial P(T) of $ is a monic irreduciblem poly- nomialm with coefficients in whose roots all have real absolute value qm . According ∕2 to Honda’s theorem (Hondaℤ 1968; Tate 1968), P(T) is the characteristic polynomial of an abelian variety A over q . Let B be the abelian variety over q obtained from A by restriction of the base field.F m The eigenvalues of the Frobenius Fmap on ( ) are the l1 th-roots of the eigenvalues of the Frobenius map on (A), and it followsH thatB V is a l1 m m H Tate substructure of l ( ). ✷ H B Theorem 4.5. Let X be an algebraic variety over , and let l be a prime ≠ p. If the Frobe- nius map acts semisimply on H (X) and the TateF conjecture holds for l and all varieties l∗ of the form X × A with A an abelian variety, then the Hodge standard conjecture holds for X and l.

Proof. According to 4.4, there exists an inclusion H (X) ↪ H (A) of Tate structures l∗ l∗ with A an abelian variety. This map is defined by a cohomological correspondence on X × A fixed by the Galois group. Any algebraic correspondence sufficiently close to this correspondence defines an inclusion H (X) ↪ H (A). Now we can apply Proposition l∗ l∗ 4.2. ✷

Corollary 4.6. If the Tate and semisimplicity conjectures hold for all algebraic varieties over and some prime number l, then both the full Tate and Grothendieck standard con- jecturesF hold for all algebraic varieties over and all l. F Proof. Immediate consquence of the theorem. ✷ 5 AN UNCONDITIONAL VARIANT 9

5 An unconditional variant

We use Schäppi 2020 to replace some of the above statements by unconditional variants.

Characteristic zero Let k be an algebraicallyclosed field of characteristiczero, and fix an embedding k ↪ . Let H denote the Weil cohomology theory X ⇝ H (X( ), ), and let Mot (k) denoteℂ ∗ H the category of motives defined using almost-algebraic classesℂ ℚ as correspondences. It is a graded pseudo-abelian rigid tensor category1 over . According to Schäppi 2020, §3, the fibre functor !ℚH ∶ MotH(k) → -Vec factors in ℚ a canonical way through a “universal” graded tannakian category ℳHℤ(k) over , ℚ ! MotH(k) [−] ℳH(k) -Vec , ⟶ ⟶ℤ ℚ where ! is a graded fibre functor.2 We define the algebraic* classes on an algebraic variety X over k to be the elements of Hom(11, [ℎ(X)]). TheLefschetzstandardconjectureholdsforalgebraic*classes (Schäppi 2020, §3; alternatively, apply Corollary 6.5 below). Now !H is a functor from MotH(k) into the category Hdg of polarizable rational Hodge structures. This factors through ℳH(k), ℚ

! MotH(k) [−] ℳH(k) Hdg , ⟶ ⟶ ℚ where ! is a functor of graded tannakian categories. Therefore algebraic* classes on X are Hodge classes relative to the given embedding of k into . It follows that Grothen- dieck’s standard conjecture of Hodge type holds for algebraic*ℂ classes. Moreover, all algebraic* classes on abelian varieties are absolutely Hodge (Deligne 1982, 2.11). The same proof as for almost-algebraic classes (see §1) shows that the Hodge conjecture holds for algebraic* classes on abelian varieties over , i.e., all Hodge classes on abelian varieties over are algebraic*. As a consequence, forℂ abelian varieties satisfying the Mumford-Tate conjecture,ℂ the Tate conjecture holds for algebraic* classes.

Characteristic p

Fix a prime number p, and let denote an algebraic closure of p. For l ≠ p, we let Motl( ) denote the category ofF motives over defined using algebraicF classes modulo l-adicF homological equivalence as correspondences.F It is a graded pseudo-abelian rigid tensor category over . ℚ According to Schäppi 2020, §3, the graded tensor functor !l ∶ Motl( ) → -Vec l ℚ factors in a canonical way through a graded tannakian category ℳl( ), F ℤ F ! Motl(k) [−] ℳl( ) -Vec , ⟶ F ⟶ℤ ℚ where ! is a graded fibre functor. Unfortunately, we do not know that End(11)= in 3 ℳl( ), only that it is a subfield of l. ℚ F ℚ 1tensor category (functor) = symmetric monoidal category (functor) 2fibre functor = exact faithful tensor functor 3André’s category of motivated classes in characteristic p has the same problem. 6 STATEMENTS IMPLYING THE LEFSCHETZ STANDARD CONJECTURE 10

Let X be an algebraic variety over . We deﬁne the algebraic* classes on X to be the elements of Hom(11, [ℎ(X)]). As before,F the Lefschetz standard conjecture holds for algebraic* classes. Therefore Proposition 2.1 holds unconditionally for algebraic* classes: let f ∶ X → S be as in the proposition, and let t be a global section of the sheaf R rf (r); if t is algebraic* for one s ∈ S( ), then it is algebraic* for all s. 2 l s ∗ℚ F Remark 5.1. Until it is shown that End(11)= in ℳl( ), this categoryis of only mod- est interest. For abelian motives, what is neededℚ is a proofF of the rationality conjecture (Milne 2009, 4.1).4

Mixed characteristic

Fix a prime w of dividing p and a prime number l≠ p. Theorem 2.6 holds uncondi- al tionally for algebraic*ℚ classes: let A be an abelian variety over with good reduction al at w to an abelian variety A over , and let t be an absoluteℚ Hodge class (e.g., an al- 0 gebraic* class) on A; then (tl) is anF algebraic* class on A . The proof is the same as before, using the * version of Proposition0 2.1. 0 We deduce, as in the proof of Theorem 3.2, that the Tate conjecture holds for algebraic* classes on abelian varieties over ,i.e., that l-adic Tate classes on abelian varieties over are algebraic*. F LetF ℳ ( ) denote the tannakian subcategory of ℳ ( ) generated by abelian ¨H al H al varieties withℚ good reduction at w. There is a canonical tensorℚ functor ℳ ( ) → ¨H al ℳl( ). ℚ F 6 StatementsimplyingtheLefschetzstandardcon- jecture

Conjecture D and the Lefschetz standard conjecture Let H be a Weil cohomology theory. The next statement goes back to Grothendieck.

Proposition 6.1. Assume that H satisﬁes the strong Lefschetz theorem. Conjecture D(X) implies A(X,L) (all L); in the presence of the Hodge standard conjecture, A(X,L) (one L) implies D(X).

Proof. Conjecture D(X) says that the pairing

x, y ↦ x ⋅ y ∶ Ai (X)× Ad i(X) → Ad (X)≃ (4) H H− H ⟨ ⟩ ℚ is nondegenerate for all i ≤ def= dim(X). Therefore, dim Ai (X) = dim Ad i(X). As H H− the map Ld i ∶ Ai (X) → Add i(X) is injective, it is surjective, i.e., A(X,L) holds. The −2 H H− converse is equally obvious. ✷

Corollary 6.2. Conjecture D(X × X) implies B(X). 4 A ℚ A F Let be an abelian variety over al with good reduction to an abelian variety over ; the cup product of the specialization to A of any absolute Hodge class on A with a product of divisors0 of complementary codimension lies in ℚ . 0 6 STATEMENTS IMPLYING THE LEFSCHETZ STANDARD CONJECTURE 11

Proof. Indeed, A(X × X,L⊗ 1 + 1 ⊗L) implies B(X) (Kleiman 1968, Theorem 4-1).✷

Remark 6.3. If Conjecture D(X × X) holds whenever X is an abelian scheme over a complete smooth curve over , then the Hodge conjecture holds for abelian varieties. ℂ Does Conjecture C imply Conjecture B? Kleiman (1994) states eight versions of Grothendieck’s standard conjecture of Lefschetz type. He proves that six of the eight are equivalent and that a seventh is “practically equivalent” to the others, but he states that the eighth version, Conjecture C, “is, doubt- less, truly weaker”. In this subsection we examine whether Conjecture C is, in fact, equivalent to the remaining conjectures. Let be a Weil cohomology theory on the algebraic varieties over an algebraically closed fieldH . Assume that satisfies conjecture C, and let MotH(k) denote the category of motivesk defined usingH algebraic classes modulo homological equivalence as the correspondences. It is a graded pseudo-abelian rigid tensor category over equipped with a graded tensor functor !H ∶ MotH → -VecQ, where Q is the coefficientℚ field of . ℤ H Proposition 6.4. Assume that satisfies the strong Lefschetz theorem in addition to Conjecture C. If ! is conservative,H then satisfies the Lefschetz standard conjecture. H H Proof. Let ∶ (X) → (X)(1) be the Lefschetz operator defined by a hyperplane r r+2 section of X.L ByH assumptionH

L ∶ (X)(i) → (X)( − ) (5) d−2i 2i 2d−2i H H d i is an isomorphism for all 2 ≤ def= dim(X). As ! is conservative, i d H l ∶ ℎ (X)(i) →ℎ (X)( − ) (6) d−2i 2i 2d−2i d i is an isomorphism for all 2 ≤ . On applying the functor Hom(11,−) to this isomorphism, we get an isomorphismi d

L ∶ A (X) →A (X). d−2i i d−i H H Thus, Conjecture A(X,L) is true. ✷

Corollary 6.5. Assume that satisﬁes the strong Lefschetz theorem and Conjecture C. If Mot (k) is tannakian, then Hsatisﬁes Conjecture . H H B Proof. Fibre functors on tannakian categories are conservative. ✷

Proposition 6.1 shows that a Weil cohomology theory satisfying both the strong Lef- schetz theorem and Conjecture also satisﬁes Conjecture . Here we prove a stronger result. D B

Proposition 6.6. Suppose that there exists a Weil cohomology theory ℋ satisfying both the strong Lefschetz theorem and Conjecture . Then every Weil cohomology theory satisfying the strong Lefschetz theorem and ConjectureD C also satisﬁes Conjecture B. H 6 STATEMENTS IMPLYING THE LEFSCHETZ STANDARD CONJECTURE 12

Proof. Let ℋ and be Weil cohomology theories satisfying the strong Lefschetz theorem and assume thatH ℋ (resp. ) satisfies Conjecture (resp. Conjecture C). Then ℋ satisfies the Lefschetz conjectureH (6.1), in particular, ConjectureD C. Let Motnum(k) = Motℋ(k) be the category of motives defined using algebraic cycles modulo numerical equivalence as correspondences. Then Motnum is a semisimple tannakian category over (Jannsen, Deligne), and there is a quotient functor q ∶ Mot → Motnum. For each M ℚin Mot , the map End(M) → End(qM) is surjective with kernelH the radical of the ring End(MH), and this radical is nilpotent (Jannsen 1992). The conditions on ℋ imply that it satisfies Conjecture B (Proposition 6.1). This means that for each i≤d def= dim(X), there exists a morphism ℎ2 − (X)(d−i) → ℎ (X) numd i numi inducing the inverse of the map

L − ∶ ℋ (X) → ℋ2 − (X)(d − i). d i inum numd i Write for the morphism ℎ (X) → ℎ2 − (X)(d − i) in Mot (k) inducing the isomor- i d i phism H L − ∶ (X) → 2 − (X)(d − i). (7) d i i d i H H According to the last paragraph, there exists a morphism ∶ ℎ2 − (X)(d − i) → ℎ (X) d i i i such that q( ◦ ) = id (X). Now ◦ = 1 + n in End(ℎ (X)), where n is nilpotent. numi On replacing with (1ℎ − n + n2 − ⋯)◦ , we ﬁnd that ◦ = 1 in End(ℎi(X)). Hence the inverse of the map (7) is algebraic, as required. ✷

Proposition 6.7. If there exists one Weil cohomology theory satisfying the strong Lef- schetz theorem and Conjecture D, then every Weil cohomology theory satisfying Conjecture D also satisﬁes the strong Lefschetz theorem

Proof. If there exists a Weil cohomology theory satisfying the strong Lefschetz theorem and Conjecture D, then in Motnum(k),

ld−i ∶ ℎi(X) →ℎ2d−i(X)(d − i) is an isomorphism for i≤d. Let be a Weil cohomology theory satisfying Conjecture . On applying to this isomorphism,H we get an isomorphism D H d−i i 2n−r ∶ (X) → (X)(n − r). ✷ L H H

Remark 6.8. Because Motnum is Tannakian, there exists a field Q of characteristic zero and a Q-valued fibre functor !. Then ℋ ∶ X ⇝ !(X, , 0) is a Weil cohomology ⨁i i theory satisfying Conjecture D. It remains to show that ! can be chosen so that ℋ satisfies the strong Lefschetz theorem. This comes down to showing that ld−i ∶ ℎi(X) → 2d−i ℎ (X)(d − i) is an isomorphism in Motnum(k).

Remark 6.9. Every Weil cohomology theory satisfying the weak Lefschetz theorem also satisﬁes the strong Lefschetz theorem (Katz and Messing 1974, Corollaries to The- orem 1). BIBLIOGRAPHY 13

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