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Theorem 1. Suppose that the standard conjecture B holds on X and the standard conjecture of Hodge type holds on X × X. Then for any r ∈ Q>0, the above homological correspondence

γr of X is algebraic and represented by a rational algebraic n-cycle Gr on X × X; moreover, there exists a constant C > 0 independent of r, so that for any polarized endomorphism f of ∗ X (i.e., f HX ∼ qHX for some q ∈ Z>0), we have

kGr ◦ fk ≤ C deg(Gr ◦ f). (1.1)

Remark 2. (1) Serre [Ser60] proved a result involving eigenvalues of pullbacks on cohomology by polarized endomorphisms of compact Kähler manifolds. If the analog of Serre’s result holds in positive characteristic, then Weil’s Riemann hypothesis follows. Grothendieck [Gro69] and Bombieri independently proposed the so-called standard conjectures in order to solve this positive characteristic analog of Serre’s result (see also Kleiman’s survey articles [Kle68,Kle94] for details on the standard conjectures). It was Deligne [Del74] who ingeniously solved Weil’s Riemann hypothesis and also gener- alized it in a form which can be applied to many exponential sums [Del80] - the latter being of great interest in analytic number theory. Deligne’s proofs, however, are different from what envisioned by Grothendieck. As of today, the standard conjectures, and also the positive char- acteristic analog of Serre’s result, are still widely open. For example, the standard conjecture D is only known in few cases (including the codimension-1 case, Abelian varieties [Clo99]), and the standard conjecture of Hodge type is known only for surfaces and Abelian 4-folds [Anc21]. (2) The authors of this note conjectured in [HT21] the inequality (1.1) in Theorem 1 (in the more general setting of effective correspondences), whose validity will imply, among other things, an earlier conjecture by the second author (see [Tru16, Question 2]). The latter contains as a special case the positive characteristic analog of Serre’s result (and hence Weil’s Riemann hypothesis as well). We have shown in [HT21] that (1.1) indeed holds for Abelian varieties (in all dimensions and for all effective correspondences). It also has certain descent properties for generically finite surjective morphisms or even some dominant rational maps, and hence holds for instance for Kummer surfaces. It then has been argued in [HT21] that this inequality could be a simpler alternative way in solving the positive characteristic analog of Serre’s result consid- ering the difficulty of the standard conjectures. Our Theorem 1 confirms that this is indeed the case: for polarized endomorphisms, our inequality (1.1) follows from the standard conjectures. We thus wonder if the general version of the inequality (1.1) for effective correspondences is also a consequence of the standard conjectures.

2. PROOF OF THEOREM 1

Recall that X is a smooth projective variety of dimension n over an algebraically closed

field k of arbitrary characteristic and HX is a fixed ample divisor on X. We also fix a Weil cohomology theory H•(X) with a coefficient field F of characteristic zero (see [Kle94, §3]). In particular, we have a cup product ∪, Poincaré duality, the Künneth formula, the cycle class ANINEQUALITYONPOLARIZEDENDOMORPHISMS 3 map clX , the Lefschetz trace formula, the weak Lefschetz theorem, and the hard Lefschetz theorem. Examples of classical Weil cohomology theories include:

• • de Rham cohomology HdR(X(C), C) if k ⊆ C, • 1 • étale cohomology Hét(X, Qℓ) with ℓ =6 char(k) if k is arbitrary, • • crystalline cohomology Hcrys(X/W (k)) ⊗ K, where K is the field of fractions of the Witt ring W (k).

For the fixed ample divisor HX on X and for 0 ≤ i ≤ 2n − 2, we let L: Hi(X) → Hi+2(X), (2.1) α 7→ clX (HX ) ∪ α be the Lefschetz operator. By the hard Lefschetz theorem, for any 0 ≤ i ≤ n, the (n−i)-th iterate Ln−i of the Lefschetz operator L is an isomorphism ∼ Ln−i : Hi(X) −−→ H2n−i(X). However, Ln−i+1 : Hi(X) → H2n−i+2(X) may have a nontrivial kernel. Denote by P i(X) the set of elements α ∈ Hi(X), called primitive, satisfying Ln−i+1(α)=0, namely, P i(X) := Ker(Ln−i+1 : Hi(X) → H2n−i+2(X)) ⊆ Hi(X). (2.2) This gives us the following primitive decomposition (a.k.a. Lefschetz decomposition): Hi(X)= LjP i−2j(X), (2.3)

Mj≥i0 where i0 := max(i − n, 0). Definition 3 (cf. [Kle68, §1.4]). For any α ∈ Hi(X), we write

j i−2j α = L (αj), αj ∈ P (X). (2.4)

jX≥i0 Then we define an operator ∗ as follows: ∗: Hi(X) → H2n−i(X), − − (i 2j)(i 2j+1) n−i+j (2.5) α 7→ (−1) 2 L (αj).

jX≥i0

It is easy to check that ∗2 = id. The standard conjecture B(X) predicts that the above homological correspondence ∗ is algebraic (cf. [Kle68, Proposition 2.3]). For any correspondence g of X, denote by g′ its adjoint with respect to the following non- degenerate bilinear form Hi(X) × Hi(X) −→ F (2.6) (α, β) 7→ hα, βi := α ∪∗β.

1The hard Lefschetz theorem turns out to be very difficult (see [Del80, Theorem 4.1.1]). 4 FEIHUANDTUYENTRUNGTRUONG

In other words, we have g′ = ∗◦ gT ◦∗ by definition, where gT denotes the canonical transpose of g by interchanging the coordinates. For any 0 ≤ k ≤ n, let Ak(X) ⊆ H2k(X) denote the Q-vector space of cohomology classes

generated by algebraic cycles of codimension k on X under the cycle class map clX , i.e.,

k k 2k A (X) := Im(clX : Z (X)Q −→ H (X)). The standard conjecture of Hodge type predicts that when restricted to Ak(X) the bilinear form (2.6) is positive definite (see [Kle68, §3] for details).

i 2n−i Lemma 4. Let πi ∈ H (X) ⊗ H (X) be the i-th Künneth component of the diagonal class, which corresponds to the projection operator H•(X) → Hi(X) via the pullback. Then for any ∗ polarized endomorphism f of X (i.e., f HX ∼ qHX for some q ∈ Z>0), we have

′ i (πi ◦ f) ◦ (πi ◦ f) = q πi as homological correspondences.

Proof. Note that for any α ∈ Hi(X) with the above primitive decomposition (2.4),

j j ∗ ∗ i−2j L (q f αj) with f (αj) ∈ P (X)

jX≥i0 is the primitive decomposition of f ∗(α). It follows that

′ ∗ ∗ ((πi ◦ f) ◦ (πi ◦ f) ) (α)= ∗◦ (πi ◦ f)∗ ◦∗◦ (πi ◦ f) (α) ∗ = ∗◦ (πi ◦ f)∗ ◦∗◦ f (α) − − ∗ (i 2j)(i 2j+1) − ∗ 2 n i+j j = ∗◦ π2n−i ◦ f∗ (−1) L (q f αj) jX≥i0 − − (i 2j)(i 2j+1) n−i+j ∗ 2 j = ∗ (−1) f∗(HX ∪ q f αj) jX≥i0 − − (i 2j)(i 2j+1) n−i+j 2 j = ∗ (−1) f∗HX ∪ q αj jX≥i0 − − (i 2j)(i 2j+1) − n−i+j 2 i j j = ∗ (−1) q HX ∪ q αj jX≥i0 − − (i 2j)(i 2j+1) i n−i+j = ∗ (−1) 2 q L (αj)

jX≥i0 = qi ∗2 α = qiα,

∗ ∗ i 2n−i where πi and (πi)∗ = π2n−i are projections to H (X) and H (X), respectively, the third equality follows from the definition of the ∗ operator, the fifth one follows from the projection formula, and the last one follows from the fact that ∗2 = id. This yields the lemma. 

Proof of Theorem 1. Since the standard conjecture B(X) implies the standard conjecture C(X),

the algebraicity of γr follows from [HT21, Lemma 4.4]. Also, by assumption, the bilinear form ANINEQUALITYONPOLARIZEDENDOMORPHISMS 5

n (2.6) is a Weil form; see [Kle68, Theorem 3.11]. In particular, if we let ∆i ∈ Z (X × X)Q

represent πi and let fi denote the composite correspondence ∆i ◦ f, then the square root of

′ ∗ ′ ∗ • Tr((fi ◦ fi ) |H (X)) = Tr((fi ◦ fi ) |Hi(X)) ∈ Q>0

∗ gives us a norm of f |Hi(X). On the other hand, it follows from Lemma 4 that

′ ∗ i Tr((fi ◦ fi ) |Hi(X))= q bi(X),

i where bi(X) := dimF H (X) is the i-th Betti number of X. Putting together, we thus obtain that ∗ i/2 f |Hi(X) = bi(X) q . p

Now, we let g denote Gr ◦ f. By assumption, the standard conjecture D holds on X × X (see [Kle68, Corollaries 3.9, 2.5, and 2.2]) and hence the cycle class map induces an injective map n 2n .(N (X × X) ⊗Z F ֒→ H (X × X It thus follows that

kgk . k clX×X (g)k. 2n 2n i Here the right-hand side denotes a norm on H (X × X) ≃ i=0 EndF(H (X)) which is equivalent to L ∗ max g |Hi(X) . 0≤i≤2n

Note that the above equivalence part depends on the choices of norms. Also, by the definitions ∗ i ∗ of Gr and f we have that g |Hi(X) = r f |Hi(X) and n n n n deg(g)= g · Hn = deg (g)= r2kqk. X×X k k k Xk=0 Xk=0 If i =2k is even, then we have

∗ 2k ∗ 2k k g |Hi(X) = r f |H2k(X) = r b2k(X) q = b2k(X) degk(g). p p When i =2k +1 is odd, similarly, we also have

∗ 2k+1 ∗ g |Hi(X) = r f |H2k+1(X)

2k+1 2k+1 = r b2k+1(X) q p 2k k 2k+2 k+1 ≤ b2k+1(X)(r q + r q )/2 p 2k k 2k+2 k+1 ≤ b2k+1(X) max{r q ,r q } p ≤ b2k+1(X) max{degk(g), degk+1(g)}. p So overall, there is a constant C depending only on the Betti numbers bi(X) of X, the dimen- sion n of X, and norms we have chosen, but independent of f nor r, such that kgk ≤ C deg(g). We thus prove Theorem 1.  6 FEIHUANDTUYENTRUNGTRUONG

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DEPARTMENT OF MATHEMATICS, UNIVERSITY OF OSLO, NIELS HENRIK ABELS HUS, MOLTKE MOES VEI 35, 0851 OSLO,NORWAY

Email address: [email protected]

DEPARTMENT OF MATHEMATICS, UNIVERSITY OF OSLO, NIELS HENRIK ABELS HUS, MOLTKE MOES VEI 35, 0851 OSLO,NORWAY

Email address: [email protected]