A Proof of the Hodge Conjecture

Kazuhisa MAEHARA∗ Nov.14, 2006

1 Introduction

In this paper we show that the Hodge conjecture and a part of the Tate conjecture hold. Since it is difficult to find algebraic cycles in general, the strategy is to proceed by induction argument to vanish a certain subspace of a cohomology of an open affine subvariety of an affine variety which is obtained excluding a general hyperplane from a given variety.

2 In Case of Non Sigular Varieties

Let k beaﬁeldwithanalgebraicclosurek and X a smooth geometrically irreducible variety over k. There exists the canonical cycle map for =char k

r r 2r cl :CH(X) −→ Het (Xk, Q(r))

This image is included in the fixed part 2r Gk Het (Xk, Q(r)) where Gk = Gal(k/k).Tatesconjecturesaysthatifk is finitely generated r 2r Gk as a field, the image of cl generates Het (Xk, Q(r)) . Fix an isomorphism ι : Q → C. Let k = C. One has the canonical cycle map

clr :CHr(X) −→ H2r(X(C), Q(2π i)r) This image is included into

H2r(X(C), Q(2π i)r) ∩ Hr,r(X(C))

∗Associate Professor, General Education and Research Center, Tokyo Polytechnic University Received Sept. 28, 2006 2 ACADEMIC REPORTS Fac. Eng. Tokyo Polytech. Univ. Vol. 29 No.1 (2006)

Hodge conjecture says that the image of clr generates H2r(X(C), Q(2π i)r) ∩ Hr,r(X(C)). Let U be a smooth quasiprojective variety over k. The images of the canonical cycle maps are 2r Gk Het (Uk, Q(r)) for ﬁnitely generated k r 2r 2r F H (U, C) ∩ W2rH (U, Q(r)) for k = C

Let U be a smooth quasi-projective variety over k and X a smooth projective compactiﬁcation of U. One denotes by cl∗ the following cycle maps clDR, cl, clH ; 2r 2r 0 2r (a) ΓDR(HDR(U)(r)) = W0(HDR(U)(r)) ∩ F (HDR(U)(r))

2r 2r Gk 2r (b) Γ(H (U)(r)) = H (U)(r) ∩ W0(H (U)(r))) 2r 2r 0 2r (c) ΓH (Hσ (U)(r)) = W0(Hσ (U)(r) ∩ F (Hσ (U)(r)) ⊗ C)

Lemma 1 It suﬃces to prove it for a smooth aﬃne variety over k.

Proof. It is well known that it is enough to treat it in the case of dim X =2d. Choose a smooth irreducible hyperplane Y on X.ThusX −Y is a smooth aﬃne variety. One obtains the following commutative diagram:

d−1 2d CH (Y) −→ Γ∗H∗,Y(X)(d) ↓↓ d 2d CH (X) −→ Γ∗H∗ (X)(d) ↓↓ d 2d CH (X − Y) −→ Γ∗H∗ (X − Y)(d) ↓↓ 00

2d ∼ Here the vertical sequences are exact. By duality, one has H∗,Y(X)(d) = 2d−2 H∗ (Y)(d − 1). Assume conjectures hold for X − Y . Induction hypothesis for d 2d−2 CH (Y) −→ Γ∗H∗ (Y)(d − 1) implies conjectures. Take general strict normalcrossing divisors D1, ··· ,D2d on X excluding Y . Since X − Y is affine, let A denote Γ(X − Y,O), which is geometrically regular commutative kalgebra of finite type. Thus X − Y =SpecA.LetI1, ··· ,I2d be ideals of A such that V (I1)=D1, ···, V (I2d)=D1 ∩···∩D2d.ByNöther’s nor- malization lemma, one obtains algebraically independent elements x1, ··· ,x2d of A such that

(a) A is integral over k[x1, ··· ,x2d].

(b) Ii ∩ k[x1, ··· ,x2d]=(x1, ··· ,xi).

Let K = k(x1, ··· ,x2d)andL = Q(A) the total fraction ﬁeld of A.LetK ⊂ L be a ﬁnite extension and M the smallest quasi-Galois extension of L/K . Take the integral closure B of A over k[X]inM .LetG =Gal(M/K). Lemma 2 The action of G on M keeps B to be invariant. A Proof of the Hodge Conjecture 3

Proof. Since A is regular; hence normal, A coincides with the integral closure of k[x1, ··· ,x2d]inL .Leta ∈ A with the minimal polynomial hand let σ ∈ G. Then hσ = h. Take any b ∈ B , which satisfies the minimal polynomial g(b)=0 with coefficients in A .Thusgσ has its coefficients in A. Hence gσ(bσ)=0, which implies b is an integral element of M. Therefore bσ ∈ B .

Let K betheﬁeldoftheinvariantsofM by G and C the integral closure of k[x1, ··· ,x2d]. . Note that K is a radical extension of K of ﬁnite degree.

Lemma 3 (i) BG = C

(ii) Spec C → Spec k[x1, ··· , x2d] is a ﬁnite, surjective and radical morphism, which is a universal homeomorphism.

G G Proof. Since M ⊃ B ⊃ A ⊃ C ⊃ k[x1, ··· ,x2d] , one has K = M ⊃ B ⊃ G G A ⊃ C = C.SinceB is a finite k[x1, ··· ,x2d]module and k[x1, ··· ,x2d]isa Nötherian ring, every submodule of B is a finite k[x1, ··· ,x2d] module. Every element of BG is integral over C. Hence BG = C since C is integrally closed. φ :SpecC→ Spec k[x1, ··· , x2d] For every point x of Spec C one has κ(x)isa radical extension of κ(φ(x)); thus φ :SpecC→ Spec k[x1, ··· , x2d]isaradical morphism. It is clear that the mophism is finite and surjective; hence a universal homeomorphism. Let F be a suitable smooth sheaf on X. One has a trace map : trL/K : L → K , which naturally extends to a map trA/C :A→ C. If f ∈ C, one further has 1 1 a map trA[ 1 ]/C[ 1 ] : A[ ] → C[ ] and a cohomological map f f f f j 1 j 1 trA[ 1 ]/C[ 1 ] :H Spec A[ ], F → H Spec C[ ], F . f f f f

1 1 If f ∈ k[x1, ··· ,x2d], φ :SpecC[f ] → Spec k[x1, ··· , x2d, f ] is a ﬁnite, surjective and radical morphism; hence a universal homeomorphism. Thus one has an isomorphism j 1 j 1 H Spec C[ ], F → H Spec k[x1, ··· , x2d, ], F . f f

Let H =Gal(M/L).

Lemma 4 Assume that dimSpecA=n=2d≥ 4.Letf1, ··· ,fn be k-linear combinations of x1, ··· ,xn in k[x1, ··· ,xn]. Assume that the intersection locus V (f1, ··· ,fn) is void in Spec k[x1, ··· , xn], i.e., the hyperplanes intersect one n 1 point in inﬁnity. One obtains Γ∗H Spec A[ ], F =0. f1···fn Proof. If n>2, by the aﬃne vanishing theorem, one has

H2n−1 (Spec A, F)=0, H2n−2 (Spec A, F)=0. 4 ACADEMIC REPORTS Fac. Eng. Tokyo Polytech. Univ. Vol. 29 No.1 (2006)

One has a Cechˇ Spectral sequence: p,q q 1 p+q E1 = ⊕|I|=p+1H ∩i∈ISpec A[ ], F ⇒ H (Spec A − V(f1, ··· , fn), F) . fi

pq Note that E1 =0forq>nby aﬃne vanishing theorem and that p>n− 1by Cechˇ cohomology theory. One obtains

n−1,n 2n−1 E∞ =H (Spec A − V(f1, ··· , fn), F)=0,

n−2,n n−2,n n−2 2n−2 E2 = E∞ = Gr H ((Spec A, F)=0,

n−4,n+1 n,n−1 since E2 = E2 = 0. Note that there exists the following exact sequence:

Γ 2d−2 F − −→ ∗H∗ (V(fik ), )(d 1)

2d 1 2d 1 Γ∗H (Spec A[ ], F)(d) −→ Γ∗H (Spec A[ ], F)(d) −→ 0. ∗ ··· ∗ ··· fi1 fik−1 fi1 fik Thus, n 1 n 1 ⊕|I|=n−2Γ∗H ( Spec A[ ], F →⊕|I|=n−1Γ∗H ( Spec A[ ], F i∈I fi i∈I fi

and n 1 n 1 ⊕|I|=n−1Γ∗H ( Spec A[ ], F →⊕|I|=nΓ∗H ( Spec A[ ], F i∈I fi i∈I fi

n−3,n+1 n+1,n−1 n,n are surjections, respectively. Using E2 = E2 = E1 = 0, one has n−1,n n−3,n+1 n−1,n n+1,n−1 E3 =H E2 → E2 → E2 =

n−1,n n−2,n n−1,n n,n n−1,n E2 =H E1 → E1 → E1 =E∞ =0.

Hence, the homomorphism

n−2,n n−1,n E1 → E1

is a surjection. n−4,n+1 n,n−1 Applying E2 = E2 = 0, one has n−2,n n−4,n+1 n−2,n n,n−1 E3 =H E2 → E2 → E2 = A Proof of the Hodge Conjecture 5

n−2,n n−3,n n−2,n n−1,n n−2,n E2 =H E1 → E1 → E1 =E∞ =0.

Moreover, since n−2,n n 1 E2 =H ⊕|I|=n−2H (Spec A[ ], F) −→ i∈I fi

n 1 n 1 ⊕|I|=n−1H (Spec A[ ], F) →⊕|I|=nH (Spec A[ ], F) =0 i∈I fi i∈I fi and the homomorphism

n 1 n 1 ⊕|I|=n−2Γ∗H (Spec A[ ], F) →⊕|I|=n−1Γ∗H (Spec A[ ], F) i∈I fi i∈I fi is surjective and a functor Γ∗ is exact, it implies that

n 1 n 1 ⊕|I|=n−1Γ∗H (Spec A[ ], F) →⊕|I|=nΓ∗H (Spec A[ ], F) i∈I fi i∈I fi is a zero map. On the other hand,

n 1 n 1 ⊕|I|=n−1Γ∗H (Spec A[ ], F) →⊕|I|=nΓ∗H (Spec A[ ], F) i∈I fi i∈I fi is a surjection. Thus one concludes that

n 1 n 1 Γ∗H (Spec A[ ], F)=⊕|I|=nΓ∗H (Spec A[ ], F)=0. f1 ···fn i∈I fi

Theorem 5 For 0 ≤ k ≤ n the images of

d 1 2d 1 CH (Spec A[ ]) −→ Γ∗H (Spec A[ ])(d) f1 ···fk f1 ···fk generate the targets. In particular, the image of

d 2d CH (Spec A) −→ Γ∗H∗ (Spec A)(d) generates the target.

Proof. One continues to proceed by induction argument: Hn (Spec A, F) → V(f1) Hn (Spec A, F) → Hn Spec A[ 1 ], F f1 ··· 6 ACADEMIC REPORTS Fac. Eng. Tokyo Polytech. Univ. Vol. 29 No.1 (2006)

Hn Spec A[ 1 ], F → Hn Spec A[ 1 ], F → Hn Spec A[ 1 ], F V(fn) f1···fn−1 f1···fn−1 f1···fn One has the following commutative diagram, whose vertical sequences are exact:

d−1 2d−2 CH (V(fk)) −→ Γ∗H∗ (V(fk))(d − 1) ↓↓ d 1 2d 1 CH (Spec A[ ]) −→ Γ∗H (Spec A[ ])(d) f1···fk−1 ∗ f1···fk−1 ↓↓ d 1 2d 1 CH (Spec A[ ]) −→ Γ∗H (Spec A[ ])(d) f1···fk ∗ f1···fk ↓↓ 00

In Case of Singular Varieties

Uwe Jannsen proved that the Hodge conjecture and the Tate conjecture for singular varieties are deduced by the original conjectures. For the readers con- vienience we explain it. For a smooth variety X of dimension d one has the ∼ 2d−2i Poincar´e duality H2d(X, i) = H (X, d − i). There is no such duality in general for non smooth varieties. Fundamental classes induce a cycle map: cli : Zi(X) −→ H2i(X, i), which factors through the canonical cycle map that Fulton deﬁnes CHi(X) −→ H2i(X, i). The Hodge conjecture for singular varieties says that for all i ≥ 0themap

−i −i cli ⊗ Q : Zi(X) ⊗ Q −→ Γ∗H2i(X, Q)(i) = (2πi) W−2iH2i(X, Q) ∩ F H2i(X, C)

is surjective.

Theorem 6 The Hodge conjecture is true for singular varieties.

Proof. By Chow’s lemma and Hironaka’s resolution of singularities for a singular non complete variety X there exist π : X → X a projective and sujective morphism with X quasi-projective and smooth and α : X → X an open immersion with X projective and smooth forming a diagram of varieties

α X −→ X π ↓ X

∗ ∗ α π Zi(X ) ⊗ F −→ Zi(X ) ⊗ F −→ Zi(X) ⊗ F

↓ cli ↓ cli ↓ cli

∗ ∗ Γα Γπ ΓH2i(X , i) −→ ΓW0H2i(X , i) −→ ΓW0H2i(X, i) A Proof of the Hodge Conjecture 7

Since H2i(X , i) is a semi-simple object, W0H2i(X, i) is a direct factor of ∗ ∗ H2i(X , i) via π∗ ◦ α and so Γπ∗ ◦ Γα is surjective. Thus Zi(X) ⊗ F → ΓW0H2i(X, i) is surjective.

Appendix

2 Theorem 7 Let f0 :: X0 → Y0 a projective morphism, ∈ H (X0, Q(1)) the ﬁrst Chern class of an f0-ample invertible sheaf and F0 a perverse sheaf over X0 for i ≥ 0. The following map is an isomorphism

i p −i ∼ p i : H f∗F0 = H f∗F0(i)

Lemma 8 It suﬃces to prove the Hodge conjecture in case of i =2d =dimX.

Proof. By the strong Lefschetz theorem it reduces to the case i =2p>2d. Let Y be a general hyperplane section of X. By the weak Lefschetz one has an exact sequence Hi−2(Y, F) → Hi(X, F) → Hi(X − Y, F) = 0. The following commutative diagram completes the proof:

p−1 2p−2 CH (Y) −→ Γ∗H (Y)(p − 1) ↓↓ p 2p CH (X) −→ Γ∗H∗ (X)(p) ↓↓ p 2p CH (X − Y) −→ Γ∗H∗ (X − Y)(p) ↓↓ 00

Theorem 9 Let f : X → Y be an aﬃne morphism. The functor

b b Rf∗ : Dc(X, Q) → Dc(Y,Q) is right t-exact. In particular, Let k be an algebraically closed ﬁeld and F an etale sheaf on X. Hi(X, F)=0for i>dim X.

References

[BBD] Beilinson A.A., Bernstein I.N., Deligne P., Faisceaux pervers, Anal- yse et Topologie sur les espaces singuliers(I), Conférence de Luminy, juillet 1981, Astérisque 100(1982). [Ber] Berthelot, P., Altérations de variétés algébraiques d’après A.J. DE JONG,Séminaire Bourbaki 48, n◦ 815, pp.273-311(1997). [Bour] Bourbaki N., El´´ ements de mathématique Algèbre commutative Ch.5,Ch.6, Fasc.XXX Hermann 1964, pp.207. 8 ACADEMIC REPORTS Fac. Eng. Tokyo Polytech. Univ. Vol. 29 No.1 (2006)

[Del] Deligne P. La conjecture de Weil I, II, Publ.Math. I.H.E.S., 43(1974), 52(1981), 273-307, 313-428.

[Fj] Fujita, T., On K¨ahler ﬁbre spaces over curves., J. Math.Soc. Japan 30, pp. 779-794(1978).

[Mot] Jannsen U., Kleiman S., Serre J-P. ed. Motives, Proceedings of Sym- posia in Pure Mathematics, Vol.55 Part1,2, American Mathematical Society, Providence, Rhode Island, pp.747, pp.676, 1994. [Grif] Griffiths P.A., A theorem concerning the differential equations satis- fied by normal functions associated to algebraic cycles, Amer.J.Math. 101(1979), 94-131. [Gr] Grothendieck A., Standard conjectures on algebraic cycles, Algebraic Geometry-Bombay Colloquium, 1968, Oxford, 1969, 193-199. [SGA] Grothendieck A. et al., Sémiaire de géométrie algébrique du Bois- Marie, SGA1, SGA4 I,II,III, SGA41/2, SGA5, SGA7 I,II, Lec- ture Notes in Mat., vols. 224,269-270-305,569,589,288-340, Springer- Verlag, New York, 1971-1977. [Dix] Grothendieck A., Giraud J., Kleiman S., Raynaud M., Tate J., Dix exposés sur la cohomologie des schémas, Advanced studies in pure math., vol. 3, North-Holland, Amsterdam, Paris, 1968, pp.386. [Har] Hartshorne R., Residues and Duality, Lecture Notes in math., vol. 20, Springer-Verlag, New York, 1966. [Hod] Hodge W.V.D., The topological invariants of algebraic varieties, Proc. Int. Congr. Math. 182-192(1950). [Iit] Iitaka S., Introduction to birational geometry., Graduate Textbook in Mathematics, Springer-Verlag, p. 357 (1976). [Ish] Ishida M., Torus embeddings and dualizing complexes, Tohoku Math. J. 32(1980), 111-146. [Jan] Jannsen U., Mixed Motives and Algebraic K-Theory, Springer Lec- ture Notes, 1400(1990). [Lm] Laumon, G., Moret-Bailly, L., Champs algébraiques.,Universitéde Paris-sud Mathématiques, p. 94 (1992). [Kat] Kato, K., Logarithmic structures of Fontaine-Illusie., Algebraic Analysis, Geometry and Number Theory, The Johns-Hopkins Univ. Press, Baltimore, MD, pp. 191-224(1989). [Katz] Deligne P., Katz N., Groupes de Monodromie en Géométrie Algébrieque, SGA7II, Lecture Notes in Mathematics 340(1973). A Proof of the Hodge Conjecture 9

[Kaw] Kawamata, Y., Minimal models and the Kodaira dimension of al- geraic ﬁbre spaces., J. Reine Angew. Math. 363, pp. 1-46 (1985).

[KKMS] Kemf, G., Knudsen, F., Mumford, D., Saint-Donat, B., Troidal embeddings I, Springer Lecture Notes, 339(1973).

[Km1] Maehara, K., Diophantine problems of algebraic varieties and Hodge theory in International Symposium Holomorphic Mappings, Diophantine Geometry and related Topics in Honor of Professor Shoshichi Kobayashi on his 60th birthday. R.I.M.S., Kyoto Univer- sity October 26-30, Organizer: Junjiro Noguchi(T.I.T), pp. 167-187 (1992). [Km2] Maehara, K., Algebraic champs and Kummer coverings. A.C.Rep. T.I.P. Vol.18, No.1, pp.1-9(1995). [Vw] Viehweg, E. Quasi-projective Moduli for Polarized Manifolds. Ergeb- nisse der Mathematik und ihrer Grenzgebiete, 3.Folge.Band 30, p. 320 (1991). [Sw] Schmid, W., Variation of Hodge structure: the singularities of the period mapping., Inventiones math., 22., pp. 211-319(1973). [Sh] Shioda T., What is known about the Hodge conjectures?, Advanced Studies in Pure Mathematics 1, 1983 [Algebraic Varieties and Analytic Varieties], 55-68. [Ta] Tate J., Algebraic cycles and poles of zeta functions, Arithmetic Algebraic Geometry (O.F.G. Schilling, ed), Harper and Row, New York, 1965, 93-110. [Zuk] Zuker S., The Hodge conjecture for cubic fourfolds, Compositio Mathematica, Vol. 34, Fasc. 2(1977), 199-209.