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6 Conclusion 27

1 Introduction

The Hodge conjecture attempts to build a bridge between complex differential geometry and algebraic geometry on K¨ahler manifolds. More precisely, it wants to show a connection between topology (Betti cohomology classes, i.e. cohomology with rational coefficients), complex geometry (Hodge decomposi- tion theorem for the De Rham cohomology in terms of Dolbeault cohomologies) and algebraic geometry (the algebraic projective subvarieties of a complex projective algebraic variety).

The conjecture was formulated by W. Hodge during the 1930s, when he studied the De Rham cohomology for complex algebraic varieties. Hodge presented it during the 1950 International Congress of Mathematicians, held in Cambridge, Massachusetts, ([Ho52]). Before that date it had received little attention by the mathematical community. The current statement reads as follows (cf. [De06]):

Conjecture 1 (Hodge). Let X be a projective non-singular (i.e. without isolated points) algebraic variety over C and, for any k 0,...,n : dim X the Hodge class of degree 2k on X is defined as “ “ Hdgk X : H2k X, Q Hk,k X, C . Then, any Hodge class on X is a rational linear combination of p q “ p qX p q classes of algebraic cycles.

In Hodge’s original conjecture the coefficients were not rational but integer. This version of the conjecture was proven false by Atiyah and Hirzenbruch [AtHi62] with a first counterexample. Totaro ([To97]) reinterpreted their result in the framework of cobordism and constructed many others. Hodge’s conjecture is false in the category of K¨ahler manifolds, as Grothendieck ([Gr69]) and Zucker ([Zu77]) have recognized. For example, it is possible to construct a K¨ahler manifold, namely a 2-dimensional complex torus T 2, whose only analytic submanifolds are isolated points and the torus itself. Hence, the Hodge conjecture cannot hold for Hdg1 T 2 . Voisin ([Vo02]) proved that even more relaxed versions of p q the Hodge conjecture for K¨ahler manifolds, allowing instead of fundamental classes for Chern classes of vector bundles or Chern classes of coherent sheaves on X cannot hold true, by proving that the Chern classes of coherent sheaves give strictly more Hodge classes than the Chern classes of vector bundles, and that the Chern classes of coherent sheaves are insufficient to generate all the Hodge classes.

In a nutshell, Hodge’s conjecture postulates a characterization cohomology classes generated over Q (i.e. algebraic classes) by classes of algebraic subvarieties of a given dimension of a complex projective manifold X, more precisely by rational cohomology classes of degree 2k which admit de Rham repre-

2 sentatives which are closed forms of type k, k for the complex structure on X (i.e. Hodge classes). p q Note that the integration over a complex submanifold of dimension n k annihilates forms of type p, q ´ p q with p, q n k,n k . p q ‰ p ´ ´ q The first result on the Hodge conjecture is due to Lefschetz who proved it for 2-Hodge classes with integer coefficients in [Lef24]. Combined with the Hard Lefschetz theorem, (see [Vo10], page 148), formulated by Lefschetz in 1924 and proved by Hodge in 1941, it implies that the Hodge conjecture is true for Hodge classes of degree 2n 2, proving the Hodge conjecture when dim X 3. Cattani, ´ ď Deligne and Kaplan provide positive evidence for the Hodge conjecture in [CDK95], showing roughly that Hodge classes behave in a family as if they were algebraic.

For a thorough treatment of Hodge theory and complex algebraic geometry see [Vo10]. For the official statement of the Hodge conjecture for the Clay Mathematics Institute see [De06]. For the current state of the research and the possible generalizations of the conjecture see [Vo11, Vo16]. For a presentation of many specific known cases of the Hodge conjecture see [Lew99].

This paper is structured as follows. In Section 2 we review the definitions of projective algebraic varieties, Hodge classes, Dirac bundles, and Dirac operators, showing that the Clifford operator on an oriented K¨ahler manifold is the Dirac operator for the antiholomorphic bundle, and the Hodge Laplacian is the Dirac Laplacian. In Section 3 we study the Green function for the Dirac Laplacian on a compact manifold with boundary, and prove a representation theorem expressing the values of the sections of the Dirac bundle over the interior in terms of the values on the boundary. This result holds true for the Hodge Laplacian over a compact oriented K¨ahler manifold. In Section 4 we first review the Nash-Moser generalized inverse function theorem, applying it to our geometric set-up by proving the existence of complex submanifolds of a compact oriented variety satisfying globally a certain partial differential equation. In Section 5 we prove the existence of complex submanifolds of a compact oriented variety whose fundamental classes span the Hodge classes. This is equivalent to the Hodge conjecture for non singular projective algebraic varieties. As expected, the presented proof cannot be extended to the category of K¨ahler manifolds.

2 Definitions

We first review some standard facts about the complex projective space, establishing the necessary notation.

3 Definition 1. Let N N1. The complex projective space is the quotient space P

CP N : CN`1 0 (1) “ zt u {„ ` ˘ for the equivalence relation in CN`1 0 , defined as „ zt u

a b : λ C : a λb, (2) „ ôD P “ for a,b CN`1 0 . The quotient map P zt u

q :CN`1 CP N ÝÑ (3) a q a : a ÞÝÑ p q “r s

N`1 induces an holomorphic atlas Ui, Φi i 0,...,N on C given by tp qu “

N Φi :Ui C ÝÑ (4) a1 ai´1 ai`1 aN a Φi a : ,..., , ,..., r s ÞÝÑ pr sq “ a a a a ˆ i i i i ˙

N`1 N N for the open set Ui : a C ai 0 . Any a C is mapped to a point in CP identified by “ r sP P | ‰ P its homogeneous coordinates (

´1 Φ a a1,...,ai 1, 1,ai 1,...,aN . (5) i p q“r ´ ` s

´1 For any i, j 0 ...,N the change of coordinate maps Φ Φj : C C is biholomorphic and the “ i ˝ Ñ complex projective space has thus the structure of a complex manifold without boundary.

Proposition 2.1. A compact complex manifold X of complex dimension N has a finite atlas Vk, Ψk k 0,...,K p q “ N such that for every k the set V k is compact and every chart Ψk : Vk Ψk Vk C has a continuous Ñ p qĂ N N extension Ψk : V k Ψk V k C with image in compact subset of C . Ñ p qĂ

Proof. It suffices to refine any finite atlas Ui, Φi i 0,...,N , which exists because X is compact. Every p q “ Ui can be represented as the union of open subsets of Ui, such that their closure is still contained in Ui:

Ui V, (6) “ V open VďĂUi ViĂUi

Those V s form an open cover of X. Since X is compact, it must exist a finite subcover Vk k 0,...,K t u “

4 for a K N0. The closure Vk is compact and is the domain of the continuous extension Ψk of the well P defined Ψk : Φi V , for Vk Ui. “ | k Ă

Definition 2. A complex/real analytic/differentiable submanifold Y of complex/real/real di- mension n of a complex/real analytic/differentiable manifold X of complex/real/real dimension N is a subset Y X such that for the atlas Uι, Φι ι I of X there exist analytic/real analytic/differentiable Ă p q P N N N N N N homeomorphisms ϕι : C R R ã C R R ι I such that p { { Ñ { { q P

n ˆN´n n ˆN´n n ˆN´n ϕι Φι Uι Y C 0 R 0 R 0 (7) p p X qq Ă ˆt u { ˆt u { ˆt u for all ι I. The compatibility condition reads P

´1 ´1 ´1 ´1 Φ ϕ Φ Φ ϕ Φ , (8) ι ˝ ι | ι˝ϕιpUιXUκq “ κ ˝ κ | κ˝ϕκpUκXUιq for all ι, κ I. The subset Y of X is a complex/real analytic/differentiable manifold complex/real/real P N N N n n n dimension n with atlas Uι Y, Π ϕι Φι ι I , where Π : C R R C R R denotes the p X ˝ ˝ q P { { Ñ { { projection onto the first n dimensions of CN RN RN . { {

It is possible to define submanifolds of a manifold by specifying appropriate change of coordinate maps

Proposition 2.2. Let X be a complex/real analytic/differentiable manifold of complex/real/real di- mension N with atlas Uι, Φι . The analytic/real analytic/differentiable local homeomorphisms p q

N N N N N N ϕι : C R R ã C R R ι I (9) p { { Ñ { { q P define a analytic/real analytic/differentiable submanifold Y of X of complex/real/real dimension n by

´1 ´1 Y : Φ ϕ Vι (10) “ ι ˝ ι p q ιPI ď

n ˆN´n n ˆN´n n ˆN´n for Vι : ϕι Φι Uι C 0 R 0 R 0 “ p p qq X ˆt u { ˆt u { ˆt u if and only if for all ι, κ` I ˘ P ´1 ´1 ´1 ´1 Φ ϕ V V Φ ϕ V V (11) ι ˝ ι | ιX κ “ κ ˝ κ | κX ι

Proof. It suffices to prove that the compatibility condition is satisfied.

Following the clear and concise exposition of chapter 1 in [Pe95] we have

5 Definition 3. If Y is a n-dimensional complex submanifold of the N dimensional complex manifold X,

then the Jacobian of the defining functions ϕι in (7) is constantly equal to n for all charts. If we drop the condition about the Jacobian, Y is termed analytic subset of X, which is called irreducible if it is not the union of non-empty smaller analytic subsets. An irreducible analytic subset is also called an analytic subvariety and the terms smooth subvariety and non-singular subvariety mean the same as submanifold.

Definition 4. An affine algebraic set is the zero set of a collection of polynomials. An affine variety is an irreducible affine algebraic set, i.e. an affine algebraic set which cannot be written as the union of two proper algebraic subsets. A projective algebraic set is an affine algebraic set, which is a N subset of the complex projective space CP for some N N1. A projective algebraic variety is an P irreducible projective algebraic set. If it is an complex submanifold of the complex projective space, it is termed projective manifold.

Remark 2.1. On the complex projective space we consider homogeneous polynomials of degree d for N any d N0. The evaluation of a polynomial is not well defined on CP , but, if it is homogeneous, its P zero set is.

Theorem 2.3 (Chow). Any analytic subvariety of the complex projective space is a projective variety.

Proof. See [Mu76].

Remark 2.2. By Chow’s theorem on a complex projective variety, algebraic cycles are the same as closed complex submanifolds.

Corollary 2.4. Any complex submanifold of a projective manifold is a projective (sub)manifold.

Definition 5. (Dirac Bundle) The quadruple V, , , ∇,γ , where p x¨ ¨y q

(1) V is a complex (real) vector bundle over the Riemannian manifold X,g with Hermitian (Rie- p q mannian) structure , , x¨ ¨y (2) ∇ : C8 X, V C8 X,T ˚X V is a connection on X, p q Ñ p b q (3) γ : Cl X,g Hom V is a real algebra bundle homomorphism from the Clifford bundle over X p q Ñ p q to the real bundle of complex (real) endomorphisms of V , i.e. V is a bundle of Clifford modules,

is said to be a Dirac bundle, if the following conditions are satisfied:

(4) γ v ˚ γ v , v TX i.e. the Clifford multiplication by tangent vectors is fiberwise skew- p q “ ´ p q @ P adjoint with respect to the Hermitian (Riemannian) structure , . x¨ ¨y

6 (5) ∇ , 0 i.e. the connection is Leibnizian (Riemannian). In other words it satisfies the product x¨ ¨y “ rule: d ϕ, ψ ∇ϕ, ψ ϕ, ∇ψ , ϕ, ψ C8 X, V . x y“x y`x y @ P p q

(6) ∇γ 0 i.e. the connection is a module derivation. In other words it satisfies the product rule: “

∇ γ w ϕ γ ∇gw ϕ γ w ∇ϕ, ϕ, ψ C8 X, V , p p q q“ p q ` p q @ P p q w C8 X, Cl X,g . @ P p p qq

The Dirac operator Q : C8 X, V C8 X, V is defined by p q Ñ p q

∇ C8 X, V C8 X,T ˚X V p q ÝÝÝÝÑ p b q Q:“γ˝p7b1q˝∇ 7b1

8 § γ 8 § C §X, V C X,TX§ V pđ q ÐÝÝÝÝ p đ b q and its square P : Q2 : C8 X, V C8 X, V is called the Dirac Laplacian. “ p q Ñ p q Definition 6. A K¨ahler manifold is a Riemannian manifold X,g of even dimension n such that p q there exists an almost complex structure J on TX, that is Jx : TxX TxX, for all x X, real linear Ñ P with J 2 1, for which g Ju,Jv g u, v and J is preserved by the parallel transport induced by “ ´ p q “ p q the Levi-Civita connection ∇g.

Remark 2.3. The complex projective space carries a (K¨ahler) metric, called the Fubini–Study metric. All complex submanifolds of CP N are examples of K¨ahler manifolds

A generic complex manifold of a K¨ahler manifold is not K¨ahler (see f.i. [KoSt82] for a counterexample), so we do not have a result analogous to Corollary 2.4. Nevertheless, by Definition 6, we directly obtain following result.

Proposition 2.5. A complex submanifold Y of a K¨ahler manifold X is K¨ahler if the tangential bundle T Y is invariant under the almost complex structure J on X.

Proposition 2.6. (Antiholomorphic Bundle as a Dirac Bundle). Let X,g,J be a K¨ahler p q manifold of even dimension n with Riemannian metric g and almost complex structure J Hom TX P p q satisfying J 2 1. The antiholomorphic bundle can be seen as a Dirac bundle V, , , , ∇ after the “´ p x ¨ ¨y q following choices:

• V : Λ T 0,1X ˚: antiholomorphic bundle over X. “ p q

0,1 ˚ • , : gΛpT Xq . x¨ ¨y “

7 0 1 ΛpT , Xq˚ • ∇ : ∇g . “ • By means of interior and exterior multiplication, we can define

TX TX1,0 TX0,1 Hom V γ : “ ‘ ÝÑ p q (12) v v1,0 v0,1 γ v : ?2 ext v1,0 int v0,1 . “ ‘ ÞÝÑ p q “ p p q´ p qq

Since γ2 v g v, v 1, by the universal property, the map γ extends uniquely to a real algebra p q“´ p q bundle endomorphism γ : Cl X,g Hom V . p q ÝÑ p q

The Dirac operator Q in the case of antiholomorphic bundles over K¨ahler manifolds X, g, Ω,J is p q ˚ the Clifford operator ?2 , while the Dirac Laplacian P : Q2 is the Hodge Laplacian pB ` B q “ ˚ ˚ ∆¯ : 2 . B “ pBB ` B Bq The cohomology group of X with complex coefficients lie in degrees 0 through 2n and there is a decomposition Hk X, C Hp,q X, C , (13) p q“ p`q“k p q à where Hp,q X, C is the subgroup of cohomology classes, which are represented by harmonic forms of p q type p, q . p q Definition 7 (Hodge classes). Let X be a complex manifold of complex dimension n. For k 0,...,n “ the Hodge class of degree 2k on X is defined as

Hdgk X : H2k X, Q Hk,k X, C . (14) p q “ p qX p q

The Hodge conjecture can by Remark 2.2 be restated as

Conjecture 2 (Hodge). On a non-singular complex projective manifold X any Hodge class is a rational linear combination of the cohomology classes of complex subvarieties of X.

3 Green Function for the Hodge Laplacian

Definition 8. A Green function for the Dirac Laplacian P on the Dirac bundle V, , , , ∇ under the p x ¨ ¨y q Dirichlet boundary condition is given by the smooth section

G : X X ∆ V b V ˚, (15) ˆ z Ñ

8 locally integrable in X X, which satisfies in a weak sense, the following boundary problem: ˆ

PyG x, y δ y x 1V p q“ p ´ q y (16) $ G x, y 0, for x X y , & p q“ PB zt u % for all x y X, where ∆ : x, y X X x y , and V b V ˚ is the fibre bundle over X X such ‰ P “ tp qP ˆ | “ u ˆ that the fibre over x, y is given by Hom Vx Hom Vy . p q p qb p q In other words, we have

G x, y ψ0, P ϕ dvoly X ψ0, ϕ x x, (17) x p q y P “x p qy żX 8 for all x X, ψ0 Vx and all sections ϕ C X, V satisfying ϕ X 0, the Dirichlet boundary P P P p q |B “ condition .

Proposition 3.1. Let V, , , , ∇ be a Dirac bundle over the compact manifold X. Then, the Dirac p x ¨ ¨y q Laplacian P has a Green function under the Dirichlet boundary condition.

Proof. The proof formally follows the steps of the proof for the Laplace Beltrami operator as in chapter 4 of [Au82]. See [Ra11] for a proof for the Atyiah-Singer operator under the chiral bag boundary condition, which can be easily modified for the Dirac Laplacian under the Dirichlet boundary condition.

Theorem 3.2. Let V, , , , ∇ be a Dirac bundle over the compact manifold X with non-vanishing p x ¨ ¨y q boundary X , Q the Dirac operator and P the Dirac Laplacian. Then, any section ϕ C8 X, V B ‰ H P p q satisfying P ϕ 0 can be written in terms of its values on the boundary as “

5x ϕ x γ ν QyG x, y ψ y, , ϕ y y dvoly X , (18) p q“´ x p q p q p ¨q p qy PB „żBX  where G is Green function of P under the Dirichlet boundary condition, ψ ψ y, any ψ C8 X, V “ p ¨q P p q ˚ such that ψ x, vx vx for all vx Vx, and : V V the bundle isomorphism induced by the p q “ P 5 Ñ Hermitian (Riemannian) structure , . The vector field ν denotes the inward pointing unit normal on x¨ ¨y X. B

Proof. By definition of Green function we have for any ψ, ϕ C8 X, V P p q

PyG x, y ψ y , ϕ y dvoly X δ y x ψ y , ϕ y dvoly X ψ x , ϕ x x. (19) x p q p q p qy P “ x p ´ q p q p qy P “x p q p qy żX żX

9 By partial integration the l.h.s. of (19) becomes

PyG x, y ψ y , ϕ y dvoly X x p q p q p qy P “ żX QyG x, y ψ y ,Qyϕ y dvoly X γ ν QyG x, y ψ y , ϕ y dvoly X “ x p q p q p qy P ´ x p q p q p q p qy PB “ żX żBX (20) G x, y ψ y , Pyϕ y dvoly X γ ν G x, y ψ y ,Qyϕ y dvoly X “ x p q p q p qy P ´ x p q p q p q p qy PB ` żX żBX “0 “0 l jh n l jh n γ ν QyG x, y ψ y , ϕ y dvoly X . ´ x p q p q p q p qy PB żBX By comparing (19) with (20) we obtain

ψ x , ϕ x x γ ν QyG x, y ψ y , ϕ y dvoly X , (21) x p q p qy “´ x p q p q p q p qy PB żBX which is equivalent to (18).

Theorem 3.2 can be seen as a generalization of the mean value property for harmonic functions.

The restriction of a Dirac bundle to a 1-codimensional submanifold is again a Dirac bundle, as following theorem (cf. [Gi93] and [B¨a96]) shows.

Theorem 3.3. Let V, h , i , ∇,γ be a Dirac bundle over the Riemannian manifold M,g and let p ¨ ¨ q p q N M be a one codimensional submanifold with normal vector filed ν. Then N,g N inherits a Dirac Ă p | q bundle structure by restriction. We mean by this that the bundle V N , the connection ∇ 8 , the | |C pN,V |N q real algebra bundle homomorphism γN : γ ν γ , and the Hermitian (Riemannian) structure “´ p q |ClpN,g|N q h , i N satisfy the defining properties (iv)-(vi). The quadruple V N , h , i N , ∇ 8 ,γN is called ¨ ¨ | p | ¨ ¨ | |C pN,V |N q q the Dirac bundle structure induced on N by the Dirac bundle V, h , i , ∇,γ on M. p ¨ ¨ q

4 Nash-Moser Generalized Inverse Function Theorem

The generalization of the inverse function and implicit function theorems of calculus and the associated equation solution theorems have been pioneered by Nash and Moser, who applied this technique to prove the Riemannian manifold embedding theorem ([Na56]) and to solve small divisors problems [Mo61, Mo61, Mo66]. Later, the technique was improved by H¨ormander ([H¨o76]) and Zehnder ([Ze76]).

Definition 9. The family Xs s 0 is a decreasing family of Banach spaces if and only if Xs, s is p q ě p }¨} q

10 a Banach space for all s 0, and for all 0 s t ě ď ď

x s x t for all x Xt. (22) } } ď} } P

We introduce the notation X : s 0Xs 8 “X ě

Definition 10. Let Xs s 0 and Ys s 0 be two families of decreasing Banach spaces. The map p q ě p q ě Φ: Xs Ys satisfies Ñ

• (A1): There exists a bounded open neighbourhood U of u0 Xs0 for some s0 0, such that for P ě all u U X the map Φ is twice Fr´echet-differentiable in u and fulfills the tame estimate P X 8

2 D Φ u . v1, v2 s C v1 s r v2 s0 v1 s0 v2 s r v1 s0 v2 s0 1 u u0 s t , (23) } p q p q} ď r} } ` } } `} } } } ` `} } } } p `} ´ } ` qs

for all s 0, all v1, v2 X and some fixed r, t 0. The constant C 0 is bounded for s ě P 8 ě ą bounded.

• (A2): There exists a bounded open neighbourhood U of u0 Xs0 for some s0 0, such that for P ě all u U X there exits a linear map Ψ : Y X such that DΦ u Ψ u 1 and fulfills the P X 8 8 Ñ 8 p q p q“ tame estimate

Ψ u .v s C v s p v s0 u u0 s q , (24) } p q } ď r} } ` `} } } ´ } ` s for all s 0, all v X and some fixed p, q 0. The constant C 0 is bounded for s bounded. ě P 8 ě ą

Definition 11. The decreasing family of Banach spaces Xs s 0 satisfies the smoothing hypothesis if p q ě there exists a family Sθ θ 1 of operators Sθ : X0 X such that p q ě Ñ 8

pβ´αq` Sθ u β Cθ u α α, β 0 } p q} ď } } p ě q β´α Sθ u u β Cθ u α α β 0 (25) } p q´ } ď } } p ą ě q d β´α Sθ u Cθ u α α, β 0 , dθ p q β ď } } p ě q › › › › where α : max a, 0 . The constants› › in the inequalities are uniform with respect to α, β whenα, β ` “ t u belong to some bounded interval.

Theorem 4.1 (Nash-Moser). Let Xs s 0 and Ys s 0 be two families of decreasing Banach spaces p q ě p q ě each satisfying the smoothing hypothesis, and Φ: Xs Ys satisfying assumptions A1 and A2 . Let Ñ p q p q s s0 max r, t max p, q . Then: ě ` t u` t u

11 (i) There exists a constant ε 0, 1 such that, if f Ys r 1 with Ps s P ` `

f Φ u0 s r 1 ε (26) } ´ p q} ` ` ď

the equation Φ u f (27) p q“

has a solution u Xs in the sense that there exists a sequence un n 0 X such that for n P p q ě Ă 8 Ñ8

un u in Xs and Φ un f in Ys p (28) Ñ p q Ñ `

1 (ii) If there exists s s such that f Ys1 r 1, then the solution constructed u Xs1 . ą P ` ` P

Proof. See [Be06] and [Se16].

Definition 12. For any s R the Sobolev space of complex valued functions over the Euclidean P space is defined as s W s Rn, CN : u 1 x 2 2 u x L2 Rn, CN , (29) p q “t | p ` | | q p qP p qu where denotes the Fourier transform, and carries the scalarp product and norm ¨

s s 2 p 2 2 2 2 n N u, v s : 1 x u x , 1 x v x R , C p q “ ` | | q p q p ` | | q p q L p q (30) u s : ` u,u s. ˘ } } “ p q p p a Let s R. If V is a vector bundle over the compact manifold X Sobolev space of sections of V over P X is denoted by W s X, V and defined by local trivializations and a partition of unit of X. p q

s n N Lemma 4.2. For any s 0 the Sobolev space W R , C , , s is a Hilbert space and a Banach ě p p q p¨ ¨q q space. There exists a constant Cs 0 and a s0 with 0 s0 s such that ą ď ă

uv s Cs u s v s0 u s0 v s . (31) } } ď p} } } } `} } } } q

Proof. We just prove inequality (31), because the remaining results are standard in functional analysis. n We assume first that s is a non negative integer. For any s0 0,...,s 1 we have for any α N such “ ´ P that α s | |ď

α α α uv α´βu βv » fi α´βu βv , (32) B p q“ β pB qpB q“ ` β pB qpB q βďα ˆ ˙ βďα βďα ˆ ˙ ÿ —|βÿ|ďs0 |βÿ|ąs0ffi — ffi – fl 12 from which (31) follows. The general case for a real s 0 is proved by norm interpolation (cf. [Tr77]). ě

We can now prove a technical Lemma which will be essential in the proof the of the Hodge conjecture in the next section, showing the existence of two submanifolds of a complex projective of a non-singular compact oriented complex projective manifold X without boundary satisfying a certain PDE.

The generic set up is given by the atlas Ui, Φi i 0,...K for X as in Proposition 2.1 and two differen- p q “ tiable submanifolds Y1 Y0 X of real codimension 1 and 2 as in Definition 2, and two differentiable Ă Ă submanifolds with boundary B0,1 X, such that B0,1 Y0,1 given by Ă B “

Ui X, Π2n ϕi Φi i 0,...,K : Atlas for X p X ˝ ˝ q “

Ui Y0, Π2n 1 ϕi Φi i 0,...,K : Atlas for Y0 p X ´ ˝ ˝ q “

Ui Y1, Π2n 2 ϕi Φi i 0,...,K : Atlas for Y1 (33) p X ´ ˝ ˝ q “ ` Ui B0, Π ϕi Φi i 0,...,K : Atlas for B0 p X 2n ˝ ˝ q “ ` Ui B1, Π ϕi Φi i 0,...,K : Atlas for B1, p X 2n´1 ˝ ˝ q “ where

2n k ` 2n k Πk : R R Π : R R 0, Ñ k Ñ ˆr `8r (34) a1,...,a2n a1,...,ak a1,...,a2n a1,...,ak, 1 ak 1 p q ÞÑ p q p q ÞÑ p r0,`8rp ` qq 2n k k 2n denote the projections of R onto the subspace R and the half-space R 0, , and ϕi : R ã ˆr `8r Ñ 2n R are local diffeomorphisms for all i. In order for Y0,1 and B0,1 to be well defined we need ϕi i 0,...,N p q “ the assumptions of Proposition 2.2, the compatibility conditions.

Lemma 4.3. . Let X be a non-singular compact oriented complex projective manifold without boundary and n be the complex dimension of X. For any ϕ ϕ0,...,ϕK let Pt u

ω BB0 BB1 BB0 BB1 X BB0 X BB0 Ξ ϕ : γ ν Q ζ γ ν Q ζ ω µ B1 . (35) p q “ p q p q ´ B

Then,

8 (i) For any chart U, Φ in the atlas Ui, Φi i 0,...,K there exist a C differential form-valued function p q p q “ F ω F ω γ, Γ for γ R2n and Γ R2nˆ2n such that “ p q P P

Ξω ϕ x F ω ϕ x , Dϕ x for all x Φ U . (36) p qp q“ p p q p qq P p q

13 Moreover, F ω is an affine functional of ω.

(ii) Let F ω,U : ϕ :Φ U ϕ Φ U diffeomorphism DF ω ϕ, Dϕ is injective (37) “t p q Ñ p p qq | p q u

2n be a set of local diffeomorphisms in R . Then, there exists local diffeomorphisms ϕi i 0,...,N p q “ such that for any

n´1,n´1 ω,Ui ω ω Ω X, C ϕi F for all i 0,...,K , (38) Pt P p q| P “ u

the local diffeomorphisms ϕi i 0,...,N define an oriented 0-codimensional submanifold of X, B0 p q “ with boundary B0, and an oriented 0-codimensional submanifold of B0, B1 with boundary B1, B B B such that the equality

BB0 BB1 BB0 BB1 X BB0 X BB0 γ ν Q ζ γ ν Q ζ ω µ B1 , (39) p q p q “ B

holds true, where we have:

• The antiholomorphic bundle on X is a Dirac bundle by Proposition 2.6 and is denoted by V, γ, , , ∇ p x¨ ¨y q with corresponding Dirac operator, the Clifford operator QX .

BB0 • The operator Q is the Dirac operator on B0 corresponding to the Dirac bundle structure B induced by Theorem 3.3 by the Dirac bundle structure on X, and νBB0 the unit normal vector field

to B0 in X. B

BB1 • The operator Q is the Dirac operator on B1 corresponding to the Dirac bundle structure B BB1 induced by Theorem 3.3 by the Dirac bundle structure on B0, and ν the unit normal vector B field to B1 in B0, B B • the n 1,n 1 -differential form µ B1 is the volume form on B1. p ´ ´ q B B

• the Green functions for the Hodge Laplacians on X B0 and B0, and, respectively B0 B1 and B1 z z as in Proposition 3.1 are denoted by GXzB0 and GB0 , and, respectively, by GB0zB1 and GB1 .

14 • The operators

BB0 XzB0 B0 ζ y : G y, x G y, x dvolX p q “ ˜ XzB0 p q` B0 p q¸ ż ż (40)

BB1 B0zB1 B1 ζ y : G y, x G y, x dvol B0 p q “ p q` p q B ˜żBB0zB1 żB1 ¸

0,1 ˚ 0,1 ˚ are bundle homomorphisms on Λ TX B0 and Λ TB B1 , respectively. pp q |B q pp 0 q |B q

Moreover, B1 is a complex projective submanifold of the projective manifold X. B

Proof. ω (i): Any ϕ in ϕi i 0,...,K is contained in the defining expression for Ξ ϕ by means of the vector p q “ p q field tangential to the coordinate lines B ,..., B ,..., B ,..., B and the corresponding o.n. system Bz1 Bzn Bz1 Bzn

defined by means of the Gram-Schmidt orthogonalization procedure e1,...,en, e1,..., en. More exactly:

• The o.n. frames are C8 functions of ϕ, Dϕ : p q

ǫ1,...,ǫ2n : e1,...,en, e1,..., en for TX U t u “t u |

ǫ1,...,ǫ2n 1 : e1,...,en, e1,..., en 1 for T B0 U (41) t ´ u “t ´ u B |

ǫ1,...,ǫ2n 2 : e1,...,en 1, e1,..., en 1 for T B1 U . t ´ u “t ´ ´ u B |

• The real algebra bundle homomorphisms are C8 functionals of ϕ, Dϕ : p q

γX v ?2 ext v1,0 int v0,1 for v v1,0 v0,1 TX p q“ p p q´ p qq “ ‘ P BB0 X BB0 X γ v γ ν γ v for v T B0 (42) p q“´ p q p q P B BB1 BB0 BB1 BB0 γ v γ ν γ v for v T B1. p q“´ p q p q P B

• The lifts of the Levi-Civita connections are C8 functionals of ϕ, Dϕ : p q

1 j ∇X dX ωX , where ωX X Γ γX ǫ γX ǫ ǫ5 (43) 4 k,i j k i “ ` “ i,j,k p q p q ÿ 1 j ∇BB0 dBB0 ωBB0 , where ωBB0 BB0 Γ γBB0 ǫ γBB0 ǫ ǫ 5 (44) 4 k,i j k i “ ` “ i,j,k p qp q ÿ 1 j ∇BB1 dBB1 ωBB1 , where ωBB1 BB1 Γ γBB1 ǫ γBB1 ǫ ǫ 5 (45) 4 k,i j k i “ ` “ i,j,k p qp q ÿ

where ωX ,ωBB0 ,ωBB1 are the local connection homorphisms depending on the Christoffel symbols

15 X j BB0 j BB1 j Γk,i, Γk,i, Γk,i, which are a functional of the first derivatives of the Riemannian metrics and its inverse. (see[BoWo93], page 15).

• The Dirac operators are C8 functional of ϕ, Dϕ : p q

n n X X X X X Q γ ei ∇ei γ e¯i ∇e¯i “ i“1 p q ` i“1 p q ÿn ÿ n´1 BB0 BB0 BB0 BB0 BB0 Q γ ei ∇ei γ e¯i ∇e¯i (46) “ i“1 p q ` i“1 p q nÿ´1 ÿn´1 BB1 BB1 BB1 BB1 BB1 Q γ ei ∇ei γ e¯i ∇e¯i . “ i“1 p q ` i“1 p q ÿ ÿ

• The Green functions for the Hodge Laplacians are C8 functionals of ϕ, Dϕ : p q

B0 B0 G G x, y for y B0 and x X “ p q PB P XzB0 XzB0 G G x, y for y X B0 and x X “ p q P Bp z q P (47) B1 B1 G G x, y for y B1 and x B0 “ p q PB PB BB0zB1 BB0zB1 G G x, y for y B0 B1 and x B0. “ p q P Bp z q PB

• The bundle homorphisms are therefore C8 functionals of ϕ, Dϕ : p q

BB0 BB0 ζ ζ y for y B0 “ p q PB (48) BB1 BB1 ζ ζ y for y B1. “ p q PB

• The volume form is a C8 functional of ϕ, Dϕ : p q

µBB1 det h BB1 B ,..., B , B ,..., B dz1 dzn´1 dz¯1 dz¯n´1 (49) “ | z1 zn 1 z¯1 z¯n 1 ^¨¨¨^ ^ ^¨¨¨^ ˆ"B B ´ B B ´ *˙

where h B1 is the Riemannian metric on B1. |B B

8 We conclude that for any chart U, Φ in the atlas Ui, Φi i 0,...,K there exist a C differential form- p q p q “ valued function F ω F ω γ, Γ for γ R2n and Γ R2nˆ2n such that “ p q P P

Ξω ϕ x F ω ϕ x , Dϕ x for all x Φ U . (50) p qp q“ p p q p qq P p q

Note that F ω is an affine functional of ω.

16 (ii): For any ϕ in ϕi i 0,...,K we have to solve the equation p q “

Ξω ϕ 0, (51) p q“

We proceed now to verify the fulfillment of the Nash-Moser inverse function theorem, making sure that our construction of the manifolds Y0,1 and B0,1 is well defined. We pick two fix i, j 0,...,K . Pt u

• Functional between Banach spaces: We consider the two family decreasing Banach spaces given by

s 2n s 2n Xs : W Φi Ui , R W Φj Uj , R “ p q ‘ p q 2 2 2 ϕi, ϕj `: ϕi ˘ϕj ` ˘ }p q}s “} }s `} }s (52)

s 0,1 ˚ s 0,1 ˚ s 2n Ys : W Ui, Λ TX W Ui, Λ TX W Φi Ui Φj Uj , R “ pp q q ‘ pp q q ‘ p qX p q 2 2 2 2 λi, λj ,ν ` : λi λj˘ ν ` , ˘ ` ˘ }p q}s “} }s `} }s `} }s

with the corresponding Sobolev norm s defined for any s 0 on the appropriate spaces. The }¨} ě families Xs, s s 0 and Ys, s s 0 satisfy the smoothing hypothesis (cf. [Ra89] page 25). p }¨} q ě p }¨} q ě Note that, by the Sobolev embedding theorem,

s 2n s 2n 8 2n 8 2n X8 W Φi Ui , R W Φj Uj , R C Φi Ui , R C Φj Uj , R . (53) “ sě0 p q ‘ p q “ p q ‘ p q č ` ˘ ` ˘ ` ˘ ´ ¯ Equation (51) for the two charts i, j and the compatibility of the definitions on the intersection

of Φi Ui with Φj Uj can be expressed as p q p q

ω ω ´1 ´1 ´1 Θ ϕi, ϕj : Ξ ϕi , Ξ ϕj , ϕ Φi Φ ϕ 0 Y . (54) p q “ p p q p q i ´ p ˝ j q˝ j q“ P 8

ω,Ui • Assumption (A1): Let S be a bounded open neighbourhood of IdR2n X0, such that S F . P Ă ω Such a S exists by continuity. For all ϕi S X the map Ξ ϕi is twice Fr´echet-differentiable P X 8 p q in ϕi, and for any s 0, and v1, v2 X we have ě P 8

2 ω D Ξ ϕi . v1, v2 p q p q“ ω B B Ξ ϕi t1v1 t2v2 “ t1 t2 p ` ` q“ B ˇt1:“0 B ˇt2:“0 (55) ˇ ω ˇ ω D.11ˇF ϕi, Dϕˇ i . v1, v2 D.22F ϕi, Dϕi . Dv1, Dv2 “ ˇ p ˇ q p q` p q p q` ω ω 0,1 ˚ D.12F ϕi, Dϕi . v1, Dv2 D.21F ϕi, Dϕi . Dv1, v2 Λ TX , ` p q p q` p q p qP pp q q

17 and, for a fixed s0 s, by applying Lemma 4.2 twice ă

2 ω ω i j ω i j D Ξ ϕi . v1, v2 s C DF ϕi s v1v2 0 DF ϕi s0 v1v2 s } p q p q} ď i,j } p q} } } `} p q} } } ď ÿ ! ” ı) (56)

Cs ϕi v1 s v2 s0 v1 s0 v2 s , ď p qr} } } } `} } } } s

where Cs ϕi Cs S , which remains bounded when s remains bounded. Hence, (A1) is fulfilled p qď p q ω ω for Ξ ϕi , and, analogously for Ξ ϕj . So, the first two components of Θ are under control. For p q p q the third component we have

2 D Θ3 ϕi, ϕj . v1, v2 x B B Θ3 ϕi t1v1 t2v2 x p q p qp q“ t1 t2 p ` ` qp q“ B ˇt1:“0 B ˇt2:“0 ˇ ´1 ´ˇ 1 2 ´1 ´1 Dϕˇ i ϕ ˇ D ϕi ϕ . Dϕi ϕ .v1, v2 “ ´r ˇ p i qsˇ p i q p p i q q` (57) 2 ´1 ´1 2 D Ai,j ϕ Dϕj ϕ v1, v2 ´ p j qqpr p j qs q` ´1 ´1 ´2 2 ´1 ´1 DAi,j ϕ Dϕj ϕ D ϕj ϕ . Dϕj ϕ .v1, v2 , ` p j qqr p j qs p j q p p j qq q

´1 2 2 2 where Ai,j : Φi Φ and D ϕi, D ϕj , D Ai,j are vector valued bilinear forms. By p¨q “ ˝ j p¨q ´1 ´1 Proposition 2.1 the domain of definition of Ai,j and the images of ϕi and ϕj are contained in a ´1 compact set. Therefore, the Sobolev norms of the linear operator DAi,j ϕ s and of the bilinear } p j q} 2 ´1 ´1 ´1 ´1 2 ´1 operator D Ai,j ϕ s, as well as those of Dϕi ϕ s, Dϕi ϕ s and D ϕi ϕ s } p j q} }r p i qs } } p i q} } p i q} remain bounded for all ϕi, ϕj S X . This step is crucial, holds for varieties but cannot be p q P X 8 extended to all K¨ahler manifolds. Hence, (A1) is fulfilled for Θ3 ϕi, ϕj and for the whole Θ. p q

ω,Ui • Assumption (A2): Let S be a bounded open neighbourhood of IdR2n X0 such that S F . P Ă ω Such a S exists by continuity. The first Fr´echet-derivative of Ξ ϕi reads p q

ω ω ω DΞ ϕi .v B Ξ ϕi tv DF ϕi, Dϕi .v p q “ t p ` q“ p q “ B ˇt:“0 (58) ˇ ω ω 0,1 ˚ D.1ˇF ϕi, Dϕi .v D.2F ϕi, Dϕi .Dv Λ TX , “ ˇ p q ` p q P pp q q

where v X , and, for any s 0 P 8 ě

ω DΞ ϕi .v s Cs ϕi v s, (59) } p q } ď p q} }

ω s ω which means that DΞ ϕi is a bounded linear operator on the Sobolev space W . Since DF ϕi p q p q ω ´1 is injective, by the bounded inverse operator theorem Ψ ϕi : DΞ ϕi is a bounded linear p q “r p qs operator on W s

Ψ ϕi .v s C˜s ϕi v s, (60) } p q } ď p q} }

18 where C˜s ϕi C˜s S , which remains bounded when s remains bounded. Hence, (A2) is fulfilled p qď p q ω ω for Ξ ϕi , and, analogously for Ξ ϕj . So, the first two components of Θ are under control. For p q p q the third component we have

´1 ´1 ´1 ´1 ´1 DΘ3 ϕi, ϕj . v1, v2 x Dϕi ϕ .v1 DAi,j ϕ Dϕj ϕ .v2, (61) p q p qp q“r p i qs ´ p j r p j qs

and, for any s 0 ě DΘ3 ϕi, ϕj . v1.v2 s Cs ϕi, ϕj v1, v2 s, (62) } p q p q} ď p q}p q}

s which means that DΘ3 ϕi, ϕj is a bounded linear operator on the Sobolev space W . Since p q ω ω DΞ ϕi and DΞ ϕj are injective, so is DΘ3 ϕi, ϕj , which is hence invertible. By the bounded p q p q p q ´1 s inverse operator theorem Υ ϕi, ϕj : DΘ3 ϕi, ϕj is a bounded linear operator on W p q “r p qs

Υ ϕi, ϕj . v1, v2 s C˜s ϕi, ϕj v1, v2 s, (63) } p q p q} ď p q}p q}

where C˜s ϕi, ϕj C˜s S , which remains bounded when s remains bounded. Hence, (A2) is p q ď p q fulfilled for Θ3 ϕi, ϕj and for the whole Θ. p q

2n By Theorem 4.1 (i) and (ii) we infer the existence of a local diffeomeorphisms ϕi,j :Φi,j Ui,j R , p q Ñ 8 2n which lie in C Φi,j Ui,j , R , defining locally B0 and B1. This construction is globally well defined p p q q on X and leads to closed manifolds B0, B1 which are the boundaries of two compact submanifolds of

X, namely Y0 and Y1, such that equation (39) is satisfied on every local chart.

8 8 Hence, B0 and B1 are C submanifolds of X of real codimension 0 and 1, and B1 is a C submanifold 8 of B0 of real codimension 1. The inclusion maps iY0 1 : Y0,1 X are embeddings, and Y0,1 are C - , Ñ 8 submanifolds of X satisfying equation (39). The C atlas on Y1 can be refined to a real analytic ´

atlas. By Royden’s theorem (cf. Theorem 5 in [Ro60]) the maps iY0,1 are homotopic to real analytic

embeddings. By Corollary 2.4, Y1 B1 is a complex projective submanifold of the complex projective “B manifold X. The proof is completed.

Remark 4.1. We can not extend Lemma 4.3 to the category of compact oriented K¨ahler manifolds, because we cannot prove the tame estimate (A1) in the generic K¨ahler case, as mentioned during the proof, where we utilize Corollary 2.4, which holds true for complex projective manifolds but not for K¨ahler manifolds: complex submanifolds of K¨ahler manifolds are in general not K¨ahler.

19 5 Proof of the Hodge Conjecture

First we want to find a basis of the Dolbeault cohomology, whose elements are fundamental classes of the underlying manifold. Later, we will specialize on the Hodge cohomology.

Corollary 5.1. Let X a non-singular complex projective n-dimensional oriented compact manifold without boundary. For all k 0,...,n, for every cohomology class in Hk,k X, C there exists a basis “ p q k,k k,k ω1 ,..., ωC , where Cn,k : dimC H X, C with representants ωj Ω X, C and complex tr s r n,k su “ p p qq P p q k k projective subvarieties Z Z ωj X of dimension k such that j “ p qĂ

˚ α ωj iZk α (64) X ^ ˚ “ Zk j ż ż j

for all α Ωk,k X, C . The map P p q iZ : Z X (65) Ñ

denotes the embedding of any Z Z1,...,ZC into X and Pt u

i˚ :Ωk,k X, C Ωk,k Z, C Z p q Ñ p q ˚ (66) α i α : α TiZ . ,...,TiZ. ÞÑ Z “ p p¨q p¨qq 2k times l jh n the pull back of k, k -forms on X on k, k -forms on Z. p q p q

Proof. First let us assume that X is connected and analyze the different cases k 0,...,n: “

• k 0: we can choose Z0 : X, because H0,0 X, C C as Corollary 5.8 in [BoTu82] carried “ 1 “ p q “ 0,0 over from the De Rham to the Dolbeault cohomology shows, and, hence, H X, C 1 C p q “ xr sy 0,0 and ω1 : 1 Ω X, C satisfies (86). ” P p q

n n,n • k n: we can choose Z : p for a p X, because H X, C µX C, where µX denotes “ 1 “t u P p q“ xr sy n,n the volume form on X, and ω1 : µX Ω X, C satisfies (86). “ P p q

• k n 1: let B0 a 0-real-codimensional submanifold of X, which has a boundary B0, a 1- “ ´ B real-codimensional submanifold of X. Let ω , α Hn´1,n´1 X, C . We apply Theorem 3.2 to r s r sP p q

20 obtain

α ω α, ω dvolX α, ω dvolX ^ ˚ “ x y “ x y “ żX żX żpXzB0qYB0

XzB0 α y ,γ ν Qy G y, x dvolXzB0 ω y volBpXzB0 q “´ BpXzB0q p q p q ˜ XzB0 p q ¸ p q ` ż A ż E (67) B0 α y ,γ ν Qy G y, x dvolB0 ω y volBB0 ´ BB0 p q p q B0 p q p q “ ż A ˆż ˙ E

α y ,γ ν Qyζ y ω y vol B0 , “ x p q p q p q p qy B żBB0 where

– the hermitian structure in the antiholomorphic bundle over X as in Proposition 2.6 is denoted by , : , x¨ ¨y “¨^ ˚¨ – the Dirac operator on X is denoted by Q,

– the Green functions for the Hodge Laplacians on X B0 and B0 as in Proposition 3.1 are z denoted by GXzB0 , and, respectively by GB0 ,

– the operator

XzB0 B0 ζ y : G y, x G y, x dvolX (68) p q “ p q` p q ˜żXzB0 żB0 ¸ 0,0 ˚ is a bundle homomorphism on Λ T X B0 . |B

Note that the unit normal fields on the boundaries of X B0 and B0 are in opposite directions. z

Let now B1 be 0-real-codimensional submanifold of B0, and let us apply Theorem 3.2 a second B time to (67) and obtain

α ω α, ω1 dvolBB0 X ^ ˚ “ B0 x y “ ż żB (69) BB0 BB0 BB0 1 α y ,γ ν Q ζ y ω y vol B1 , “ x p q p q y p q p qy B żBB1 where

1 BB0 – the differential form ω y : γ ν Qyζ y ω y is defined for y B0 p q “ p q p q p q PB

– the hermitian structure in the antiholomorphic bundle over B0 as in Proposition 2.6 is B denoted by , , , x ¨ ¨y BB0 – the Dirac operator on B0 is denoted by Q , B

21 – the Green functions for the Hodge Laplacians on B0 B1 and B1 as in Proposition 3.1 are B z denoted by GXzBB0 , and, respectively by GBB0 ,

– the operator

BB0 BB0zB1 B1 ζ y : G y, x G y, x dvol B0 (70) p q “ p q` p q B ˜żBB0zB1 żB1 ¸

0,0 ˚ is a bundle homomorphism on Λ T B0 B1 . B |B

n´1,n´1 Let ω1 ,..., ωC 1 be a basis of the Dolbeault cohomology H X, C . Assuming that tr s r n,n´ su p q

for all ω ω1,...,ωC 1 Pt n,n´ u

n´1,n´1 ω,Ui ω ω Ω X, C ϕi F for all i 0,...,K , (71) Pt P p q| P “ u

we can now apply Lemma 4.3 to solve equation (69) to find the submanifolds B0 and B1. Iffora ωj , n´2,n´2 condition (71) is not fulfilled, we replace ωj with ωj ¯ηj for an appropriate ηj Ω X, C `BB P p q such that

n´1,n´1 ω,Ui ωj ¯ηj ω Ω X, C ϕi F for all i 0,...,K . (72) `BB Pt P p q| P “ u

n´1,n´1 Such a ηj exists by continuity. Note that ωj ωj ¯ηj H X, C . The complex r s“r `BB s P p q n´1 submanifolds Z : Z ωj : B1 ωj have complex dimension n 1 and are complex projective j “ p q “B p q ´ varieties.

• k n 2,..., 1: for ω , α Hk,k X, C we continue applying Theorem 3.2 till a complex “ ´ r s r s P p q k-codimensional submanifold appears:

α ω α y ,γ ν Qyζ y ω y vol B0 ^ ˚ “ x p q p q p q p qy B “ żX żBB0 BB0 BB0 BB0 1 α y ,γ ν Q ζ y ω y vol B1 “ x p q p q y p q p qy B “ żBB1 (73)

“¨¨¨“

BU2k´2 BU2k´2 BU2k´2 2k´1 α y ,γ ν Q ζ y ω y vol U2 1 , “ x p q p q y p q p qy B k´ żBB2k´1

where

– the submanifolds B0,B1,...,B2k 1 of X have real codimensions 0, 1, 2,..., 2k 1, ´ ´ 2 3 2k´1 BB2k 3 B k´ BB2k 3 – the differential form ω y : γ ν ´ Qy ζ ´ y ω y is defined for y B2k 1 p q “ p q p q p q PB ´

– the Dirac structure on Bk induced by the Dirac structure on Bk 1 by Theorem 3.3 has B ´ Dirac operator QBBk .

22 – the Green functions for the Hodge Laplacians on Bk Bk 1 and Bk as in Proposition 3.1 are B z ` denoted by GBk zBBk`1 , and, respectively by GBBk ,

– the operator

BBk BBk zBk`1 Bk`1 ζ y : G y, x G y, x dvol B (74) p q “ p q` p q B k ˜żBBkzBk`1 żBk ¸

k,k ˚ is a bundle homomorphism on Λ T Bk B 1 . B |B k` Therefore, in view of equations (69) and (73), for any k n 2,..., 1 we have to find an unknown “ ´ k,k Z : B2k 1 such that, for all α Ω X, C “B ´ P p q

BB2k´2 BB2k´2 BB2k´2 2k´1 ˚ γ ν Q ζ y ω y , α z volZ i α . (75) x p q y p q p q p qy “ Z p q żZ żZ

Since iZ IdZ , TziZ 1, equation (75) holds true for all α if and only if “ “

BB2k´1 BB2k´1 BB2k´1 BB2k´2 BB2k´2 BB2k´2 2k´2 γ ν Q ζ γ ν Q ζ y ω y µZ (76) p q y p q y p q p q“

Equation (76) has been solved for k n 1. Assuming that its has been solved for k 1, “ ´ ´ 2k´2 the differential form ω is well defined, Let ω1 ,..., ωC be a basis of the Dolbeault tr s r n,k su k,k cohomology H X, C . Assuming that for all ω ω1,...,ωC p q Pt n,k u

2k´2 k,k ω ,Ui ω ω Ω X, C ϕi F for all i 0,...,K , (77) Pt P p q| P “ u

we can now apply Lemma 4.3 to solve equation (76) to find the submanifolds B2k´1 and B2k´2.

If for a ωj , condition (77) is not fulfilled, we replace ωj with ωj ¯ηj for an appropriate ηj `BB P Ωk´1,k´1 X, C such that p q

2k´2 k,k ω ,Ui ωj ¯ηj ω Ω X, C ϕi F for all i 0,...,K . (78) `BB Pt P p q| P “ u

k,k Such a ηj exists by continuity. Note that ωj ωj ¯ηj H X, C . The complex r s“ r ` BB s P p q k submanifolds Z : Zk ωj : B2k 1 ωj have complex dimension k and are complex projective j “ p q “B ´ p q varieties.

If X is not connected, then it can represented as disjoint union of its connected components Xι I . p P q Since for any k 0,...,n “

k,k k,k k,k k,k H X, C H Xι, C and Ω X, C Ω Xι, C , (79) p q“ ιPI p q p q“ ιPI p q à à 23 the result follows from the connected case and the proof is completed.

Remark 5.1. Corollary 5.1 cannot hold for all ω Hk,k X, C as the simple counterexample ω : 0 P p q “ shows.

Definition 13. If the complex compact manifold X is boundaryless, the expression

α , ω : α ω (80) pr s r sq “ ^ ˚ żX by Stokes’s theorem does not depend on the choice of the representatives for the cohomology classes α and ω , and, hence, defines a scalar product in Hk,k X, C . By Riesz’s Lemma, r s r s p q

Hk,k X, C ˚ Hk,k X, C , (81) p q “ p q where the isomorphism is induced by the scalar product in (80)

5 5 Ωk,k X, C Õ Ωk,k X, C ˚ Hk,k X, C Õ Hk,k X, C ˚, (82) p q 7 p q p q 7 p q

which does not depend on the choice of the representatives in the cohomology classes.

Definition 14. Let Z be an oriented k-dimensional closed submanifold of the n dimensional complex manifold X. The expression 7 Z : i˚ (83) r s “ Z p¨q «ˆżZ ˙ ff defines cohomology class induced in Hk,k X, C by Z, which is termed the fundamental class. p q Remark 5.2. The definition of Z carries over for any closed smooth submanifold Z of X for any r s closed manifold X.

Proposition 5.2. Let X be an oriented compact manifold without boundary and Z a real r-codimensional submanifold of X. The coholomology class Z defined in (83) satisfies r s

´1 Z JZ T 1 , (84) r s“ p p qq

r 0 r k where T : H X,X Z, Z H Z, Z is the Thom isomorphism and jZ : H X,X Z, Z H X, Z p z q– p q p z q Ñ p q the natural map.

Proof. It is a reformulation of Corollary 11.15 in [Vo10] page 271.

24 Corollary 5.1, reformulated using Definition 14, leads to

Theorem 5.3. Let X be an oriented compact n-dimensional complex projective manifold without bound- ary. Then, for any k 0,...,n the cohomology Hk,k X, C is generated by cohomology classes induced “ p q k,k by k-dimensional subvarieties. More exactly, there exist C : dimC H X, C k-dimensional subvari- “ p q eties of X, V1,...,VC , such that

k,k H X, C V1 ,..., VC C. (85) p q“xr s r sy

Similarly, we obtain

Corollary 5.4. Let X a non-singular complex projective n-dimensional oriented compact manifold without boundary. For all k 0,...,n, for every cohomology class in Hdgk X there exists a basis “ p q k k,k ω1 ,..., ωQ , where Qn,k : dimQ Hdg X with representants ωj Ω X, C with ωj Q tr s r n,k su “ p p qq P p q Y P k k for all k-dimensional submanifolds and complex projective subvarieties Z Z ωj X of dimensionş j “ p qĂ k such that ˚ α ωj iZk α (86) X ^ ˚ “ Zk j ż ż j for all α Ωk,k X, C . P p q

Proof. The proof mimicks the proof of Corollary 5.1. Only the following part needs to be adapted for k 1,...,n 1 for the parts around formulae (77) and (78) as follows. “ ´ k Let ω1 ,..., ωQ be a basis of the Hodge cohomology Hdg X . Assuming that for all ω tr s r n,k su p q P ω1,...,ωQ t n,k u

2k´2 k,k ω ,Ui ω ω Ω X, C ϕi F for all i 0,...,K Pt P p q| P “ u (87) ω Ωk,k X, C ω Q for all k-dimensional submanifolds Y , X P p q P " ˇ żY * ˇ ˇ we can now apply Lemma 4.3 to solveˇ equation (76) to find the submanifolds B2k´1 and B2k´2. Iffora k´1,k´1 ωj , condition (87) is not fulfilled, we replace ωj with ωj ¯ηj for an appropriate ηj Ω X, C `BB P p q such that

2k´2 k,k ω ,Ui ωj ¯ηj ω Ω X, C ϕi F for all i 0,...,K `BB Pt P p q| P “ u (88) ω Ωk,k X, C ω Q for all k-dimensional submanifolds Y , X P p q P " ˇ żY * ˇ ˇ ˇ k Such a ηj exists by continuity and because Q is dense in R. Note that ωj ωj ¯ηj Hdg X . r s“r `BB sP p q

25 k The complex submanifolds Z : Zk ωj : B2k 1 ωj have complex dimension k and are complex j “ p q “B ´ p q projective varieties.

Otherwise the proof remain formally unchanged.

Theorem 5.5. Conjecture 2 holds true for any compact oriented complex projective manifold X. More k exactly, there exist Q : dimQ Hdg X k-dimensional subvarieties of X, Z1,...,ZQ, such that “ p p qq

k Hdg X Z1 ,..., ZQ Q. (89) p q“xr s r sy

Proof. The 2k-Hodge class group, defined as

Hdgk X : H2k X, Q Hk,k X, C , (90) p q “ p qX p q

can be represented as k Hdg X ω1 ,..., ωQ Q, (91) p q“xr s r sy

k where Q : dimQ Hdg X and ω1,...,ωQ are rational k, k differential forms on X, i.e. “ p p qq p q

ωm Q (92) P żY for all k-complex dimensional submanifolds Y of X and all m 1,...,Q. “ By Corollary 5.4 and Definition 14 we can choose the rational differential forms ω1,...ωQ so that t u there exist Q oriented k-dimensional subvarieties Z1,...,ZQ of X such

ωm Zm , (93) r s“r s for all m 1,...,Q, and, hence “

k Hdg X Z1 ,..., ZQ Q. (94) p q“xr s r sy as Conjecture 2 states.

Remark 5.3. The Hodge classes Z1 ,..., ZQ are not a complex basis of the Dolbeault cohomology, r s r s and the Dolbeault classes V1 ,..., VC are not a rational basis of the Hodge cohomology. The Hodge r s r s classes Z1 ,..., ZQ can be completed to a complex basis of the Dolbeault cohomology, by adding r s r s appropriate complex linear independent fundamental classes VQ 1 ,..., VC . r ` s r s

26 We can now infer the proof of the Hodge conjecture.

Theorem 5.6. Conjecture 1 holds true for any non singular projective algebraic variety.

Remark 5.4. We can not extend the proof of Conjecture 1 to the category of compact oriented K¨ahler manifolds, because we Lemma 4.3 does not hold for that category, as mentioned in Remark 4.1, which is in line with the counterexamples in [Zu77] and [Vo02].

Remark 5.5. For any field F there is a natural non singular pairing

Hk,k X, F Hn´k,n´k X, F F (95) p qˆ p q Ñ given by the cup product. For F R, C it takes the form of “

α ω α ω, (96) r sˆr s ÞÑ ^ żX and for F Q we need to tensor the rational cohomology groups with C: “

C Hk,k X, Q C Hn´k,n´k X, Q C p b p qq ˆ p b p qq ÝÑ (97) a α ,b ω aα bω. p br s br sq ÞÑ ^ żX Moreover, if Tors G denotes the torsion subgroup of a group G, there is a natural non singular pairing p q

Hk,k X, Z Tors Hk,k X, Z Hn´k,n´k X, Z Tors Hn´k,n´k X, Z Z (98) p q{ p p qq ˆ p q{ p p qq Ñ given by the cup product, which takes the form

α ω α ω Z, (99) r sˆr s ÞÑ ^ P żX where α Hk,k X, Z Tors Hk,k X, Z and ω Hn´k,n´k X, Z Tors Hk,k X, Z . Without the r s P p q{ p p qq r s P p q{ p p qq factorization of the torsion subgroups the pairing is singular. This is why the construction presented so far does not work for F Z, as the counterexamples of Atiyah-Hirzebruch [AtHi62] and Totaro [To97] “ demonstrate.

6 Conclusion

A K¨ahler manifold can be seen as a Riemannian manifold carrying a Dirac bundle structure whose Dirac operator is the Clifford operator. Utilizing a theorem for the Green function for the Dirac Laplacian

27 over a manifold with boundary, the values of the sections of the Dirac bundle can be represented in terms of the values on the boundary, extending the mean value theorem of harmonic analysis. This representation, together with Nash-Moser generalized inverse function theorem, leads to a technical result stating the existence of complex submanifolds of a compact oriented variety satisfying globally a certain partial differential equation. This is the key to prove the existence of complex submanifolds f a compact oriented variety whose fundamental classes span the Hodge classes, proving the Hodge conjecture for non singular algebraic varieties.

Acknowledgement

I would like to express my gratitude to Claire Voisin and Pierre Deligne for highlighting parts of a previous version of this paper which needed important corrections. The possibly remaining mistakes are all mine.

References

[AtHi62] M. ATIYAH and F. HIRZEBRUCH, Analytic Cycles on Complex Manifolds, Topology 1, (25-45), 1962.

[Au82] T. AUBIN, Nonlinear Analysis on Manifolds. Monge-Amp`ere Equations, Grundlehren der math- ematischen Wissenschaften 252, 1982.

[B¨a96] C. BAR,¨ Metrics with Harmonic Spinors, Geometric and Functional Analysis 6, (899-942), 1996.

[Be06] M. BERTI, Lecture Notes on Nash-Moser Theory and Hamiltonian PDEs, School on Nonlinear Differential Equations, ICTP, Trieste, 23 October 2006.

[BoWo93] B. BOOSS/BAVNBEK and K.P.WOJCIECHOWSKI. Elliptic Boundary Problems for Dirac Operators, Birkh¨auser, 1993.

[BoTu82] R. BOTT and L. W. TU, Differential Forms in Algebraic Topology, Graduate Texts in Math- ematics 82, Springer, 1982.

[CDK95] E. CATTANI, P. DELIGNE and A. KAPLAN, On the locus of Hodge classes, J. Amer. Math. Soc. 8 (2), (483-506), 1995.

[De06] P. DELIGNE, The Hodge Conjecture, Clay Math. Institute, 2006.

28 [Gi93] P. B. GILKEY, On the Index of Geometric Operators for Riemannian Manifolds with Boundary, Adv. in Math. 102, (129-183), 1993.

[Gr69] A. GROTHENDIECK, Hodge’s General Conjecture is False for Trivial Reasons, Topology 8, (299–303), 1969.

[Ho52] W. HODGE, The Topological Invariants of Algebraic Varieties, Proceedings of the International Congress of Mathematicians, Cambridge, Mass., 1950, Vol. 1, pp. 182-192. Amer. Math. Soc., Providence, R. I., 1952.

[H¨o76] L. HORMANDER¨ , The Boundary Problems of Physical Geodesy, Arch. Rat. Mech. Anal., 62, (1-52), 1976.

[KoSt82] B. KOSTANT and S. STERNBERG, Symplectic Projective Orbits New Directions in Applied Mathematics (81-84), Springer, 1982

[Lef24] S. LEFSCHETZ, L’Analysis situs et la g´eom´etrie alg´ebrique, Collection de Monographies publi´ee sous la Direction de M. Emile´ Borel, Gauthier-Villars, Paris, 1924.

[Lew99] J. LEWIS, A survey of the Hodge Conjecture, second edition with an appendix B by B. Brent Gordon, CRM Monograph Series, 10, AMS, Providence (RI), 1999.

[Mo61] J. MOSER, A new technique for the Construction of Solutions of Nonlinear Differential Equa- tions, Proc. Nat. Acad. Sci., 47, (1824-1831), 1961.

[Mo62] J. MOSER, On Invariant Curves of Area Preserving Mappings on an Annulus, Nachr. Akad. G¨ottingen Math. Phys., n. 1, (1-20), 1962.

[Mo66] J. MOSER,A Rapidly Convergent Iteration Method and Non-Linear Partial Differential Equa- tions, Ann. Scuola Normale Sup., Pisa, 3, 20, (499-535), 1966.

[Mu76] D. MUMFORD, Algebraic Geometry I, Complex Projective Varieties, Grundl. Math. Wissensch. 221, Springer Verlag, Berlin etc., 1976.

[Na56] J. NASH , The Embedding Problem for Riemannian Manifolds, Annals of Math. 63, (20-63), 1956.

[Pe95] C. PETERS, An Introduction to Complex Algebraic Geometry with Emphasis on the Theory of Surfaces, Cours de l’institut Fourier, no. 23, 1995.

[Ra11] S. RAULOT, Green Functions For The Dirac Operator Under Local Boundary Conditions And Applications, Annals of Global Analysis and Geometry, Springer Verlag, 39 (4), (337-359), 2011.

29 [Ra89] X. S. RAYMOND, Simple Nash-Moser Implicit Function Theorem, L’Enseignement Math´ematique, 35, (217-226), 1989.

[Ro60] H.L. ROYDEN, The Analytic Approximation of Differentiable Mappings, Mathematische An- nalen 139 (3), (171 - 179), 1960.

[Se16] P. SECCHI, On the Nash-Moser Iteration Technique, in Recent Developments of Mathematical Fluid Mechanics, Editors: Herbert Amann, Yoshikazu Giga, Hideo Kozono, Hisashi Okamoto, Masao Yamazaki, Springer Basel 2016.

[To97] B. TOTARO, Torsion Algebraic Cycles and Complex Cobordism, Journal of the American Math- ematical Society, 10 (2), (467–493), 1997.

[Vo02] C. VOISIN, A Counterexample to the Hodge Conjecture Extended to K¨ahler Varieties, n 20, (1057-1075), IMRN 2002.

[Vo10] C. VOISIN, Hodge Theory and Complex Algebraic Geometry I,II, Cambridge University Press, 2010.

[Vo11] C. VOISIN, Lectures on the Hodge and Grothendieck–Hodge Conjectures, School on Hodge The- ory, Rend. Sem. Mat. Univ. Politec. Torino, Vol. 69, 2, (149–198), 2011.

[Vo16] C. VOISIN, The Hodge Conjecture, in Open Problems in Mathematics, Editors: John Forbes Nash, Jr., Michael Th. Rassias, Springer 2016.

[Tr77] H. TRIEBEL, Interpolation Theory, Function Spaces, Differential Operators, North-Holland Publishing Company, 1977.

[Ze76] E. ZEHNDER, Generalized Implicit Function Theorems with Applications to some Small Divisors Problems I-II, Comm. Pure Appl. Math., 28, (91-140), 1975; 29, (49-113), 1976.

[Zu77] S. ZUCKER, The Hodge Conjecture for Cubic Fourfolds, Comp. Math., (199–209), 1977.

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