NNONCOMMUTATIVE HODGE CONJECTURE
XUN LIN
Abstract.
The paper generalizes the classical Hodge conjecture to smooth and proper dg categories. The noncommutative Hodge conjecture is proved to be additive for semi-orthogonaldecompositions. We obtain examples of evidence of the Hodge conjecture by techniques ofnoncommutative geometry. Finally, we show that the noncommutative Hodge conjecture forsmooth proper connective dg algebras is true. Contents
1. Introduction 1Notation 52. Preliminary 52.1. The classical Hodge conjecture 52.2. Noncommutative geometry 73. Hodge conjecture and geometric semi-orthogonal decompositions 94. Noncommutative Hodge conjecture 114.1. Formulation 124.2. Application to geometry and examples 174.3. Connective dg algebras 22References 241.
Introduction
Recently, G. Tabuada proposed a series of noncommutative counterparts of the celebratedconjectures, for example, Grothendieck standard conjecture of type C and type D , Voevod-sky nilpotence conjecture, Tate conjecture, Weil conjecture, and so on. After proposing thenoncommutative counterparts, he proved additivity with respect to the SOD s (semi-orthogonaldecomposition, see the notation Section 1) for most of these conjectures. Then, he was able togive new evidence of the conjectures by a good knowledge of the semi-orthogonal decomposi-tions of derived category of varieties. For the details, the reader can refer to “Noncommutativecounterparts of celebrated conjectures” [Tab19].In this paper, the author generalizes the rational Hodge conjecture to the smooth and propersmall dg categories. This new conjecture is equivalent to the classical Hodge conjecture when the dg category is Per dg ( X ), where X is a projective smooth variety. The ideas to establish the theory a r X i v : . [ m a t h . AG ] F e b XUN LIN are as follows : we regard 0 th Hochschild homology as the noncommutative Hodge cohomology,periodic cyclic homology as de Rham cohomology. In order to specify the noncommutativerational Betti cohomology subspace in periodic cyclic homology and then in 0 th Hochschildhomology, we use Anthony Blanc’s topological K -theory on noncommutative spaces [Bla16].A. Perry has established a version of integral and rational Hodge conjecture for admissiblesubcategory in the derived category of varieties, [Per20, Conjecture 5.11]. Furthermore, healso provides a criterion to examine the integral Hodge conjecture for certain CY-2 categories.For example, the Kuznetsov components in cubic fourfold or Gushel–Mukai fourfold are CY-2categories, and it turns out that the integral Hodge conjecture is true for these components andthe usual integral Hodge conjecture of weight 2 is true for cubic fourfold and Gushel–Mukaifourfold.A. Perry regards the topological K theory of spaces as the Betti cohomology and its intersectionwith 0 th Hochschild homology as abstract Hodge classes. However, this paper’s idea is to findthe abstract Hodge classes in 0 th Hochschild homology, which is naturally inside the periodiccyclic homology. The two versions of the rational Hodge conjecture turn out to be equivalentfor admissible subcategories, but we also deal with general smooth proper dg categories in thispaper.Firstly, the image of the usual Chern character are of type ( p , p ) in de Rham cohomology,that is, they belong to 0 th Hochschild homology. We obtain a noncommutative analog of thisby using the noncommutative Hodge to de Rham spectral sequence which degenerates.
Theorem 1.1. (=Theorem 4.3). Let A be a smooth proper dg category. The Chern character Ch : K ( A ) −→ HC per ( A ) factors through HH ( A ) via embedding HH ( A ) (cid:44) → HC per ( A ) . Next, we obtain a noncommutative analog of rational Betti cohomology HC per ( A ) Q as theimage of rational topological Chern character in periodic cyclic homology. Then, we are able todefine the noncommutative analog of the rational Hodge classes as follows. Definition 1.2. (=Definition 4.5) Let A be a smooth proper dg category. We define thenoncommutative rational Hodge classes as HH ( A ) Q := HH ( A ) ∩ HC per ( A ) Q ⊂ HC per ( A ) , where HC per ( A ) Q := Im Ch top Q ( A ).Finally, since the topological Chern character factors through the algebraic Chern character[Bla16, Theorem 1.1], K ( A ) (cid:47) (cid:47) Ch (cid:37) (cid:37) K top ( A ) Ch top (cid:15) (cid:15) HC per ( A )the image of the algebraic Chern character is in the noncommutative rational Hodge classes.We propose the noncommutative Hodge conjecture as follows. ONCOMMUTATIVE HODGE CONJECTURE 3
Conjecture 1.3. (Noncommutative Hodge conjecture, =Conjecture 4.6). Let A be a smoothproper small dg category. The Chern character Ch : K ( A ) −→ HH ( A ) maps K ( A ) Q surjec-tively to the noncommutative rational Hodge classes HH ( A ) Q . We prove that the noncommutative Hodge conjecture is equivalent to the classical Hodgeconjecture when the dg category is Per dg ( X ). Theorem 1.4. (=Theorem 4.8). Let X be a smooth projective variety.Hodge conjecture for X ⇔ Noncommutative Hodge conjecture for
Per dg ( X ) . The author also proves that the Hodge conjecture is additive for geometric semi-orthogonaldecomposition.
Theorem 1.5. (=Theorem 4.13). Suppose we have a nontrivial semi-orthogonal decompositionof derived category D b ( X ) = (cid:104)A , B(cid:105) such that A and B are geometric, that is, B ∼ = D b ( Y ) and A ∼ = D b ( Z ) for some varieties Y and Z . Then, Hodge conjecture is true for X if and only if it istrue for Y and Z .Remark . I use this to obtain some results to prove that the commutative Hodge conjectureis a birational invariant for 4 and 5 dimensional varieties, see Theorem 4.20, which may beclassically known for the experts, see also [Men19].After establishing the language of noncommutative Hodge conjecture, the author proves thatthe conjecture is additive for general
SOD s and the noncommutative motives.
Theorem 1.7. (=Theorem 4.13). Suppose we have a
SOD , D b ( X ) = (cid:104)A , B(cid:105) . There are natural dg liftings A dg , B dg of A , B corresponding to dg lifting Per dg ( X ) of D b ( X ) .Hodge conjecture for X ⇔ Noncommutative Hodge conjecture for A dg and B dg . Theorem 1.8. (=Theorem 4.21) Let A , B and C be smooth and proper dg categories. Supposethere is a direct sum decomposition : U ( C ) Q ∼ = U ( A ) Q ⊕ U ( B ) Q , see section 4.2 for the definitionof U ( • ) and U ( • ) Q . We have the following.Noncommutative Hodge conjecture for C ⇔
Noncommutative Hodge conjecture for A and B . Let A be a sheaf of Azumaya algebras on X . Using work of G. Tabuada and Michel Van denBergh on Azumaya algebras [TVdB15, Theorem 2.1], U ( Per dg ( X , A )) Q ∼ = U ( Per dg ( X )) Q . Wehave the following. Theorem 1.9. (=Theorem 4.25) Noncommutative Hodge conjecture for
Per dg ( X , A ) ⇔ Non-commutative Hodge conjecture for
Per dg ( X ) . This formulation of the noncommutative Hodge conjecture is compatible with the semi-orthogonal decompositions. Therefore, good knowledge of semi-orthogonal decomposition ofvarieties can simplify the Hodge conjecture, and gives new evidence of the Hodge conjecture.
XUN LIN
The survey “Noncommutative counterparts of celebrated conjectures” [Tab19, Section 2] pro-vides many examples of the applications to the geometry for some conjectures via this approach.The examples also apply to the noncommutative Hodge conjecture, and we give some furtherexamples which are combined in the theorem below.
Theorem 1.10.
Combining Theorem 1.7, Theorem 1.8, and Theorem 1.9, we have (1)
Fractional Calabi–Yau categories.
Let X be a hypersurface of degree ≤ n +1 in P n . There is a semi-orthogonal decomposition Perf ( X ) = (cid:104)T ( X ) , O X , · · · , O X ( n − deg ( X )) (cid:105) . T ( X ) is a fractional Calabi–Yau of dimension ( n +1)( deg − deg ( X ) [Kuz19, Theorem 3.5]. Wewrite T dg ( X ) for the full dg subcategory of Per dg ( X ) whose objects belong to T ( X ) . ThenHodge conjecture of X ⇔ Noncommutative Hodge conjecture of T dg ( X ) . (2) Twisted scheme. (A). Let X be a cubic fourfold containing a plane. There is a semi-orthogonal decompo-sition Perf ( X ) = (cid:104) Perf ( S , A ) , O X , O X (1) , O X (2) (cid:105) . S is a K surface, and A is a sheaf of Azumaya algebra over S [Kuz10, Theorem 4.3].Since the noncommutative Hodge conjecture is true for Per dg ( S , A ) by Theorem 1.9, hencethe Hodge conjecture is true for X .(B). Let f : X −→ S be a smooth quadratic fibration, for example, smooth quadric inrelative projective space P n +1 S [Kuz05]. There is a semi-orthogonal decomposition Perf ( X ) = (cid:104) Perf ( S , Cl ) , Perf ( S ) , · · · , Perf ( S ) (cid:105) . Cl is a sheaf of Azumaya algebra over S if the dimension n of the fiber of f is odd.Thus, if n is odd, the Hodge conjecture of X ⇔ S . Moreover, if dim S ≤ , the Hodgeconjecture for X is true. (3) HP duality.
We write
Hodge ( • ) if the (noncommutative) Hodge conjecture is true for varieties(smooth and proper dg categories). Let Y → P ( V ∗ ) be the HP dual of X → P ( V ) , then Hodge ( X ) ⇔ Hodge ( Y ) . Choosing a linear subspace L ⊂ V ∗ . Let X L = X × P ( V ) P ( L ⊥ ) and Y L = Y × P ( V ∗ ) P ( L ) be the corresponding linear section. Assume X L and Y L are ofexpected dimension and smooth. If we assume Hodge ( X ) , then Hodge ( X L ) ⇔ Hodge ( Y L ) . We can prove (3) directly from the description of
HPD , see Theorem 4.28. For more examplesconstructed from
HPD , see Example 4.30. Motivated from the noncommutative techniques,Theorem 1.10 (3), we expect that we can establish duality of the Hodge conjecture for certainlinear section of the projective dual varieties by classical methods of algebraic geometry.
ONCOMMUTATIVE HODGE CONJECTURE 5
Conjecture 1.11. (=Conjecture 4.32) Let X ⊂ P ( V ) be a projective smooth variety. Supposethe Hodge conjecture is true for X . Let Y ⊂ P ( V ∗ ) be the projective dual of X ⊂ P ( V ) . Choosinga linear subspace L ⊂ V ∗ . Suppose the linear section X L = X ∩ P ( L ⊥ ) and Y L = Y ∩ P ( L ) are bothof expected dimension and smooth. Then, the Hodge conjecture of X L is equivalent to the Hodgeconjecture of Y L . Finally, we obtain some results by the algebraic techniques. A dg algebra A is called connectiveif H i ( A ) = 0 for i >
0. According to [RS20, Theorem 4.6], if A is a connective smooth proper dg algebra, then U ( A ) Q ∼ = U ( H ( A ) / Jac ( H ( A ))) Q ∼ = ⊕U ( C ) Q . Thus, we have the following. Theorem 1.12.
The noncommutative Hodge conjecture is true for smooth proper and connected dg algebra A , see Theorem 4.34. In particular, the noncommutative Hodge conjecture is true forsmooth and proper algebras. We also provide another proof for the case of smooth and proper algebras, see Theorem 4.35.Theorem 1.12 implies that if a variety X admits a tilting bundle (or sheaf), then the Hodgeconjecture is true for X , see the Corollary 4.39 in the text. Notation.
We assume the varieties to be defined over C . We write SOD for semi-orthogonaldecomposition of triangulated categories. We say a semi-orthogonal decomposition is geometricif its components are equivalent to some derived categories of projective smooth varieties. Wealways assume the dg categories to be small categories. We write k as the field C in some placeswithout mentioning. Acknowledgements.
The author is grateful to his supervisor Will Donovan for helpful supports,discussions, and suggestions. The author would like to thank Anthony Blanc and Dmitry Kaledinfor helpful discussions through E-mail. The author also thanks Shizhuo Zhang for informing theauthor about Alexander Perry’s work when the author finished most parts of the paper. Finally,the author is indebted to Alexander Perry for helpful comments and suggestions.2.
Preliminary
The classical Hodge conjecture.
Given a projective smooth variety X , there is a famousHodge decomposition H k ( X ( C ) , Z ) ⊗ C ∼ = ⊕ p + q = k H p ( X , Ω qX )where H p ( X , Ω qX ) can be identified with the ( p , q ) classes in H p + q ( X ( C ) , C ). We define the rational(integral) Hodge classes as rational (integral) ( p , p ) classes. By Poincar´e duality, there is a cyclemap which relates the Chow group of X with its Betti cohomology Cycle : CH ∗ ( X ) −→ H ∗ ( X ( C ) , C ) . Clearly, the image lies in the integral Hodge classes. We obtain the rational cycle map when wetensor with Q . The famous Hodge conjecture concerns whether the image of the (rational) cycle XUN LIN map is exactly the (rational) integral Hodge classes. It is well known that the integral Hodgeconjecture is not true in general [AH62], and the rational Hodge conjecture is still open. Formore introductions to the classical Hodge conjecture, the reader can refer to the survey “Someaspects of the Hodge conjecture” [Voi03].
Remark . The rational (and integral) Hodge conjecture is true for weight one by Lefschetzone-one theorem. According to the Poincar´e duality, the rational Hodge conjecture is true forweight n − n is the dimension of the variety. In particular, the rational Hodge conjecture istrue for varieties of dimension less than or equal to 3.This paper focuses on the non-weighted rational Hodge conjecture. That is, we concernwhether the rational cycle map maps CH ∗ ( X ) Q surjectively into the rational Hodge classes. Theorem 2.2. (Part of Grothendieck-Riemann-Roch [SGA6 exp.XIV] [RR71]) Let X be asmooth projective variety. There is a commutative diagram, where Ch Q are the certain Cherncharacters, K ( X ) Q is the rational th algebraic K group of the coherent sheaves. K ( X ) Q Ch Q (cid:47) (cid:47) ∼ = Ch Q (cid:15) (cid:15) H ∗ ( X , C ) CH ∗ ( X ) Q cycle (cid:56) (cid:56) The image of the Chern character is in the rational Hodge classes, and the rational Hodgeconjecture can be reformulated that Ch Q maps K ( X ) Q surjectively into the rational Hodgeclasses. Proposition 2.3.
We have the Mukai vector v ( • ) v : K ( X ) −→ ⊕ H p , p ( X ) , E (cid:55)→ Ch ( E ) (cid:112) Td ( X ) The non-weighted Hodge conjecture can be reformulated that v Q maps K ( X ) Q surjectively intothe rational Hodge classes.Proof. There is a commutative diagram : K ( X ) v (cid:47) (cid:47) Ch (cid:39) (cid:39) ⊕ H p , p ( X ) ∼ = √ Td ( X ) (cid:15) (cid:15) ⊕ H p , p ( X )Since the vertical morphism is an isomorphism which preserves the rational Hodge classes, v Q maps K ( X ) Q surjectively into the rational Hodge classes if and only if Ch Q maps K ( X ) Q surjectively into the rational Hodge classes. Thus, the statement follows from the Theorem 2.2above. (cid:3) ONCOMMUTATIVE HODGE CONJECTURE 7
Noncommutative geometry.
We briefly recall the theory of noncommutative spaces.We regard certain dg categories as noncommutative counterparts of varieties. We will recall thebasic notions. For survey of the dg categories, the reader can refer to the survey by B. Keller,“On differential graded categories” [Kel06]. Definition 2.4.
The C -linear category A is called a dg category if Mor ( • , • ) are differential Z -graded k -vector spaces. For every objects E , F , G ∈ A , the compositions Mor ( F , E ) ⊗ Mor ( G , F ) → Mor ( G , E )of complexes are associative. Furthermore, there is a unit k → Mor ( E , E ). Note that thecomposition law implies that Mor ( E , E ) is a differential graded algebra. Example 2.5.
A basic example of dg categories is C dg ( k ), whose objects are complexes of k - vector space. The morphism spaces are refined as follows :Let E , F ∈ C dg ( k ), define degree n piece of the morphism Mor ( E , F ) to be Mor ( E , F )( n ) :=Π Hom ( E i , F i + n ). The n th differential is given by d n ( f ) = d E ◦ f − ( − n f ◦ d F , f ∈ Mor ( E , F )( n ). Definition 2.6.
We call F : C −→ D a dg functor between dg categories if F : Hom ( E , G ) −→ Hom ( F ( E ) , F ( G )) is in C ( k ) (morphisms are morphism of chain complexes), E , G ∈ C . We call F to be quasi-equivalent if F induces isomorphisms on homologies of morphisms and equivalenceon their homotopic categories. Definition 2.7.
The dg functor F : A −→ B is derived Morita equivalent if it induces anequivalence of derived categories by composition F ∗ : D ( B ) ∼ = D ( A ) . Note that if dg functor A −→ B is a quasi-equivalence, then it is derived Morita equivalent, thereader can refer to “Categorical resolutions of irrational singularities” [KL15, Proposition 3.9]for an explicit proof.We consider the category of small dg categories, whose morphisms are the dg functors. It iswritten as dg − cat . According to G. Tabuada [Tab05], there is a model structure on dg − cat with derived Morita equivalent dg functors as weak equivalences. We write Hmo ( dg − cat ) as theassociated homotopy category for such model structure. Given two dg categories A and B , wehave a bijection Hom
Hmo ( A , B ) ∼ = Iso rep ( A op ⊗ L B ), where rep ( A op ⊗ L B ) is the subcategory of D ( A ⊗ L B ) with bi-module X such that X ( A , • ) is a perfect B module. Linearizing the category,we obtain Hmo whose morphism spaces become K ( rep ( A op ⊗ B )). After Q linearization andidempotent completion, we get the category of pre-noncommutative motive PChow Q . Definition 2.8.
Any functor to an additive category C , F : dg − cat −→ C , is called an additiveinvariant in the sense of G. Tabuada [Tab05] if :(1) It maps the Morita equivalences to isomorphisms.(2) For pre-triangulated dg categories A , B and X with natural morphism i : A −→ X and XUN LIN j : B −→ X which induces semi-orthogonal decomposition of triangulated categories Ho ( X ) = (cid:104) Ho ( A ) , Ho ( B ) (cid:105) , there is an isomorphism F ( X ) ∼ = F ( A ) ⊕ F ( B ) which is induced by F ( i ) + F ( j ).The following theorem is due to G. Tabuada. Theorem 2.9. (G. Tabuada [Tab05, Theorem 4.1]) The functor F in Definition 2.8 that induces Hmo −→ A is an additive invariant if and only if it factors through
Hmo −→ Hmo −→ A .That is, Hmo plays a role as the usual motives, and the additive invariants should be regardedas noncommutative Weil cohomology theories.Remark . Due to many people’s works, see a survey [Tab], the Hochschild homology, alge-braic K -theory, (periodic) cyclic homology theory are all additive invariants. The Hochschildhomology of proper smooth variety is the noncommutative counterpart of Hodge cohomology,and periodic cyclic homology corresponds to the de Rham cohomology.Given a proper smooth variety X , there is a natural dg lifting Per dg ( X ), which is a dg liftingof Perf ( X ). In this sense, the dg categories can be regarded as noncommutative counterpart ofvarieties. In order to focus on the nice spaces, for example, the Chow motive concerns the propersmooth varieties, we restrict the dg − cat to the smooth proper dg categories. Definition 2.11. Dg category A is called smooth if A is perfect A − A bi-module. It is calledsmooth and proper if A is derived Morita equivalent to a smooth dg algebra of finite type.It is well known that the property of dg categories being smooth and proper is closed underderived Morita equivalence and tensor product [Tab, Chapter 1, Theorem 1.43]. People alsodefine the properness as Hom A ( • , • ) being perfect k − mod . According to a book of G. Tabuada,“Noncommutative motive” [Tab, Proposition 1.45], such a definition of smooth and propernessis equivalent to our definition. Definition 2.12 (Noncommutative
Chow motive) . [Tab] We write Hmo sp as a full sub-categoryof Hmo whose objects are smooth proper dg categories. Q linearizing the category Hmo sp , thatis, the morphisms become K ( A op ⊗ B ) Q [Tab, Cor 1.44], we obtain Hmo sp , Q . Then, we define NChow Q to be idempotent completion of Hmo sp , Q .There is a universal additive invariant : U : dg - cat sp −→ NChow . Let C be the category with one object whose morphism space is C . Then for any A ∈ dg - cat , Hom
NChow ( U ( C ) , U ( A )) ∼ = K ( rep ( A )) ∼ = K ( A ) := K ( D c ( A )). Since we have a functorial mor-phism Hom
NChow ( U ( C ) , U ( A )) −→ Hom C ( HH ( C ) , HH ( A )), there is a Chern character map Ch : K ( A ) −→ HH ( A ) . ONCOMMUTATIVE HODGE CONJECTURE 9
Given any A module X ∈ D c ( A ) , it is defined via the following diagram of dg categories. Per dg ( A ) C X (cid:59) (cid:59) A (cid:79) (cid:79) It induces morphisms of Hochschild complexes naturally, and then an element in HH ( A ) viaisomorphism HH ( A ) ∼ = HH ( Per dg ( A )). The isomorphism is because the Yoneda embedding A −→
Per dg ( A ) is a derived Morita equivalence. Per dg ( A ) is defined as a full subcategory of dg A module whose objects are isomorphic to objects in Perf ( A ).In general, given any additive invariant F with F ( k ) ∼ = k , we have a Chern character map K ( A ) −→ F ( A ). For example, the (periodic) cyclic homology, and the Hochschild homology.It is natural to ask what are the relations between Chow motive
Chow Q and noncommutative Chow motive
NChow Q . There is a nice answer due to remarkable works of Kontsevich andG. Tabuada. Theorem 2.13. ( [Tab11b, Theorem 1.1]) There is a symmetric monoidal functor φ : SmProjec op −→ dg − cat op , X (cid:55)→ Per dg ( X ) such that the natural diagram is commutative. SmProjec op φ (cid:47) (cid:47) (cid:15) (cid:15) dg − cat sp (cid:15) (cid:15) Chow Q (cid:15) (cid:15) Hmo sp (cid:15) (cid:15) Chow Q / − ⊗ Q (1) φ (cid:48) (cid:47) (cid:47) NChow Q ⊂ Hmo ∗ , Q With this commutative diagram, G. Tabuada was able to generalize some famous conjec-tures to the noncommutative spaces, see “Noncommutative counterparts of celebrated conjec-tures” [Tab19].3.
Hodge conjecture and geometric semi-orthogonal decompositions
In this section, we prove that the Hodge conjecture is additive for the geometric semi-orthogonal decompositions. In particular, the Hodge conjecture is a derived invariant.
Theorem 3.1.
Suppose we have a nontrivial semi-orthogonal decomposition of derived categories D b ( X ) = (cid:104)A , B(cid:105) such that A and B are geometric, that is, B ∼ = D b ( Y ) and A ∼ = D b ( Z ) for somevarieties Y and Z . Then Hodge conjecture is true for X if and only if it is true for Y and Z . Proof.
Let’s assume j : D b ( Z ) (cid:44) → D b ( X ) to be an embedding with left adjoint L , i : D b ( Y ) (cid:44) → D b ( X )with right adjoint R . According to D. Orlov [Orl96, Theorem 2.2], they are all Fourier-Mukaifunctors. There is a diagram of triangulated categories : D b ( Y ) i (cid:47) (cid:47) D b ( X ) L (cid:47) (cid:47) R (cid:113) (cid:113) D b ( Z ) j (cid:114) (cid:114) with R ◦ i ∼ = id , L ◦ j ∼ = id , R ◦ j ∼ = 0 and L ◦ i ∼ = 0. Apply 0 th K -theory and 0 th Hochschild homologytheory, there are diagrams K ( D b ( Y )) i (cid:47) (cid:47) K ( D b ( X )) L (cid:47) (cid:47) R (cid:112) (cid:112) K D b ( Z ) j (cid:112) (cid:112) . HH ( Y ) i H (cid:47) (cid:47) HH ( X ) L H (cid:47) (cid:47) R H (cid:113) (cid:113) HH ( Z ) j H (cid:113) (cid:113) . Here we define HH ( • ) as a subspace of de Rham cohomology. The morphisms of HH areinduced by the Mukai vector of the corresponding kernel of functors. The morphisms of K groups are induced by the Fourier-Mukai functor. According to [Huy, Chapter 5, Section 5.2],the Mukai vector v is compatible with morphism of K -theory, namely, we have a diagram K ( Y ) i (cid:47) (cid:47) v Y (cid:15) (cid:15) K ( X ) L (cid:47) (cid:47) R (cid:114) (cid:114) v X (cid:15) (cid:15) K ( Z ) v Z (cid:15) (cid:15) j (cid:114) (cid:114) HH ( Y ) i H (cid:47) (cid:47) HH ( X ) L H (cid:47) (cid:47) R H (cid:113) (cid:113) HH ( Z ) j H (cid:113) (cid:113) The morphisms R H , i H , j H , and L H preserve rational classes. We first prove that i H + j H inducesan isomorphism of Hochschild homologies. Clearly i + j is an isomorphism of K ( X ) groups. SinceHochschild homology is an additive invariant, we have a non-canonical isomorphism HH ( X ) ∼ = HH ( Y ) ⊕ HH ( Z ), which implies dim C HH ( X ) = dim C HH ( Y ) + dim C HH ( Z ). This was provedby classical dg methods and the HKR isomorphism. The reader can also refer to A. Kuznetsov’spaper “Hochschild homology and semi-orthogonal decomposition” [Kuz09, Theorem 7.3(i)].Since i H and j H are injective, which will be proved below, therefore i H + j H being an isomorphismis equivalent to the fact that Im ( i H ) ∩ Im ( j H ) = 0. It suffices to prove that L H ◦ i H = 0. If this istrue, let α ∈ Im ( i H ) ∩ Im ( j H ), then α = i H α Y = j H α Z , therefore L H α = ( L H ◦ i H ) α Y = ( L H ◦ j H ) α Z = α Z = 0, hence α = 0. In order to prove the claim L H ◦ i H = 0, we need the following lemma. Lemma 3.2.
Suppose an object E ∈ D b ( X × Y ) induces a trivial Fourier–Mukai transform Φ E : D b ( X ) −→ D b ( Y ) , then E ∼ = 0 ∈ D b ( X × Y ) .Proof of the lemma. Given any closed point x ∈ X , we have a natural closed embedding l x : x × Y (cid:44) → X × Y , and a simple calculation shows that Φ E ( k ( x )) ∼ = L l ∗ x E via identifying x × Y with Y .Therefore, Φ E being trivial implies that L l ∗ x E is trivial. Since this is true for any closed pointsof X , support of E is empty, which implies E ∼ = 0. (cid:3) ONCOMMUTATIVE HODGE CONJECTURE 11
Back to the proof of Theorem 3.1. Since the functor L ◦ i ∼ = 0 as Fourier–Mukai functor, bylemma above the kernel corresponding to L ◦ i is trivial. In particular, its Mukai vector is trivial,hence L H ◦ i H = 0.Now it is prepared enough to prove Theorem 3.1. Suppose Hodge conjecture for X . Let α Y ∈ ⊕ H p , p ( Y , Q ), consider α = i H α Y ∈ ⊕ H p , p ( X , Q ). Since Hodge conjecture holds for X , thereexists an E ∈ K ( X ) Q such that v ( E ) = α . Let E Y = R ( E ), then the image of v ( E Y ) and α Y under i H coincide. Since R H ◦ i H = id H , then i H is an injective morphism, therefore v ( E Y ) = α Y . Thisimplies Hodge conjecture for Y . The Hodge conjecture is true for Z by the similar argument.Suppose Hodge conjecture is true for Y and Z , we prove that it is also true for X . Let α ∈ ⊕ H p , p ( X , Q ), consider R H ( α ) ∈ HH ( Y ) Q and L H ( α ) ∈ HH ( Z ) Q . Since the Hodge conjectureis true for Y and Z , there exists an E Y ∈ K ( Y ) Q and an E Z ∈ K ( Z ) Q such that v ( E Y ) = R H ( α ), v ( E Z ) = L H ( α ). Define α (cid:48) = i H ◦ R H ( α )+ j H ◦ L H ( α ). We prove that α (cid:48) = α . Since i H ⊕ j H induces anisomorphism, there exist α ∈ HH ( Y ) and α ∈ HH ( Z ) such that α = i H ( α )+ j H ( α ). Applyingmorphism R H , we obtain α = R H ( α ). Apply morphism L H , we obtain α = L H ( α ). Thus α = i H ◦ R H ( α ) + j H ◦ L H ( α ). Define E = i ( E Y ) + j ( E Z ) ∈ K ( X ) Q , then v ( E ) = v ( i ( E Y )) + v ( j ( E Z )) = i H ( R H ( α )) + j H ( L H ( α )) = α . (cid:3) Remark . The statement of the theorem is still true if there is a semi-orthogonal decomposi-tion of D b ( X ) that has more than two components. The proof is essentially the same. Corollary 3.4. If D b ( X ) ∼ = D b ( Y ) , then Hodge conjecture of X ⇔ Hodge conjecture of Y . Corollary 3.5.
Suppose D b ( X ) admits a full exceptional collection, then the Hodge conjecture istrue for X . For example, the Grassmannians [Kap85], certain homogeneous spaces (see a briefsurvey in [KP16, Section 1.1]), and smooth projective toric varieties [Kaw05, Theorem 1.1]. Example 3.6.
Let X be the projective space P n . There is a semi-orthogonal decomposition D b ( X ) = (cid:104)O , O (1) , · · · , O ( n ) (cid:105) . We assume n = 3 for simplicity. Since O ( i ) is a line bundle, c j ( O ( i )) = 0, j ≥
2. Write H as hyperplane of P , then Ch ( O ( i )) = 1 + i · H + i · H + i · H , HH ( P ) Q ∼ = Q ⊕ Q H ⊕ Q H ⊕ Q H . The vectors Ch ( O ), Ch ( O (1)), Ch ( O (2)), and Ch ( O (3)) arelinear independent which generate HH ( P ) Q .4. Noncommutative Hodge conjecture
In this section, we propose the noncommutative Hodge conjecture, and prove that the non-commutative Hodge conjecture is additive for semi-orthogonal decomposition. We obtain moreevidence of the Hodge conjecture via good knowledge of semi-orthogonal decomposition. Finally,we prove that the noncommutative Hodge conjecture is true for smooth proper connective dg algebras.For Per dg ( X ), HH ( Per dg ( X )) ∼ = ⊕ H p , p ( X , C ) by HKR isomorphism. In order to generalize theHodge conjecture, we need to find natural intrinsic rational Hodge classes in HH ( A ), and most importantly, it becomes the usual rational Hodge classes when A = Per dg ( X ). Classically, itis well known that the images of rational topological K -groups under topological Chern char-acter recovers the rational Betti cohomolgy. The topological K -theory was generalized to thenoncommutative spaces by Anthony Blanc [Bla16], it turns out that the image of rational topo-logical K -group K top ( A ) under the topological Chern character becomes the even rational Betticohomology when A = Per dg ( X ).4.1. Formulation.Lemma 4.1. (Anthony Blanc, [Bla16, Theorem 1.1]) The topological K -theory of the noncom-mutative spaces is an additive invariant and the Chern character factors through algebraic K -theory. That is, there is a functorial morphism with respect to dg category A Ch : K ( A ) (cid:47) (cid:47) K top ( A ) Ch top (cid:47) (cid:47) HC per ( A ) . When A = Per dg ( X ) , it is naturally isomorphic to the classical Chern character via identifyingperiodic cyclic homology of variety with its de Rham cohomology, and the image of topological K -groups under topological Chern character is exactly the rational Betti cohomology. The lemma above provides a rational subspace of periodic cyclic homology and it is compatiblewith the usual rational structure in the commutative cases. However, we need a certain rationalsubspace of HH ( A ). It is given by the natural embedding HH ( A ) ⊂ HC per ( A ) below. Lemma 4.2. (D. Kaledin [Kal17, Theorem 5.4]) Suppose A is a smooth proper small dg cate-gory, the noncommutative Hodge to de Rham spectral sequence degenerates. Furthermore, thereis a natural isomorphism HC per ∗ ( A ) ∼ = HH ∗ ( A )(( u )) , where u is a formal element of degree 2. The classical Chern character maps K ( X ) Q to HH ( X ) Q := ⊕ H p , p ( X , Q ). In general, it is stilltrue. Theorem 4.3.
Let A be a smooth proper dg category. The Chern Character Ch : K ( A ) −→ HC per ( A ) factors through HH ( A ) via embedding HH ( A ) ⊂ HC per ( A ) from Lemma 4.2.Proof. We may work with HC per ( Per dg ( A )) and HH i ( Per dg ( A )) because the Yoneda embedding A −→
Per dg ( A ) is a derived Morita equivalence, and then induces an isomorphism of additiveinvariants. For simplicity, we write HC per ( Per dg ( A )) and HH i ( Per dg ( A )) as HC per ( A ) and HH i ( A ).Note that the Hochschild complexes may have negative degrees, for simplicity, we write the ONCOMMUTATIVE HODGE CONJECTURE 13
Hochschild complex C ∗ and periodic cyclic bi-complex as the following diagram. . . . · · · C C C · · · C C · · · C . . . Clearly, Ch ( • ) in HC per can be represented by a class in C × C × · · · . Firstly, consider mor-phism of bi-complexes of periodic cyclic homology Tot • , • ( k ) −→ Tot • , • ( Per dg ( A )) induced by X ∈ K ( A ) := K ( D c ( A )). It induces morphism of Hodge to de Rham spectral sequences. Fur-thermore, it induces a morphism of degenerate Hodge to de Rham spectral sequences. Therefore,there is a commutative diagram : HC per ( k ) Ch X (cid:47) (cid:47) ∼ = (cid:15) (cid:15) HC per ( A ) ∼ = (cid:15) (cid:15) · · · ⊕ HH − ( k ) ⊕ HH ( k ) ⊕ HH ( k ) ⊕ · · · (cid:47) (cid:47) · · · ⊕ HH − ( A ) ⊕ HH ( A ) ⊕ HH ( A ) ⊕ · · · According to G. Tabuada, “A universal characterization of Chern maps” [Tab11a, Section 3],we can specify a generator [ u ∞ ] = [( · · · , y i , z i , · · · , y , z , y )] of HC per ( k ) where y i := ( − i (2 i )! i ! 1 , and z i := ( − i (2 i )!2( i !) 1 . Then Ch X ([ u ∞ ]) is a cycle in Tot ( Per dg ( A )) replacing id k by id X . Since Hochschild complex of k is exact except the degree 0, the image Ch X ([ u ∞ ]) only survives in HH ( A ) (cid:44) → ⊕ HH i ( A ) ∼ = HC per ( A ). Hence Ch X ([ u ∞ ]) lies in HH ( A ) under isomorphism HC per ( A ) ∼ = ⊕ HH i ( A ).The remaining thing we need to verify is that, the natural isomorphism HC per ( k ) ∼ = HH ( k )maps [ u ∞ ] to the canonical generator of HH ( k ). We firstly identify the Hochschild complexwith complex of holomorphic differential forms (with trivial differential). The double complex isnaturally isomorphic to the double complex corresponding to the holomorphic de Rham complexof k . Then clearly the generator [ u ∞ ] of HC per ( k ) maps to 1 ∈ Ω ( k ) which is the canonical generator of HH ( k ). This implies that there is a commutative diagram : K ( A ) Ch (cid:47) (cid:47) Ch (cid:37) (cid:37) HC per ( A ) HH ( A ) (cid:79) (cid:79) where the vertical map is the embedding HH ( A ) (cid:44) → ⊕ HH i ( A ) ∼ = HC per ( A ). (cid:3) Remark . It is not known that topological K -theory provides a rational structure for periodiccyclic homology in general. See the lattice conjecture in the paper [Bla16, Conjecture 4.5]. Sinceit is true for variety X , it is also true for A being an admissible subcategory of D b ( X ) becausethe topological K -theory is an additive invariant. Definition 4.5.
Let A be a smooth proper dg category. Let HC per ( A ) Q := Im Ch top Q ( A ), where Ch top Q is the rational topological Chern charater. We define the noncommutative rational Hodgeclasses as HH ( A ) Q := HH ( A ) ∩ HC per ( A ) Q ⊂ HC per ( A ) . By Theorem 4.3, the Chern character Ch : K ( A ) −→ HC per ( A ) maps K ( A ) to HH ( A ).Moreover, it maps K ( A ) to the rational subspace HC per ( A ) Q since there is a commutativediagram Lemma 4.1. K ( A ) Ch (cid:37) (cid:37) (cid:47) (cid:47) K top ( A ) Ch top (cid:15) (cid:15) HC per ( A )Combining these, Ch maps K ( A ) to the noncommutative rational Hodge classes HH ( A ) Q , thatis, Im Ch ⊂ HH ( A ) Q . It immediately follows that Im Ch Q ⊂ HH ( A ). Now, it is prepared enoughto state the noncommutative Hodge conjecture. Conjecture 4.6 (Noncommutative Hodge conjecture) . Let A be a smooth proper small dg category. The Chern character Ch : K ( A ) −→ HH ( A ) maps K ( A ) Q surjectively to noncom-mutative Hodge classes HH ( A ) Q .Remark . Note that we obtain the abstract rational Hodge classes in HH ( A ). Classically,the Hodge conjecture concerns the weight. However, to the author’s knowledge, we don’t knowhow to obtain the weight of the abstract Hodge classes. In the paper, we always assume theconjecture as a non-weighted Hodge conjecture. The Conjecture 4.6 is equivalent in the case ofadmissible subcategories of D b ( X ) to the one in A. Perry’s paper [Per20, Conjecture 5.11].Let X be a smooth projective variety. The noncommutative Hodge conjecture of Per dg ( X ) isequivalent to the commutative Hodge conjecture of X . ONCOMMUTATIVE HODGE CONJECTURE 15
Theorem 4.8.
Let X be a smooth projective variety. Hodge conjecture for X ⇔ NoncommutativeHodge conjecture for
Per dg ( X ) .Proof. The commutative Hodge conjecture claims that the Chern character Ch : K ( X ) Q −→ (cid:76) p H p , p ( X , C ) maps K ( X ) Q surjectively to the rational Hodge classes. The noncommutativeHodge conjecture claims that the map Ch Q : K ( X ) Q = K ( Per dg ( X )) Q −→ HH ( Per dg ( X )) Q issurjective.There is a commutative diagram : K ( Per dg ( X )) Ch (cid:47) (cid:47) ∼ = (cid:15) (cid:15) HH ( Per dg ( X )) (cid:44) → (cid:47) (cid:47) ∼ = (cid:15) (cid:15) HC per ( Per dg ( X )) ∼ = (cid:15) (cid:15) K ( X ) Ch (cid:47) (cid:47) (cid:76) H p , p ( X , C ) (cid:44) → (cid:47) (cid:47) H evendR ( X , C )We explain the commutative square on the right hand side. There is a natural quasi isomor-phism of double complexes Tot • , • ( Per dg ( X )) → Tot • , • ( R Γ( ⊕ Ω iX [ i ])) which is the generalized tracemap described by B. Keller in [Kel98]. It naturally induces a morphism of degenerate spec-tral sequences, which is the commutative square on the right hand side. Furthermore, afteridentifying HC per ( Per dg ( X )) with H evendR ( X , C ), the noncommutative Chern character becomes theusual Chern character. The reader can refer to C. Weibel, “The Hodge filtration and cyclichomology” [Cha, Proposition 3.8.1] or Anthony Blanc’s paper [Bla16, Proposition 4.32]. Hence,the noncommutative Chern character maps K ( X ) Q surjectively to the noncommutative rationalHodge classes if and only if the commutative Chern character maps K ( X ) Q surjectively to thecommutative rational Hodge classes. (cid:3) Remark . The theorem was also proved in A. Perry’s paper under his version of Hodgeconjecture [Per20, Lemma 5.12 and proof of Proposition 5.4]. The proof there is essentially thesame as the proof here when identifying two versions of the Hodge conjecture.
Theorem 4.10.
Suppose F : A −→ B is a derived Morita equivalence, then Hodge conjecture istrue for A if and only if it is true for B .Proof. This is true since the topological and algebraic K -theory, Hochschild homology and peri-odic cyclic homology are all additive invariants. We have a commutative diagram K ( A ) (cid:47) (cid:47) Ch (cid:15) (cid:15) K ( B ) Ch (cid:15) (cid:15) HH ( A ) (cid:47) (cid:47) (cid:15) (cid:15) HH ( B ) (cid:15) (cid:15) HC per ( A ) (cid:47) (cid:47) HC per ( B ) whose rows are isomorphisms. The lower commutative square comes from the natural morphismsof degenerate spectral sequences. The rational subspaces HC per ( • ) Q are isomorphic under suchidentification. Ch A , Q maps K ( A ) Q surjectively to HH ( A ) Q if and only if it is true for Ch B , Q . (cid:3) Corollary 4.11.
For the unique enhanced triangulated categories, we can define its Hodge con-jecture via its smooth and proper dg lifting (if it exists). The Hodge conjecture does not dependon the dg lifting.Proof. This is because two dg lifting of the unique enhanced triangulated categories are con-nected by a chain of quasi-equivalences, and the corollary follows from Theorem 4.10. (cid:3) Remark . For a projective smooth variety X , D b ( X ) ∼ = Perf ( X ) is a unique enhanced triangu-lated category. Thus, it suffices to check whether the conjecture is true for any pre-triangulated dg lifting of D b ( X ). Theorem 4.13.
Suppose we have a
SOD , D b ( X ) = (cid:104)A , B(cid:105) . There are natural dg lifting A dg , B dg of A , B corresponding to dg lifting Per dg ( X ) of D b ( X ) .Hodge conjecture for X ⇔ Noncommutative Hodge conjecture for A dg and B dg . Proof.
We still write A and B as dg categories corresponding to the natural dg lifting again. Wecan lift the semi-orthogonal decomposition to the dg world by [KL15, Proposition 4.10]. Thatis, there is a diagram B i (cid:47) (cid:47) D L (cid:47) (cid:47) R (cid:117) (cid:117) A j (cid:117) (cid:117) where D is certain gluing of A and B and it is quasi-equivalent to Per dg ( X ). Therefore, we stillhave a diagram such that i + j induces isomorphism of K group, and i H + j H induces isomorphismof Hochschild homology : K ( B ) i (cid:47) (cid:47) Ch (cid:15) (cid:15) K ( D ) L (cid:47) (cid:47) R (cid:114) (cid:114) Ch (cid:15) (cid:15) K ( A ) Ch (cid:15) (cid:15) j (cid:114) (cid:114) HH ( B ) i H (cid:47) (cid:47) HH ( D ) L H (cid:47) (cid:47) R H (cid:113) (cid:113) HH ( A ) j H (cid:113) (cid:113) Again, since topological K -theory is an additive invariant, the functors preserve the rationalsubspaces HC per ( • ) Q . Hence Ch D , Q maps K ( D ) Q surjectively to HH ( D ) Q if and only if Ch B , Q and Ch A , Q map K ( B ) Q and K ( A ) Q surjectively to HH ( B ) Q and HH ( A ) Q respectively. But thenoncommutative Hodge conjecture is true for D if and only if it is true for the Hodge conjectureof X by the Theorem 4.8 and Theorem 4.10. Thus, the statement follows. (cid:3) Remark . Similar to the geometric case 3.1, the statement is still true if there are more thantwo components for
SOD s.We immediately reprove Theorem 3.1.
ONCOMMUTATIVE HODGE CONJECTURE 17
Corollary 4.15.
Let X be a projective smooth variety, suppose there is a SOD , D b ( X ) = (cid:104) D b ( Z ) , D b ( Y ) (cid:105) . Then Hodge conjecture is true for X if and only for Z and Y . In particularHodge conjecture is a derived invariant.Proof. According to Theorem 4.13, Hodge conjecture is true for X if and only if it is true forcorresponding dg lifting of D b ( Z ) and D b ( Y ). Since D b ( Z ) and D b ( Y ) are unique enhancedtriangulated categories, hence the Hodge conjecture is true for X if and only for Z and Y . (cid:3) Corollary 4.16.
Consider blow up X of Y with smooth center Z , according to Orlov’s blow-upformula [BO02, Theorem 4.2], we have a SOD , D b ( X ) = (cid:104) D b ( Z ) , · · · , D b ( Z ) , D b ( Y ) (cid:105) . Hence theHodge conjecture is true for X if and only if for Z and Y .Remark . It was known by classical method. We can even write down the
Chow groups withrespect to the blow up, for explicit details, the reader can refer to the book of C. Voisin, “Hodgetheory and complex algebraic geometry II ” [Voi03, Theorem 9.27] Corollary 4.18.
We reprove Corollary 3.5: Suppose D b ( X ) admits a full exceptional collection,then the Hodge conjecture is true for X . For low dimensional varieties, Hodge conjecture is a birational invariant. We use the followinglemma :
Lemma 4.19. ( [AKMW99, Theorem 0.1.1]) Let X and Y be proper smooth varieties. If X isbirational to Y , then there is a chain of blow-ups and blow-downs of smooth centers connecting X and Y . X (cid:127) (cid:127) (cid:32) (cid:32) · · · X (cid:126) (cid:126) (cid:31) (cid:31) X X Y The following may be well known for the expects, see also [Men19]. Here, we use the non-commutative techniques to reprove the results.
Theorem 4.20.
Since Hodge conjecture is true for , , and dimensional varieties, theHodge conjecture is a birational invariant for and dimensional varieties.Proof. Combining Corollary 4.16 and Lemma 4.19, and observe that X and Y are connected bya chain of blow-ups of smooth center whose dimension is less or equal to 3. (cid:3) Application to geometry and examples.
The survey “Noncommutative counterpartsof celebrated conjecture” [Tab19, Section 2] provides many examples of the applications to thegeometry for some celebrated conjectures. The examples also apply to the noncommutativeHodge conjecture. In this subsection, we still show some interesting examples.There is a universal functor U : dg − cat −→ NChow . We call U ( A ) the noncommutative Chow motive corresponds to A . We write the image of U ( A )in NChow Q as U ( A ) Q . Similar to works of G. Tabuada, the noncommutative Hodge conjectureis compatible with the direct sum decomposition of the noncommutative Chow motives.
Theorem 4.21.
Let A , B and C be smooth and proper dg categories. Suppose there is a directsum decomposition : U ( C ) Q ∼ = U ( A ) Q ⊕ U ( B ) Q , then noncommutative Hodge conjecture holds for C if and only if it holds for A and B .Proof. This follows from the fact that the Hochschild homology, periodic cyclic homology andrational (topological or algebraic) K -theory are all additive invariants, and the correspondingtarget categories are idempotent complete. The proof is similar to Theorem 4.13. (cid:3) Example 4.22.
Suppose we have a semi-orthogonal decomposition : H ( C ) = (cid:104) H ( A ) , H ( B ) (cid:105) ,then U ( C ) ∼ = U ( A ) ⊕ U ( B ).4.2.1. Fractional Calabi–Yau categories.
Theorem 4.23. ( [Kuz19, Theorem 3.5]) Let X be a hypersurface of degree ≤ n + 1 in P n . Thereis a semi-orthogonal decomposition : Perf ( X ) = (cid:104)T ( X ) , O X , · · · , O X ( n − deg ( X )) (cid:105) . T ( X ) is a fractional Calabi–Yau of dimension ( n +1)( deg ( X ) − deg ( X ) . Then U ( X ) ∼ = U ( T dg ( X )) ⊕ U ( k ) ⊕ · · · ⊕ U ( k ) . Therefore, Hodge conjecture of X ⇔ Noncommutative Hodge conjecture of T dg ( X ) . Twisted scheme.
Definition 4.24.
Let X be a scheme with structure sheaf O X . A is a sheaf of Azumaya algebraover X . We call the derived category of perfect A module Perf ( X , A ) the twisted scheme. Theorem 4.25.
Noncommutative Hodge conjecture for
Per dg ( X , A ) ⇔ Noncommutative Hodgeconjecture for
Per dg ( X ) .Proof. According to [TVdB15, Theorem 2.1], U ( Per dg ( X , A )) Q ∼ = U ( Per dg ( X )) Q . Thus, by The-orem 4.21, the statement follows. (cid:3) Cubic fourfold containing a plane.
Example 4.26.
Let X be a cubic fourfold containing a plane. There is a semi-orthogonaldecomposition [Kuz10, Theorem 4.3] Perf ( X ) = (cid:104) Perf ( S , A ) , O X , O X (1) , O X (2) (cid:105) . S is a K surface, and A is a sheaf of Azumaya algebra over S . Since the noncommutative Hodgeconjecture is true for Per dg ( S , A ) which is unique enhanced, hence the Hodge conjecture is truefor X . ONCOMMUTATIVE HODGE CONJECTURE 19
Quadratic fibration.
Example 4.27.
Let f : X −→ S be a smooth quadratic fibration, for example, the smoothquadric in relative projective space P nS . There is a semi-orthogonal decomposition Perf ( X ) = (cid:104) Perf ( S , Cl ) , Perf ( S ) , · · · , Perf ( S ) (cid:105) . Cl is a sheaf of Azumaya algebra over S if the dimension n of the fiber of f is odd [Kuz05].Thus, the Hodge conjecture of X ⇔ S . Moreover, if dim S ≤
3, the Hodge conjecture for X istrue.4.2.5. HP duality.
Let X be a projective smooth variety with morphism f : X −→ P ( V ). Set O X (1) = f ∗ O P ( V ) .Assume there is a SOD D b ( X ) = (cid:104)A , A (1) , · · · , A m − ( m ) (cid:105) where A m − ⊂ · · · ⊂ A ⊂ A . Define H := X × P ( V ) Q , where Q is the incidence quadric in P ( V ) × P ( V ∗ ). Then, there is a SODD b ( H ) = (cid:104)L , A , P ( V ∗ ) (1) , · · · , A m − , P ( V ∗ ) ( m − (cid:105) . Projective smooth variety Y with morphism g : Y −→ P ( V ∗ ) is called homological projectivedual of X if there is an object E ∈ D b ( H ⊗ P ( V ∗ ) Y ) which induces an equivalence from D b ( Y )into L .We refer to [Kuz15, Section 2.3] or Kuznetsov’s original paper [Kuz07]. Let ( Y , g ) be a HP dual of ( X , f ), then1. There is a SOD D b ( Y ) = (cid:104)B n − (1 − n ) , · · · , B ( − , B (cid:105) where B n − ⊂ · · · ⊂ B ⊂ B . Moreover A ∼ = B via Fourier-Mukai functor.2. (Symmetry) ( X , f ) is a HP dual of ( Y , g ).3. For any subspace L ⊂ V ∗ , define X L = X × P ( V ) P ( L ⊥ ) and Y L = Y × P ( V ∗ ) P ( L ). If we assume thatthey have the expected dimension, dim X L = dim X − dim L , dim Y L = dim Y − (dim V − dim L ),and write dim L = r , dim V = N , then there are SOD such that L X , L ∼ = L Y , L . D b ( X L ) = (cid:104)L X , L , A r ( r ) , · · · , A m − ( m − (cid:105) . D b ( Y L ) = (cid:104)B n − (1 − n ) , · · · , B N − r ( r − N ) , L Y , L (cid:105) . Theorem 4.28.
We write
Hodge ( • ) if the (noncommutative) Hodge conjecture is true for va-rieties (smooth and proper dg categories). Then, Hodge ( X ) ⇔ Hodge ( A ) ⇔ Hodge ( B ) ⇔ Hodge ( Y ) . If we assume Hodge ( X ) , then Hodge ( X L ) ⇔ Hodge ( Y L ) .Proof. The midterm equivalence
Hodge ( A ) ⇔ Hodge ( B ) is because A ∼ = B via a Fourier-Mukai functor, and then there is an isomorphism of natural dg lifting A dg , ∼ = B dg , in Hmo ,see a proof in [BT14, Section 9]. Since L X , L ∼ = L Y , L via Fourier-Mukai functor, the statement Hodge ( X L ) ⇔ Hodge ( Y L ) follows from the same argument. (cid:3) Remark . The
HPD can be generalized to the noncommutative version, see the discussion in[Kuz15, Section 3.4] or the paper by Alexander Perry, “Noncommutative homological projectiveduality” [Per19].
Example 4.30.
One of the nontrivial examples of the Homological projective duality comesfrom the Grassmannian-Pfaffian duality. Let W be a dimension n vector space, X = Gr (2 , W ) theGrassmannian of 2-dimensional sub-vector spaces of W . Consider the projective space P ( ∧ W ∗ ),there is a natural filtration called the Pfaffian filtration : Pf (2 , W ∗ ) ⊂ Pf (4 , W ∗ ) · · · ⊂ P ( ∧ W ∗ ). Pf (2 k , W ∗ ) = { ω ∈ P ( ∧ W ∗ ) | rank ( ω ) ≤ k } The intermediate Pfaffians are no longer smooth but with singularities. The singularity of Pf (2 k , W ∗ ) is Pf (2 k − , W ∗ ). Classically, it was known that Y = Pf (2 (cid:98) n (cid:99) − , W ∗ ) is the classicalprojective dual of X = Gr (2 , W ) via the Pl¨ucker embedding. For n ≤
7, the noncommutativecategorical resolution of Pf (2 (cid:98) n (cid:99)− , W ∗ ) is the homological projective dual of Gr (2 , W ). However,it was not known for the cases n ≥
8. The interested reader can refer to a survey [Kuz15, Section4.4, Conjecture 4.4] or Kuznetsov’s original paper [Kuz06].The known nontrivial Grassmannian-Pfaffian duality are the cases n = 6 ,
7. In these cases,Hodge conjecture is true for X since it has a full exceptional collection, then the noncommutativeHodge conjecture is true for the noncommutative categorical resolution of the Pfaffians. However,the Hodge conjecture is trivial for the noncommutative category since it automatically has fullexceptional collections, or the geometric resolution of the Pfaffians are of the form P Gr (2 , W ) ( E )[Kuz06, Section 4] for some vector bundle E . It has a full exceptional collection too.We expect to obtain duality of the Hodge conjecture for X L and Y L when they are smooth,and have the expected dimension. According to the Lefschetz hyperplane theorem, there is acommutative diagram for i ≤ dim X L − CH i ( X L ) Q (cid:47) (cid:47) H i ( X L , Q ) CH i ( Gr (2 , W )) Q ∼ = (cid:47) (cid:47) (cid:79) (cid:79) H i ( Gr (2 , W ) , Q ) ∼ = (cid:79) (cid:79) The Hodge conjecture is true for weight less than dim X L . By the hard Lefschetz isomorphism,it is still true for weight greater than dim X L . Thus, if dim X L is odd, the Hodge conjecture for X L is true. I . n = 6, dim X L = 8 − dim L , dim Y L = dim L −
2. When dim L = 7, Y L = Pf (4 , ∩ P is a cubic5-fold, the Hodge conjecture is true. When dim L = 6, the expected dimension of X L is 2 whilethe expected dimension of Y L is 4. This is the duality between Pfaffian cubic fourfold and the K surface [Kuz06]. When dim L = 5, dim X L = dim Y L = 3, the Hodge conjecture is true by di-mension reason. When dim L = 4, Y L = Pf (4 , ∩ P is a cubic surface. Then X L = Gr (2 , ∩ P has a full exceptional collection. X L is a rational Fano 4-fold [Fei, Section 2.2, Theorem 2.2.1]. ONCOMMUTATIVE HODGE CONJECTURE 21
Hence, the Hodge conjecture is true for X L by weak factorization theorem [AKMW99, Theo-rem 0.1.1]. When dim L = 3, dim X L = 5, the Hodge conjecture is true for X L . We obtain a table.dim L dim X L dim Y L classically3 5 1 Known4 4 2 Known, X L is a rational Fano 4-fold5 3 3 Known, they are 3-fold6 2 4 Known, Y L is a cubic 4-fold7 1 5 Known, Y L is a cubic 5-fold II . n = 7, dim X L = 10 − dim L , dim Y L = dim L −
4. For example, take dim L = 7. Theexpected dimension of X L and Y L are both 3. The Hodge conjecture is true for them by dimensionreason. When dim L = 5, dim X L = 5, the Hodge conjecture is true for X L . We obtain a table.dim L dim X L dim Y L classically5 5 1 Known, since dimension of X L is odd6 4 27 3 3 Known by dimension reason8 2 49 1 5 Remark . We leave the blanks in the table since the author doesn’t know if the Hodgeconjecture was previously known in these cases. It would be interesting to understand whetherthe noncommutative methods give more than was known by classical methods in these cases.
III . For n ≥
8, the
HPD is not constructed. However, when n = 10, there is an interestingpicture inspired by the Mirror Symmetry which was constructed by E. Segal and RP. Thomas[ST14, Theorem A].Let L be a 5-dimensional subspace of ∧ W ∗ , L ⊥ ⊂ ∧ W . Write X = Gr (2 , ⊂ P and Y = Pf (8 , ⊂ P ; X L = P ( L ⊥ ) ∩ X , Y L = P ( L ) ∩ Y . We choose general linear subspace L suchthat both X L and Y L are smooth. In particular, Y L is quintic 3-fold and X L is a Fano 11-fold.According to E. Segal and RP. Thomas [ST14, Theorem A], there is a fully faithful embedding D b ( Y L ) (cid:44) → D b ( X L ) . Let A be the exceptional collections { Sym S , Sym S , S , O} of D b ( Gr (2 , S is thetautological bundle on Gr (2 , D b ( X L ) by techniquesin [Kuz06]. Then, let (cid:104)A , A (1) , · · · , A (4) (cid:105) be an exceptional collection in D b ( X L ). They are rightorthogonal to the above embedding of D b ( Y L ), see description in [ST14, Remark 3.8]. TheHochschild homology HH ( X L ) ∼ = C and HH ( Y L ) ∼ = C . Therefore, 0 th Hochschild homologyof the right orthogonal complement of (cid:104)A , A (1) , · · · , A (4) , D b ( Y L ) (cid:105) is trivial. Thus, the Hodgeconjecture for X L follows from the additive theory. Inspired by the example above, we expect that even though we do not have
HPD , the dualityof the Hodge conjecture between linear section of the dual varieties can be proved by classicalmethods.
Conjecture 4.32.
Let X ⊂ P ( V ) be a projective smooth variety. Suppose the Hodge conjectureis true for X . Let Y ⊂ P ( V ∗ ) be the projective dual of X ⊂ P ( V ) . Choose a linear subspace L ⊂ V ∗ . Suppose the linear sections X L = X ∩ P ( L ⊥ ) and Y L = Y ∩ P ( L ) are both of expecteddimension and smooth. Then, the Hodge conjecture of X L is equivalent to the Hodge conjectureof Y L . Connective dg algebras.
In this subsection, we prove that the noncommutative Hodgeconjecture is true for the connective dg algebras. Definition 4.33. A is called a connective dg algebra if H i ( A ) = 0 for i > Theorem 4.34. If A is a smooth and proper connective dg algebra, the noncommutative Hodgeconjecture is true for A .Proof. According to recent work of Theo Raedschelders and Greg Stevenson [RS20, Corollary4.3, Theorem 4.6], U ( A ) Q ∼ = U ( H ( A ) / Jac ( H ( A ))) Q ∼ = ⊕U ( C ) Q . Hence, the noncommutativeHodge conjecture is true for connective dg algebras. In particular, it is true for the propersmooth algebras (concentrated in degree 0). (cid:3) We provide another proof which involves more calculation for smooth and proper algebras.Clearly, proper algebras are finite dimensional algebras. Due to R. Rouquier [Rou08, section7],
Pdim A e ( A ) = Pdim ( A ), smooth algebras are finite global dimensional algebras. Considerthe acyclic quiver Q with finitely many vertices. Let A := kQ / I be the quiver algebra withrelations, where kQ is the path algebra of Q . Then, A is a smooth and proper algebra. Thenoncommutative Hodge conjecture is true for A . Theorem 4.35.
Let A = kQ / I . Consider natural Chern character map Ch : K ( A ) −→ HH ( A ) . Then,
Im Ch Q ⊗ C = HH ( A ) . In particular, the noncommutative Hodge conjecture is true for A .Proof. Firstly, for the algebra A , HH ( A ) ∼ = A / [ A , A ] ∼ = k (cid:104) e , e , · · · , e n (cid:105) where e i is vertex ofthe quiver Q . We write S i = A · e i which is considered as a left A module, [ S i ] ∈ K ( A ). Weprove that Ch ([ S i ]) = e i . According to the paper of McCarthy, “Cyclic homology of an exactcategory” [McC94, section 2], there is an natural identification of Hochschild homology : (cid:77) n Hom A ( A , A ) ⊗ · · · ⊗ Hom A ( A , A ) −→ (cid:77) X , Y , n Hom A ( X , E ) ⊗ · · · ⊗ Hom A ( E n , Y ) . It is a natural quasi-isomorphism, the left hand side is exactly the bar complexes of A . X and Y are both projective left A modules. Under this identification, the image of the Chern character ONCOMMUTATIVE HODGE CONJECTURE 23 of object [ P ] that is projective A module is the homology class of id P in the right hand sidecomplex. Consider the local picture : Bar : Hom A ( S i , A ) ⊗ Hom A ( A , S i ) −→ Hom A ( S i , S i ) ⊕ Hom A ( A , A ) . Let f ∈ Hom ( S i , A ) be the natural inclusion, e i ∈ Hom A ( A , S i ) be the multiplication by e i . Then Bar ( f ⊗ e i ) = id S i − e i . Therefore, [ e i ] = [ id S i ] in HH ( Proj A ). Hence Ch ([ S i ]) = [ e i ]. Finally, Im Ch Q ⊗ C = HH ( A ). Since Im Ch Q ⊂ HH , Q ( A ), therefore Im Ch Q = HH , Q ( A ). (cid:3) A finite dimensional algebra A is (derived) Morita equivalent to an elementary algebra whichis isomorphic to kQ / I for some quiver Q . Clearly kQ / I is smooth and proper if A is smooth andproper. Then according to Theorem 4.35, the Hodge conjecture is true for any smooth and finitedimensional algebra A . Remark . A. Perry pointed out to the author that if A is a smooth and proper algebra, Perf ( A ) can be an admissible subcategory of the Perf ( X ) which admits full exceptional collec-tions for some smooth and projective varieties X by Orlov [Orl16, section 5.1]. Therefore, thenoncommutative Hodge conjecture of A is true.Classically, given any projective smooth variety X , there is a compact generator E of D Qch ( X ).Write E again after the resolution to an injective complex. Denote A = Hom dg ( E , E ), then thereis an equivalence D per ( A ) ∼ = Perf ( X ) and chain of derived Morita equivalences between Per dg ( A )and Per dg ( X ). Thus, commutative Hodge for X ⇔ Noncommutative Hodge for dg algebra A .By the results above, suppose A is a smooth and finite dimensional algebra, then the Hodgeconjecture of A is true. Definition 4.37.
Let X be a projective smooth variety. An object T is a called tilting sheaf ifthe following property holds :(1) T classical generates D b ( X ).(2) A := Hom ( T , T ) is of finite global dimension.(3) Ext k ( T , T ) = 0 for k > Example 4.38. (Van den Bergh [Van02, theorem A]) Suppose there is a projective morphism f : X −→ Y = Spec R between noetherian schemes. Furthermore, Rf ∗ ( O X ) ∼ = O Y and the fibersare at most one dimensional. Then there is a tilting bundle E of X . Corollary 4.39.
Suppose X admits a tilting sheaf, then Hodge conjecture for X is true.Proof. Let T be a tilting sheaf of X . We write T again after resolution to an injective complex.Define A := Hom dg ( T , T ), which is quasi-isomorphic (hence derived Morita equivalent) to asmooth and finite dimensional algebra. Thus, the Hodge conjecture for X is true. (cid:3) References [AH62] M.F. Atiyah and F. Hirzebruch. Analytic cycles on complex manifolds.
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Xun Lin, Yau mathematical science center, Tsinghua university, Beijing China.
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