Differential forms and invariants of complex manifolds

2.1 Spaces

In the following, X will always denote a topological space, which will subsequently be assumed to carry more and more extra structure. Although it is unnecessary in many places, for simplicity of exposition, we will generally assume X to be simply connected and of finite type.

A basic invariant of a topological space is its singular cohomology ring with rational coefficients H sing * ( X ; Q ) , or, even more basic, the sequence of Betti numbers b k ( X ) ≔ dim H sing k ( X ; Q ) . The answer to the numerical realization question – which sequences ( b k ) k ≥ 0 of non-negative integers arise as Betti numbers of spaces? – is clearly: all of them, as one may see by considering wedges of spheres. Structurally, the question becomes: Which graded-commutative graded Q -algebras arise as the cohomology ring of spaces?

Again, the answer is: all of them. To give a precise statement, let us introduce some notation. We abbreviate graded-commutative non-negatively graded Q -algebra as rational cga. Furthermore, we call a rational cga simply connected if H 0 = Q and H 1 = 0 .

Theorem 2.1 can be deduced as a consequence of the deeper results of rational homotopy theory [76,88]. Before we state them, let us explain some context.

Recall that singular cohomology is defined to be cohomology of the complex of singular cochains C * ( X , Q ) . The multiplication is defined on the cochain level, but is not commutative there. On the other hand, if X is a manifold, one can associate with it the complex of R -valued differential forms A X , R ∗ , which does have a graded-commutative multiplication. It thus has the structure of a graded-commutative differential graded R -algebra, which will be abbreviated as real cdga. The cohomology of A X , R , the de Rham cohomology H d R ∗ ( X ; R ) ≔ H ( A X , R , d ) , is thus a real cga, and the de Rham theorem gives an isomorphism of cga’s: H sing * ( X ; R ) ≅ H d R * ( X ; R ) .

It is an insight of Sullivan [88], based on earlier work of Cartan, Quillen, Thom, Whitney and others [89], that one can extend the method of differential forms to rational coefficients and general spaces. Sullivan functorially (cf. [16]) attaches a rational cdga of A P L ( X ) to any space X , such that H sing ∗ ( X ; Q ) ≅ H * ( A P L ( X ) , d ) . Roughly speaking, A P L ( X ) consists of polynomial forms on every simplex mapping into X . Conversely, there is a “geometric realization” functor, attaching a space ⟨ A ⟩ to any rational cdga A . A map of two simply connected spaces is a rational homotopy equivalence if it induces an isomorphism in cohomology with rational coefficients. Similarly, in the category of rational cdga’s, one has a natural notion of quasi-isomorphism, namely, a map of cdga’s inducing an isomorphism in cohomology.

Theorem 2.1 then follows from the following more general result.

In a more technical language, Sullivan’s A P L -forms and the geometric realization induce an equivalence of categories between the full subcategories of the category of spaces localized at rational homotopy equivalences, resp. the category of rational cdga’s localized at quasi-isomorphisms, which are spanned by those objects that are of finite type and (cohomologically) simply connected.

In view of this theorem, it becomes a natural question to find an optimal representative for the quasi-isomorphism class of a rational cdga A . We recall two ways of doing so.

One way is by Sullivan’s minimal models: for any connected cdga A , there exists a minimal cofibrant model, i.e., a quasi-isomorphism from another cdga M A → A such that M A = Λ V and V has a well ordered basis ( v i ) with v i < v j for deg v i < deg v j such that d v i is a linear combination of products of two or more lower-order generators. Every self-homotopy equivalence of such an M A is an actual isomorphism, and so the homotopy class of A can be identified with the isomorphism class of M A .

Another way is via higher operations and homotopy transfer: to motivate this, recall the classical triple Massey products [61,62]. For any three pure degree classes [ a ] , [ b ] , [ c ] ∈ H ∗ ( A ) s.t. a b = d x , b c = d y , one obtains a new class ⟨ a , b , c ⟩ ≔ [ a y − ( − 1 ) ∣ a ∣ x c ] ∈ H ∗ ( A ) , well defined up to the ideal generated by [ a ] and [ c ] and depending only on the cohomology classes of a , b , and c . This can be considered as a partially defined ternary operation H ∗ ( A ) ⊗ 3 → H ∗ ( A ) . The theory of homotopy transfer gives a slightly more involved way of equipping H ( A ) with everywhere defined n -ary operations m k for k ≥ 1 (with m 1 = 0 and m 2 being the usual product), satisfying certain compatibility conditions making it into a so-called C ∞ -algebra. Any cdga with m 1 = d , m 2 the product and all higher operations trivial is in particular a C ∞ -algebra. Then, the homotopy class of A can be identified with the isomorphism class of the C ∞ -algebra ( H ∗ ( A ) , 0 , m 2 , m 3 , m 4 , … ) . Let us summarize this discussion in the following theorem:

2.2 Compact manifolds

Let us now consider compact smooth manifold of dimension n , without boundary. Again, one can ask the realization questions: which collections of non-negative integers are Betti numbers of compact smooth n -folds? Which rational cga’s are cohomology algebras of compact smooth n -folds?

One obvious obstruction comes from Poincaré duality: for any compact smooth n -fold, the n -dimensional cohomology is one-dimensional H n ( X ; Q ) ≅ Q and multiplication yields a perfect pairing H k ( X ; Q ) × H n − k ( X , Q ) → H n ( X ; Q ) ≅ Q . Any finite-dimensional cga satisfying these properties will be called a (rational) PD-algebra of formal dimension n . For such algebras, the linear relation b k = b n − k needs to hold. Furthermore, in even dimensions n = 2 k , the induced pairing in middle cohomology is ( − 1 ) k symmetric. In particular, on all PD-algebras of formal dimension n with n ≡ 2 mod ( 4 ) , the congruence b n ⁄ 2 ≡ 0 mod ( 2 ) holds.

Not every collection of non-negative numbers b 0 , … , b n with b 0 = 1 satisfying the aforementioned duality constraints can be obtained as the Betti numbers of a connected compact n -fold as we will see shortly. However, we have a positive answer to the linearized numerical realization problem:

If one wants to realize not just Betti numbers, but cohomology rings, there are, in dimensions n = 4 k , more subtle restrictions coming from Poincaré duality and characteristic classes. First, since Poincaré duality holds integrally, the nondegenerate symmetric pairing in middle degree needs to be rationally equivalent to one of the form ∑ ± y i 2 .

Next, recall that for any (stably) complex vector bundle V on X , one has a classifying map f V : X → B U and the cohomology of B U is a polynomial algebra on generators c i situated in degrees 2 i . The Chern classes of the vector bundle V are then the classes c i ( V ) ≔ f V * c i ∈ H 2 i ( X ; Z ) . For a real vector bundle W , one defines its Pontryagin classes by p i ( W ) ≔ ( − 1 ) i c 2 i ( W ⊗ C ) ∈ H 4 i ( X ; Z ) . In particular, for any smooth manifold X , one writes p i ( X ) ≔ p i ( T X ) . If X is compact with fundamental class [ X ] and of dimension n divisible by 4, one thus obtains for every partition τ of n 4 , say n 4 = ∑ i = 1 n 4 k i ⋅ i , the Pontryagin number

By Hirzebruch’s signature theorem, in every dimension divisible by 4, the signature σ of the middle degree pairing can be computed as a universal linear expression with rational coefficients in the Pontryagin numbers. For example, dropping evaluation on the fundamental class in the notation (taken from [68]):

This shows that there are necessary congruences for the Pontryagin classes, and hence, not all sequences of Betti numbers satisfying the necessary linear relations are realized by manifolds. For example:^[1] if there were a compact 12-fold with Betti numbers b 0 = b 6 = b 12 = 1 and 0 else, it would follow that 62 945 ∈ Z , which is absurd. With this in mind, one has the following version of Theorem 2.1 for compact smooth manifolds:

In particular, in all dimensions ≥ 5 , whether a rational homotopy type contains a simply connected smooth closed manifold only depends on the rational cohomology ring.

In fact, one can even solve the classification problem up to finite ambiguity [88]: roughly speaking, the statement is that if one takes into account integral information, one may refine the aforementioned to a statement roughly saying that the map

induces, in dimension ≥ 5 , a finite-to-one map on diffeomorphism classes.

2.3 Almost complex manifolds

A manifold X together with an endomorphism J of the tangent bundle squaring to − Id is called an almost complex manifold. By definition, the tangent bundle is a complex vector bundle, and so the real dimension of the manifold is even, say 2 n . Furthermore, one has canonically defined Chern classes c i ( X ) ≔ c i ( X , J ) ≔ c i ( T X ) ∈ H sing 2 i ( X ; Z ) . If X is, in addition, compact, one can thus form Chern numbers

for any partition τ of n , written as n = k 1 + k 2 ⋅ 2 + … + k n ⋅ n .

Again, these numbers have to satisfy certain universal congruences on all almost complex manifolds of a given dimension, characterizing the image of the map Ω U → H ∗ ( B U , Q ) , resp., in the case c 1 = 0 , the image of Ω S U → H ∗ ( B S U , Q ) . Roughly speaking, they are all integrality conditions that come out of the Atiyah-Singer index theorem. A precise statement is as follows: we call these the Stong congruences following [64]:

The first set of congruences in low dimensions are as follows [41,64]:

Then, one has the following almost complex version of the Sullivan-Barge theorem, which, after the formulation and proof had been left to the reader in [88], has been worked out fully in [64]:

We note that, as in the smooth case, the realizability of a simply connected rational homotopy type depends only on the cohomology ring. Unlike the smooth case, the aforementioned data combined with integral information do not finitely determine the almost complex manifold up to pseudoholomorphic equivalence. For instance, the action of the diffeomorphism group of a given real manifold of real dimension ≥ 4 on the space of almost complex structures has an infinite-dimensional orbit space.

3.1 Zoo of cohomology theories

On any almost complex manifold ( X , J ) , the cdga of C -differential forms A X = A X , R ⊗ C carries a bigrading A X ≔ ⊕ A X p , q . It is induced by the splitting of the bundle of C -valued 1-forms A X 1 = A X 1,0 ⊕ A X 0,1 into i - and − i -eigenbundles for the endomorphism J . With respect to this bigrading, the differential splits into components d = ∂ + ∂ ¯ of bidegree ( 1,0 ) and ( 0,1 ) , if and only if J is integrable. We restrict to this case from now on. By the equation d 2 = 0 , one has ∂ 2 = ∂ ¯ 2 = ∂ ∂ ¯ + ∂ ¯ ∂ = 0 . In other words, the complex of forms on a complex manifold is the total complex underlying a bicomplex. The multiplication is compatible with the bigrading, so that A X carries the structure of a graded-commutative bidifferential, bigraded algebra, which we will abbreviate as cbba. The antilinear conjugation action on A X interchanges A X p , q with A X q , p and ∂ with ∂ ¯ . A cbba with such an antilinear isomorphism will be called an R -cbba. A smooth map f : X → Y of complex manifolds is holomorphic iff it respects the bigrading, i.e., f * ( A Y p , q ) ⊆ A X p , q .

From A X , one can build various holomorphic invariants of X . We write down the following definitions only for A X , but they are meaningful for any bicomplex (or R -cbba), and we will use this in the next sections.

The most well-known holomorphic cohomological invariant is perhaps Dolbeault cohomology [28], controlling existence and uniqueness of the ∂ ¯ -equation x = ∂ ¯ y . It is defined as the graded-commutative bigraded algebra, obtained as the cohomology of the columns of the bicomplex ( A X , ∂ , ∂ ¯ ) :

It can be identified with the sheaf cohomology of the sheaf of holomorphic functions H ∂ ¯ p , q ( X ) = H q ( X , Ω p ) .

Next, note that on the de Rham cohomology with complex coefficients H d R ( X ) ≔ H d R ( X ; C ) , one has a multiplicative filtration, induced from the column filtration on A X , i.e.,

and another filtration F ¯ , computed analogously from the rows of A X , which is conjugate to F . These filtrations generally depend on the complex structure.

As for any bicomplex, there is a spectral sequence, the Frölicher spectral sequence [32],

which does not degenerate in general, and so the higher pages E r p , q ( X ) give additional invariants of the bihomolomorphism type of X .

On any compact X , the vector spaces H ∂ ¯ p , q ( X ) (and hence all later pages) are finite-dimensional and satisfy Serre duality, i.e., for any connected compact X of dimension n , one has H ∂ ¯ n , n ( X ) ≅ C and the multiplication

is a perfect pairing. The vector spaces H ∂ ¯ ( X ) are generally not conjugation invariant; rather, conjugation swaps H ∂ ¯ p , q ( X ) with H ∂ q , p ( X ) , where the latter vector space is defined analogously but exchanging ∂ ¯ by ∂ .

Another natural bigraded cohomology algebra, which is conjugation invariant, is the Bott-Chern cohomology

It was defined by Bott and Chern in [15] studying generalizations of Nevanlinna theory and has found many other uses since.^[4] It is an initial cohomology in the sense that it has natural maps to H ∂ ¯ , H ∂ , H d R , and all other cohomology theories discussed later on. For any holomorphic vector bundle V , the Chern classes can naturally be lifted to Bott-Chern cohomology in the following sense: there are classes c i , BC ( V ) ∈ H BC i , i ( X ) , compatible with pullback by holomorphic maps, such that under the natural map H BC ( X ) → H d R ( X ) , they coincide with the image of c i ( V ) under the map H sing ( X ; Z ) → H d R ( X ) .

Bott-Chern cohomology generally does not satisfy a Serre-type duality. Rather, it pairs nondegenerately with Aeppli cohomology [2,3]

These vector spaces are also stable under conjugation and are a “terminal” cohomology in the sense that they receive natural maps from H ∂ ¯ , H ∂ , H d R , and all other cohomology theories discussed later on.

Bott-Chern and Aeppli cohomology groups arise as hypercohomology groups in certain degrees of the complexes of sheaves [11,12,26,78]

which are closely related to the complexes computing the Deligne cohomology groups [10]. Other cohomology groups of these complexes appear, for example, in the classification of holomorphic string algebroids [33], or holomorphic higher operations [65]. Some general properties of these cohomologies are established in [85] and [70].

One can continue this list with more cohomology theories, arising naturally from certain geometric contexts. For example, if one extends J as an algebra-automorphism to the vector space A X of all forms and sets d c ≔ J − 1 d J , one may obtain new complexes ( ker d c , d ) and ( A ⁄ im d c , d ) and corresponding singly graded cohomology theories

which are also finite-dimensional and in duality, [81]. Then, there are the groups defined by Varouchas [98], and higher-page analogs of those and Aeppli and Bott-Chern cohomology [73–75] and possibly more which we have not mentioned.

All of the aforementioned cohomologies are finite-dimensional, and so in addition to the Betti and Hodge numbers b k ( X ) ≔ dim H d R k ( X ) and h ∂ ¯ p , q ( X ) ≔ dim H ∂ ¯ p , q ( X ) , one obtains a vast collection of numerical invariants from the dimensions of the respective cohomologies: h BC p , q ≔ dim H BC p , q ( X ) , h A p , q ≔ H A p , q ( X ) , etc. One also has a bigraded refinement of the Betti numbers using the filtrations, setting b k p , q ( X ) ≔ dim gr F p gr F ¯ q H d R k ( X ) .

These numbers satisfy certain universal relations. For example, on all compact complex manifolds of a given dimension n , one has linear relations induced by duality and the real structure, such as

but also other linear relations such as

There is then the following natural question, which we will come back to later:

We note that in dimension n = 2 , one also has a nonlinear polynomial relation

but it is unknown whether universal nonlinear polynomial relations exist in higher dimensions.^[5]

Furthermore, there are universally valid inequalities. Let us denote the total dimension of any of the aforementioned cohomology theories by a letter without sub- or superscripts indicating a degree, so b ≔ ∑ k = 0 2 n b k , h BC ≔ ∑ p , q = 0 n h BC p , q , e i ≔ ∑ p , q = 0 n dim E i p , q , etc.

It is an interesting question to find manifolds on which equality holds. For example, one has h ker d c = h ∂ ¯ for Vaisman manifolds, all complex surfaces, and complex parallelizable manifolds with solvable Lie algebra of holomorphic vector fields [48,73,81]. In the first two cases, in addition, h ∂ ¯ = b , while in the last case, in addition, h BC = h ker d c . There are also simply connected examples for both cases [47]. There is, however, by no means any known kind of classification of manifolds satisfying a particular case of equality. In particular, it is unknown whether there are any manifolds satisfying equality in 3.2 for r ≥ 2 , but not for r − 1 .

3.2 Compact Kähler case

An important class of compact manifolds are those admitting a Kähler metric. This includes all complex submanifolds of projective space.

In this case, the cohomological story greatly simplifies: roughly speaking, all cohomologies are determined by Dolbeault cohomology. More precisely:

A complex manifold X for which A X satisfies the aforementioned conditions is called a ∂ ∂ ¯ -manifold. Compact Kähler manifolds are ∂ ∂ ¯ -manifolds, but the converse is not true. A broader class is, for example, given by those manifolds bimeromorphic to compact Kähler manifolds, i.e., Fujiki’s class C , but also these do not exhaust all ∂ ∂ ¯ -manifolds (see, e.g., [31,47,59]).

One readily checks that all cohomologies introduced earlier are compatible with direct sums, vanish on squares, and are one-dimensional on dots. Thus, as soon as a cohomology allows us to reconstruct the information on the position of the dots, e.g., Dolbeault cohomology, or de Rham cohomology with its filtrations, it determines all other ones on ∂ ∂ ¯ -manifolds. In such cases, the answer to Question 3.1 is given by the following theorem:

It is stated here for compact Kähler manifolds as in [56], but from the proof in [56], it is clear that it remains true when considering the smaller (resp. larger), classes of projective, resp., ∂ ∂ ¯ -manifolds. The result (for Hodge numbers) was extended to polynomial relations in Paulsen and Schreieder [69] (see also [96,97] for characteristic p versions of these results).

3.3 Hirzebruch’s question

Given a biholomorphism X → Y of complex manifolds, it induces an isomorphism H ( Y ) → H ( X ) for H any of the cohomologies introduced earlier. In particular, the cohomology dimensions are invariants of the biholomorphism class of a manifold.

On the other hand, some linear combinations of these cohomology dimensions are even topological invariants. This is most obvious for h ∂ ¯ 0,0 , which counts connected components. By a spectral sequence argument, one sees that one can compute the Euler characteristic as χ = ∑ p , q ( − 1 ) p + q h ∂ ¯ p , q . More deeply, by the Hodge index theorem, for general compact complex manifolds, σ = ∑ p , q ( − 1 ) q h ∂ ¯ p , q ( X ) holds and so the combination of Hodge numbers on the right is invariant under orientation preserving homeomorphisms. In view of such relations, Hirzebruch asked the following question in 1954 [40]:

Problem 31. Are the h p , q and the Chern characteristic numbers of an algebraic variety V n topological invariants of V n ? If not, determine all those linear combinations of the h p , q and the Chern characteristic numbers which are topological invariants.

There are several remarks in order: first, one may understand “topological invariant” in at least four a priori different ways, namely, invariant under homeomorphisms or diffeomorphisms of the underlying manifold, which may or may not be required to preserve the orientation. Next, as mentioned earlier, there are universal linear relations between the Hodge and Chern numbers of compact complex manifolds in a given dimension, described for compact Kähler manifolds in Theorem 3.4, and one should therefore answer the question modulo these relations.

Some extreme cases of the problem were quickly resolved: for example, in 1958, Hirzebruch and Borel exhibited an example of diffeomorphic 5-folds with distinct c 1 5 [14], giving a negative answer to the initial question. A determination of topologically invariant Chern numbers of mere almost complex manifolds was given in [46].

Much more recently, the problem concerning the Chern numbers only was solved by Kotschick [55,57], following earlier work [53,54], and finally, the original problem for both Hodge and Chern numbers of projective varieties was solved by Kotschick and Schreieder:

It is clear from the proof of this theorem that one can replace projective manifolds by Kähler manifolds or even by ∂ ∂ ¯ -manifolds and the statement remains unaffected.

It is proved in [83] that in the aforementioned formulation, the answer remains the same when relaxing to all compact complex manifolds of a given dimension n ≥ 3 , only that the universal relations are different (e.g., h ∂ ¯ p , q ≠ h ∂ ¯ q , p in general).^[6] We will furthermore see that there are more general questions to be asked in the general compact complex realm, taking into account all cohomological invariants instead of just the Hodge numbers.

3.4 Common motives

All the previously discussed cohomologies behave similarly in many geometric situations, e.g., they can be computed as one would expect from the de Rham case for projective bundles and blow-ups [80]. However, they do not all satisfy a straightforward Künneth formula (see, e.g., [20,83,84]).

Naturally, one expects a more fundamental invariant lurking behind all these cohomology groups. In some sense, there is an obvious answer: they are all computed from the bicomplex of forms ( A X , ∂ , ∂ ¯ ) . In fact, one readily checks that they are well defined for a general bicomplex ( A , ∂ , ∂ ¯ ) and that they are naturally compatible with direct sums H ( ⊕ A i ) ≅ ⊕ H ( A i ) . Finally, they all vanish on squares as in Diagram (3.3).

These observations may be seen as a motivation for the following definitions: we denote by BiCo the category of all bicomplexes and by Ho ( BiCo ) the quotient category, where all morphisms that factor over a direct sum of squares are set to zero. This is called the homotopy category, or derived category, of bicomplexes. An additive functor H : BiCo → Ad to some additive category Ad is called a cohomological functor if it factors through this derived category. In other words, it has to commute with finite direct sums and vanish on direct sums of squares. A functor from complex manifolds to Ad is called cohomological if it factors, via X → A X through a cohomological functor. Examples of such functors are of course all the aforementioned cohomology theories and also diagrams of those, such as Diagram (3.4).

Now, the values of cohomological functors on objects A are determined by the isomorphism type of A in Ho ( BiCo ) , and so instead of studying the cohomology theories of a complex manifold one at a time, one should determine this isomorphism type for the bicomplex ( A X , ∂ , ∂ ¯ ) , i.e., determine A X “up to squares.” To do this, one needs a good understanding of what maps of bicomplexes induce isomorphisms in Ho ( BiCo ) and how to represent an isomorphism class in Ho ( BiCo ) , so let us discuss these points next.

We call a map of bicomplexes inducing an isomorphism in Ho ( BiCo ) a bigraded quasi-isomorphism. By definition, it has a quasi-inverse, i.e., a map in the other direction such that the compositions both ways differ from the identity by maps factoring through direct sums of squares. This is the bicomplex analog of a map of complexes invertible up to chain homotopy. One may characterize these maps cohomologically as follows:

Thus, a map of bounded bicomplexes is a bigraded quasi-isomorphism if and only if it is a quasi-isomorphism in row and column cohomology and, using the real structure, a map of complex manifolds f : X → Y induces a bigraded quasi-isomorphism A Y → A X if and only if it induces an isomorphism in Dolbeault cohomology. For both A and B satisfying the ∂ ∂ ¯ -Lemma, a bigraded quasi-isomorphism is the same as a map of bicomplexes, which is also a usual quasi-isomorphism (i.e., it induces an isomorphism in total cohomology). In view of the characterization in terms of Bott-Chern and Aeppli cohomology, which measure the existence and uniqueness of solutions to the ∂ ∂ ¯ -equation x = ∂ ∂ ¯ y , and to avoid confusion with other possible meanings of “bigraded quasi-isomorphism,” we will sometimes also use the name pluripotential quasi-isomorphism to denote the same concept.

To describe the pluripotential quasi-isomorphism type, or even the actual isomorphism type, of a bicomplex, the following theorem is useful:

Here, a zigzag is a bicomplex concentrated in at most two antidiagonals, s.t. all nonzero vector spaces are C , all nonzero maps are the identity, and all nonzero vector spaces are connected by a chain of maps. For example,

The total dimension of a zigzag will be called its length (so the above have length 4 and 3).^[7] Zigzags of length one are called dots, and those of length two are called lines. For every length ≥ 2 , there are, up to shift, two zigzags of that length, and then, there are zigzags that may be infinite in one or both directions.

Given some bicomplex A and a zigzag Z , denote by mult Z ( A ) the number of direct summands isomorphic to Z in a decomposition as in Theorem 3.7. This number does not depend on the chosen decomposition. Thus, any bicomplex A can be written as a direct sum of squares and zigzags A ≅ A s q ⊕ A z i g . The bigraded quasi-isomorphism type (knowing A up to squares) is uniquely determined by the isomorphism type of A z i g , which is again uniquely determined by the collection of numbers mult Z ( A ) for all zigzags A . One can encode this information in a “checkerboard” diagram. For example, suppose A is the direct sum of the two zigzags in Figure (3.5), where we assume the top-left corner of the first to sit in degree (0, 1) and the top-left corner of the second to sit in degree (3, 1). Then, we may depict this as

Now the situation discussed in the previous section has become much more transparent: The value (on objects) of a cohomological functor H is determined by its value on all zigzags. For example, we urge the reader to stop for a moment and convince themselves that the Frölicher spectral sequence(s) degenerates on the first page for every odd zigzag, while for any even zigzag of length 2 r , the de Rham cohomology vanishes and there is a nonzero differential on page r of the row- or column Frölicher spectral sequence. In particular, as a consequence of finite dimensionality of Dolbeault cohomology, the multiplicities of all zigzags (and hence the dimensions of all other cohomological functors discussed) are finite on compact complex manifolds.

Properties of the quasi-isomorphism type of A X will have shadows in the various cohomological functors. For instance, real structure and duality hold on the bicomplex level: for any bicomplex ( A , ∂ 1 , ∂ 2 ) , denote by A ¯ the conjugate bicomplex, which has the same underlying vector space and total differential, but twisted multiplication and bigrading, i.e., A ¯ p , q = A q , p with conjugate C -linear structure. Then, there is an isomorphism

Similarly, denote by D A [ n ] the n th dual bicomplex, i.e., ( D A [ n ] ) p , q = Hom ( A n − p , n − q , C ) with total differential given (up to sign) by precomposition with the differential of A . This is a module over A by precomposition with the multiplication. For any compact complex n -fold X , integration over X yields a canonical closed element ∫ X ∈ D A X [ n ] 0,0 and the composition with the module structure yields a bigraded quasi-isomorphism

Thus, for any cohomological functor, one has canonical isomorphisms

For each particular H , it is an easy exercise to work out an expression for the two terms on the right and one may thus recover all the known results about real structures and duality for the individual cohomological functors.

Any numerical invariant of compact complex manifolds of the form h = dim ∘ H for some cohomological functor H is a sum of the invariants mult Z ( _ ) by additivity of cohomological functors. Consequently, all universal relations between all such numerical invariants h on compact complex manifolds of a given dimension, it is necessary and sufficient to determine all universal relations between the numbers mult Z ( _ ) in a given dimension. The following are all known relations, which follow from classical results (in particular, Serre duality and the cohomological properties of compact complex surfaces). See [82] for a treatment in this language and references.

The following more precise version of Question 3.1 is open, but partial results are obtained in [83]:

Using the first two relations earlier, one may refine the “checkerboard” notation for compact complex manifolds of a given dimension. Namely, for a complex n -dimensional manifold, consider an ( n + 1 ) × ( n + 1 ) board and instead of single zigzag write the entire Z ⁄ 2 × Z ⁄ 2 = ⟨ τ , σ ⟩ orbit on one board. For example, in this notation, the bicomplex of forms of a connected compact complex curve Σ g of genus g looks as follows:

and that of a connected compact complex surface X looks as follows:

where σ = b 2 + − b 2 − denotes the signature and ε is zero for b 1 ( X ) even (Kähler case) and 1 otherwise.

To give a positive answer to Question 3.9, it essentially remains to solve the following construction problems (cf. [83]):

That is, one is looking for a manifold with an indecomposable summand of the following form in the bicomplex of differential forms:

For odd n ≥ 3 , these exist.

That is, one needs to construct manifolds realizing the following direct summands in their bicomplexes, where for readability we omit zigzags determined via the real structure:

For n = 3 , the first case is known to exist and the second one corresponds to a (hypothetical) threefold with a nontrivial differential E 2 0,1 → E 2 2,0 .

3.5 Topological invariants

One notes that the examples in the last section imply that for curves and surfaces, the entire bigraded quasi-isomorphism type is determined by the (oriented) topological manifold underlying X . This is far from true in dimensions ≥ 3 . For instance if one takes X = N ⁄ Γ the quotient of an even-dimensional nilpotent Lie-group modulo a lattice, there are generally many left-invariant complex structures that have distinct cohomological invariants like Hodge numbers, and hence distinct bigraded quasi-isomorphism type, but are all deformation equivalent. LeBrun’s examples [58], [63] show there can even be infinitely complex structures with pairwise distinct Hodge numbers on the same smooth manifold.

This, together with Hirzebruch’s question, begs the following more general question:

For example, the Betti numbers are a sum of the multiplicities of certain odd zigzags. On the other hand, not all multiplicities of odd zigzags are topological invariants as one may see, for example, in small deformations of the Iwasawa manifold [6,66]. In analogy with what happens in the Hodge-case, one may conjecture:

3.6 Bimeromorphism invariants

Apart from topological equivalence (i.e., homeomorphism or diffeomorphism, with or without preserving the orientation), there is another natural notion of equivalence between complex manifold, namely, that of bimeromorphic equivalence. The typical example of a bimeromorphic map is that of a blow-up X ˜ → X in some submanifold Z ⊂ X . By the deep results of [1,102], two manifolds are related by a bimeromorphic map iff they are related by a chain of roofs of blow-ups in smooth centers. Thus, to show that some number, or isomorphism class of algebraic object, is invariant under bimeromorphisms, it is necessary and sufficient to show it is does not change under blow-ups. The bigraded quasi-isomorphism class (and hence every cohomology) of the blow-up X ˜ of a smooth complex submanifold Z ⊆ X of codimension r ≥ 2 can be computed as follows [80]:

Together with relations (R1)–(R5), this implies:

The pictures given here might be helpful when reading [83, $10]. In particular, this theorem recovers the classical results of the bimeromorphic invariance of the Hodge numbers h p , 0 and h 0 , q , but also exhibits further invariants away from the boundary. Naturally, one may then ask:

Again, this question has to be read modulo the (still not completely determined) universal relations. The analogous question for Hodge numbers alone (and also including Chern numbers) has a positive answer as was confirmed by Kotschick and Schreieder [56] in the Kähler case and for general compact complex manifolds in [83]. What is missing for a general resolution is an answer to Question 3.9.

3.7 Multiplicative matters

In the previous discussion, we focused on numerical invariants. There are now many structural realization questions one could ask. For instance, which bigraded rings arise as Dolbeault cohomology or Bott-Chern cohomology rings, which diagrams of the form (3.4) arise from compact complex manifolds? Which rational homotopy types contain compact complex manifolds?

If one asks for realization by compact Kähler or projective manifolds, on the one hand, the whole cohomological situation is determined by Dolbeault cohomology; on the other hand, there is further structure, e.g., coming from the Hard Lefschetz theorem. Known necessary conditions for positive answer to the realizability of cohomology rings have been discussed, e.g., in [99–101].

In the case of general compact complex manifolds, one again faces the situation that there are many distinct cohomologies and corresponding variants of realization questions one could consider. For instance, analog of rational homotopy theory that replace ordinary cohomology by a particular complex cohomology have to some extent been developed in the Dolbeault or Bott-Chern setting (see, e.g., [5,38,67,91]).

As in the additive case, the need for a universal invariant arises. A by now obvious candidate is the bicomplex of forms together with its multiplication, turning it into a graded-commutative, bigraded, bidifferential algebra with real structure ( R -cbba) and satisfying Serre duality, up to (multiplicative, real) bigraded quasi-isomorphism.