Products of Harmonic Forms

and Rational Curves

Daniel Huybrechts

Received: May 8, 2000 Revised: October 10, 2000

Communicated by Thomas Peternell

Abstract. In general, the product of harmonic forms is not harmonic. We study the top exterior power of harmonic two-forms on compact Kähler manifolds. The non-harmonicity in this case is related to the geometry of the manifold and to the existence of rational curves in particular. K3 surfaces and hyperkähler manifolds are discussed in detail.

2000 Mathematics Subject Classification:14E05, 14J32, 32Q25 Keywords and Phrases: Harmonic forms, compact Kähler manifolds, Ricci-flat Kähler forms, rational curves

The product of closed forms is closed again. The analogous statement for harmonic forms, however, fails. A priori, there is no reason why the product of harmonic forms should be harmonic again. This phenomenon was recently studied by Merkulov [8]. He shows that it leads to a natural A_{∞}-structure on a Kähler manifold. In the context of mirror symmetry Polishchuk made use of (a twisted version of) this A_{∞}-structure on elliptic curves to confirm Kontsevich’s homological version of mirror symmetry in this case [9].

In the present paper we show that this failure of harmonicity in fact happens quite frequently. It usually is related to certain geometric properties of the manifold and to the existence of rational curves in particular. In fact, we are only interested in the product of harmonic (1, 1)-forms, as this is the geometric relevant case. We wish to emphasize that the interplay between harmonicity and geometry is by far not completely understood. The results of this paper just seem to indicate that there is in fact a relationship. As the failure of harmonicity is related to the shape of the ‘non-linear Kähler cone’ (cf. 1.2, 2.3), the results of this paper can roughly be phrased by saying that the geometry of the manifold forces the non-linear Kähler cone to be curved. For a deeper understanding of the situation one will have to study the shape of the non-linear Kähler cone or, equivalently, the non-harmonicity of products of harmonic forms. Let us briefly indicate the main results for the special case of compact Ricci-flat Kähler manifolds. For a Kähler class ω ∈ H^{2}(X, ℝ) on such a manifold there exists a unique Ricci-flat Kähler form ω̃ representing it. Let H^{1,1}(ω̃) denote the space of (1, 1)-forms harmonic with respect to ω̃. Of course, for a different Kähler class ω′ and the representing Ricci-flat Kähler form ω̃′ this space might be different. The main technical result (Prop. 2.3) says that H^{1,1}(ω̃) is independent of ω if and only if the top exterior power of any harmonic form α ∈ H^{1,1}(ω̃) is again harmonic. This can be used to interpret the failure of harmonicity of the top exterior power geometrically. Proposition 3.2 asserts that there always exist harmonic (1, 1)-forms with non-harmonic top exterior power, whenever the Kähler cone (or ample cone) does not form a connected component of the (integral) cone of all classes α ∈ H^{1,1}(X, ℝ) with ∫_{X} α^{N} > 0. Note that there are many instances where the Kähler cone is strictly smaller. E.g. this is the case for any Calabi-Yau manifold that is birational to a nonisomorphic Calabi-Yau manifold (Prop. 6.1). In Section 4 we apply the result for K3 surfaces. One finds that on any K3 surfaces containing a rational curve there exists a harmonic (1, 1)-form α such that α^{2} is not harmonic. This can be extended to arbitrary K3 surfaces by using the existence of rational curves on nearby K3 surfaces.

1 Preparations

Let X be a compact Kähler manifold. Then K_{X} ⊂ H^{1,1}(X, ℝ) denotes the Kähler cone, i.e. the open set of all Kähler classes on X. For a class α ∈ H^{1,1}(X, ℝ) we usually denote by α̃ ∈ A^{1,1}(X)_{ℝ} a closed real (1, 1)-form representing α. Let us recall the Calabi-Yau theorem [3].

Theorem 1.1 — Let X be an N-dimensional compact Kähler manifold with a given volume form vol ∈ A^{N,N}(X)_{ℝ}. For any Kähler class ω ∈ K_{X} there exists a unique Kähler form ω̃ ∈ A^{1,1}(X)_{ℝ} representing ω, such that ω̃^{N} = c · vol, with c ∈ ℝ.

Since ω̃^{N} is harmonic with respect to ω̃, this can be equivalently expressed by saying that any Kähler class ω can uniquely be represented by a Kähler form ω̃ with respect to which the given volume form is harmonic. Note that the constant c can be computed as c = ∫_{X} ω^{N}/vol(X).

Definition 1.2 — For a given volume form vol ∈ A^{N.N}(X)_{ℝ} we let K̃_{X} ⊂ A^{1,1}(X)_{ℝ} be the set of Kähler forms ω̃ with respect to which vol is harmonic.

By the Calabi-Yau theorem the natural projection K̃_{X} → K_{X} is bijective. The Kähler cone K_{X} is an open subset of H^{1,1}(X, ℝ), whereas K̃_{X} is in general not contained as an open subset in a linear subspace of A^{1,1}(X) (cf. 2.1). Thus it might be appropriate to call K̃_{X} the non-linear Kähler cone. Let ω̃ ∈ K̃_{X} and c = ∫_{X} ω^{N}/vol(X). The tangent space of K̃_{X} at ω̃ can be computed as follows. Firstly, we may write K̃_{X} = ℝ_{+} × K̃_{X}^{c}, where K̃_{X}^{c} = {ω̃ ∈ K̃_{X}|ω̃^{N} = c · vol}. Secondly, the infinitesimal deformations of ω̃ in the direction of K̃_{X}^{c} are of the form ω̃ + εṽ, where ṽ is a closed real (1, 1)-form and such that (ω̃ + εṽ)^{N} = ω̃^{N}. The latter condition gives ω̃^{N} + Nεω̃^{N−1}ṽ = ω̃^{N}, i.e. ṽ is primitive. As any closed primitive (1, 1)-form is harmonic, this shows that the tangent space of K̃_{X}^{c} at ω̃ is the space H^{1,1}(ω)_{ℝ,prim} of real ω̃-primitive ω̃-harmonic (1, 1)-forms. Thirdly, the ℝ_{+}-direction corresponds to the scaling of ω̃ and this tangent direction is therefore canonically identified with ℝω̃. Altogether, one obtains that Tω̃K̃_{X} = H^{1,1}(ω̃)_{ℝ} is the space of real ω̃-harmonic (1, 1)-forms. In particular, K̃_{X} is a smooth connected submanifold of A^{1,1}(X)_{ℝ}. To make this approach rigorous, one first completes A^{1,1}(X) in the L_{k}^{2}-topology, where k > N. The Sobolev embedding theorem then shows that L_{k}^{2}(Λ_{ℝ}^{1,1})_{cl} → L_{k}^{2}(Λ_{ℝ}^{N,N}) given by α ↦ α^{N} is a well-defined continuous multi-linear map and hence differentiable. Then, K̃_{X} is contained as an open subset in the fibre over vol. It inherits the differentiable structure and the above calculation then shows that it is smooth. Also note that the projection from the closed L_{k}^{2} forms onto cohomology is differentiable. Hence, K̃_{X} → K_{X} is a differentiable map. Moreover, again due to the description of the tangent space, this map is in fact a diffeomorphism. In particular, the bijection K̃_{X} → K_{X} yields a differentiable map K_{X} → A^{2}(X) (in the L_{k}^{2}-topology). This fact is used in 2.2.

Definition 1.3 — Let X be a compact Kähler manifold with a given volume form. Then one associates to a given Kähler class ω ∈ K_{X} the space H^{p.q}(ω) := H^{p,q}(ω̃) of (p, q)-forms that are harmonic with respect to the unique ω̃ ∈ K̃_{X} representing ω.

Note that two different Kähler forms ω̃_{1} and ω̃_{2} representing the same Kähler class ω_{1} = ω_{2} always have different spaces of harmonic (1, 1)-forms. Indeed, ω̃_{1} and ω̃_{2} are ω̃_{1}-harmonic respectively ω̃_{2}-harmonic. Since any class, in particular ω_{1} = ω_{2}, is represented by a unique harmonic form and ω̃_{1} ≠ ω̃_{2}, this yields H^{1,1}(ω̃_{1}) ≠ H^{1,1}(ω̃_{2}). But one might ask whether H^{1,1}(ω̃_{1}) and H^{1,1}(ω̃_{2}) can be equal for two Kähler forms ω̃_{1}, ω̃_{2} not representing the same class, e.g. ω̃_{1}, ω̃_{2} ∈ K̃_{X}. It is quite interesting to observe that the dependence of H^{1,1}(ω̃) on the Kähler class ω is related to the problem discussed in the introduction. This is explained in the next section.

2 How ‘harmonic’ depends on the Kähler form

As before, we consider a compact Kähler manifold X with a fixed volume form and we let K̃_{X} be the associated non-linear Kähler cone. Let us begin with the following fact which relates the shape of K̃_{X} to the dependence of H^{1,1}(ω) on ω.

Proposition 2.1 — The subspace H^{1,1}(ω) ⊂ A^{1,1}(X) is independent of ω if and only if K̃_{X} spans an ℝ-linear subspace of dimension h^{1,1}(X).

Proof. Let H^{1,1}(ω) ⊂ A^{1,1}(X) be independent of ω ∈ K_{X}. Since for any ω ∈ K_{X} the unique ω̃ ∈ K̃_{X} representing it is ω̃-harmonic, the assumption immediately yields K̃_{X} ⊂ H^{1,1}(ω)_{ℝ} for any ω ∈ K_{X}. Conversely, if K̃_{X} spans an ℝ-linear subspace of dimension h^{1,1}(X), then this subspace coincides with the tangent space of K̃_{X} at every point ω̃ ∈ K̃_{X}. But the latter was identified with H^{1,1}(ω)_{ℝ}. Hence, the linear subspace equals H^{1,1}(ω)_{ℝ} for any ω ∈ K_{X} and H^{1,1}(ω), therefore, does not depend on ω. □

Remark 2.2 — The assertion might be rephrased from a slightly different point of view as follows. Use the differentiable map K_{X} → A^{2}(X). The proposition then just says that this map is linear if and only if the Gauss map is constant. It might be instructive to rephrase some of the results later on in this spirit, e.g. Proposition 3.2.

The next proposition states that the ‘global’ change of H^{1,1}(ω) for ω ∈ K_{X} is determined by the ‘harmonic’ behavior with respect to a single ω ∈ K_{X}.

Proposition 2.3 — Let X be a compact Kähler manifold of dimension N with a fixed Kähler form ω̃_{0} and volume form ω̃_{0}^{N}/N!. Then the following statements are equivalent:

i) The linear subspace H^{1,1}(ω)_{ℝ} ⊂ A^{1,1}(X)_{ℝ} does not depend on ω ∈ K_{X}. ii) For all α ∈ H^{1,1}(ω_{0}) one has α^{N} ∈ H^{N,N}(ω_{0}).

Proof. Let us assume i). By the previous proposition the lifted Kähler cone K̃_{X} spans the ℂ-linear subspace H^{1,1}(ω_{0}). Since K̃_{X} is open in H^{1,1}(ω_{0})_{ℝ} and all α ∈ K̃_{X} satisfy the ℂ-linear equation

α^{N} = (∫_{X} α^{N}/ ∫_{X} ω_{0}^{n}) · ω_{0}^{N}  (1)

which is an algebraic condition, in fact all α ∈ H^{1,1}(ω_{0}) satisfy (1). Hence, for all α ∈ H^{1,1}(ω_{0}) the top exterior power α^{N} is harmonic, i.e. ii) holds true. Let us now assume ii). If α ∈ H^{1,1}(ω_{0}), such that its cohomology class ω := [α] is a Kähler class, let ω̃ ∈ K̃_{X} denote the distinguished Kähler form representing ω. If α itself is strictly positive definite, then the unicity of ω̃ and ii) imply α = ω̃. Thus, the intersection of the closed subset H^{1,1}(ω_{0})_{ℝ} with the open cone of strictly positive definite real (1, 1)-forms is contained in K̃_{X}. This intersection is non-empty, as it contains ω̃_{0}. Since K̃_{X} is a closed connected subset of this open cone of the same dimension as H^{1,1}(ω_{0})_{ℝ} this yields K̃_{X} ⊂ H^{1,1}(ω_{0})_{ℝ}. By Prop. 2.1 one concludes that H^{1,1}(ω) does not depend on ω ∈ K_{X}. □

3 The positive cone

The next proposition is a first step towards a geometric understanding of the failure of harmonicity of α^{N} for a harmonic form α. To state it we recall the following notation.

Definition 3.1 — For a compact Kähler manifold X the positive cone C_{X} ⊂ H^{1,1}(X, ℝ) is the connected component of {α ∈ H^{1,1}(X, ℝ) | ∫_{X} α^{N} > 0} that contains the Kähler cone.

Note that by definition K_{X} ⊂ C_{X}. Also note that the positive cone C_{X} might not be convex. However, for hyperkähler manifolds also C_{X} is convex, as it coincides with the cone defined by the Beauville-Bogomolov quadratic form (see Sect. 5).

Proposition 3.2 — If X is a compact Kähler manifold such that K_{X} is strictly smaller than C_{X}, then for any Kähler form ω̃ there exists a ω̃-harmonic (1, 1)-form α such that α^{N} is not ω̃-harmonic.

Proof. Assume that there exists a Kähler form ω̃_{0} such that for all α ∈ H^{1,1}(ω̃_{0}) also α^{N} is ω̃_{0}-harmonic. We endow X with the volume form ω̃_{0}^{N}/N!. By Prop. 2.3 the lifted Kähler cone K̃_{X} is contained in H^{1,1}(ω̃_{0}). Since K_{X} is strictly smaller than C_{X} there exists a sequence ω_{t} ∈ K_{X} converging towards a ω ∈ C_{X} \ K_{X}. As K̃_{X} is contained in the finite-dimensional space H^{1,1}(ω̃_{0}) the lifted Kähler forms ω̃_{t} ∈ K̃_{X} will converge towards a form (!) and just only a current ω̃ ∈ H^{1,1}(ω̃_{0}) \ K̃_{X}. As a limit of strictly positive definite forms ω̃ is still semi-positive definite. Moreover, ω̃ is strictly positive definite at x ∈ X if and only if ω̃^{N} does not vanish at x. By assumption ω̃^{N} = c · ω̃_{0}^{N} with c = ∫_{X} ω^{N}/ ∫_{X} ω_{0}^{N}. Since ω ∈ C_{X}, the scalar factor c is strictly positive. Hence, ω̃^{N} is everywhere non-trivial. Thus ω̃ is strictly positive definite. This yields the contradiction. □

The interesting thing here is that the proposition in particular can be used to determine the positivity of a class with positive top exterior power just by studying the space of harmonic forms with respect to a single given, often very special Kähler form:

Corollary 3.3 — Let X be a compact Kähler manifold with a given Kähler form ω̃_{0}. If for all ω̃_{0}-harmonic (1, 1)-forms α the top exterior power α^{N} is also ω̃_{0}-harmonic, then any class ω ∈ C_{X} is a Kähler class. □

Here is another version of the same corollary in a more algebraic spirit.

Corollary 3.4 — Let X be a compact Kähler manifold with a given Kähler form ω̃_{0}, such that for every ω̃_{0}-harmonic (1, 1)-form α the top exterior power α^{N} is also ω̃_{0}-harmonic. Then, a line bundle L on X is ample if and only if c_{1}(L) ∈ C_{X}. □

We conclude this section with a few examples, where the assumption of the corollary is met a priori. In the later sections we will discuss examples where K_{X} is strictly smaller than C_{X} and where Prop. 3.2 can be used to conclude the ‘failure’ of harmonicity.

Examples 3.5 — i) If X is a complex torus and ω is a flat Kähler form, then harmonic forms are constant forms and their products are again constant, hence harmonic. In particular, one recovers the fact that on a torus the Kähler cone and the positive cone coincide. ii) If for two Kähler manifolds (X, ω̃) and (X′, ω̃′) with b_{1}(X) · b_{1}(X′) = 0 the top exterior power of any harmonic (1, 1)-forms on X or on X′ is again harmonic, then the same holds for the product (X × X′, ω̃ × ω̃′). The additional assumption on the Betti-numbers is necessary as the product of two curves of genus at least two shows. Indeed, any φ ∈ H^{1,0}(X), for a curve X, is harmonic, but φ ∧ φ̄ is not. Hence, α = φ × φ̄+ φ̄× φ is a harmonic (1, 1)-form on X × X′ with non-harmonic α^{2}. iii) Let X be a Kähler manifold and X → X′ a smooth finite quotient. Consider the non-linear Kähler cone on X with respect to the pull-back of a volume form on X′. Then H^{1,1}(ω) with ω ∈ K_{X} does not depend on ω if and only if the same holds true for X′. iv) For hermitian symmetric spaces of compact type it is known that the space of harmonic forms equals the space of forms invariant under the real form. As the latter space is invariant under products, the Kähler cone of an irreducible hermitian symmetric space coincides with the positive cone.

4 K3 surfaces

As indicated earlier the behavior of the Kähler cone is closely related to the geometry of the manifold. We shall study this in more detail for K3 surfaces. The next proposition follows directly from the well-known description of the Kähler cone of a K3 surface.

Proposition 4.1 — Let X be a K3 surface containing a smooth rational curve. Then for any Kähler form ω̃ there exists an ω̃-harmonic form (1, 1)-form α such that α^{2} is not harmonic.

Proof. If X contains a smooth rational curve, then K_{X} is strictly smaller than C_{X} and we apply Prop. 3.2. Indeed, a smooth rational curve C ⊂ X determines a (−2)-class [C], whose perpendicular hyperplane [C]^{⊥} cuts C_{X} into two parts and K_{X} is contained in the part that is positive on C. □

If the harmonicity of the top exterior powers fails for a Kähler manifold with a given Kähler form (X, ω̃) then it should do so for any small deformation of (X, ω̃). For a Ricci-flat Kähler structure on a K3 surface the argument can be reversed and one can use the existence of rational curves on arbitrarily near deformations to prove the assertion in the above proposition on any K3 surface with respect to a Ricci-flat Kähler form.

Corollary 4.2 — Let X be an arbitrary K3 surface. If ω̃ is any hyperkähler form on X, then there exists an ω̃-harmonic (1, 1)-form α such that α^{2} is not ω̃-harmonic.

Proof. Let H^{0}(X, Ω_{X}^{2}) = ℂσ. Then

\begin{aligned}H^{2}(ω̃) & = & H^{1,1}(ω̃) ⊕ H^{2,0}(ω̃) ⊕ H^{0,2}(ω̃) \\ & = & H^{1,1}(ω̃) ⊕ ℂσ ⊕ ℂσ̄ \end{aligned}

As the space of harmonic forms only depends on the underlying hyperkähler metric g, the space H^{1,1}(ω̃) ⊕ ℂσ ⊕ ℂσ̄ contains H^{1,1}(ω̃_{aI+bJ+cK}) for all (a, b, c) ∈ S^{2}. Here, I, J, and K are the three complex structures associated with the hyperkähler metric g (cf. Sect. 5). Assume α^{2} is g-harmonic for all α ∈ H^{1,1}(ω̃). Since σ = ω̃_{J} + iω̃_{K} (up to a scalar factor) and since the product of a harmonic form with the Kähler form is again harmonic, also σσ̄ is harmonic. This implies that α^{2} is harmonic for all α ∈ H^{2}(ω̃), as σ^{2} = σ̄^{2} = ασ = ασ̄ = 0 for α ∈ H^{1,1}(ω̃). Thus, α^{2} is g-harmonic for all α ∈ H^{1,1}(ω̃_{aI+bJ+cK}) and all (a, b, c) ∈ S^{2}. On the other hand, it is well-known that for a non-empty (dense) subset of S^{2} the K3 surface (X, aI + bJ + cK) contains a smooth rational curve. Indeed, if e ∈ H^{2}(X, ℤ) is any (−2)-class, then the subset of the moduli space of marked K3 surfaces for which e is of type (1, 1) is a hyperplane section. This hyperplane section, necessarily, cuts the complete curve given by the base ℙ^{1} = S^{2} of the twistor family. Hence, on one of the K3 surfaces (X, aI + bJ + cK) the class e is a (−2)-class of type (1, 1) and, thus, X contains a smooth rational curve. This yields a contradiction to Prop. 4.1. □

Remark 4.3 — What are the bad harmonic (1, 1)-forms? Certainly ω̃^{2} is harmonic and for any harmonic form α also ω̃α is harmonic. So, if there is any bad harmonic (1, 1)-form there must be also one that is ω̃-primitive. Most likely, it is even true that the square of any primitive harmonic form is not harmonic. The proof of it should closely follow the arguments in the proof of Proposition 3.2, but there is a slight subtlety concerning the existence of sufficiently many (−2)-classes, that I do not know how to handle. We sketch the rough idea: Assume there exists a ω̃-harmonic ω̃-primitive real (1, 1)-form α such that α^{2} is ω̃-harmonic. As an ω̃-harmonic ω̃-primitive (1, 1)-form, the form α is also of type (1, 1) with respect to any complex structure λ = aI + bJ + cK induced by the hyperkähler metric corresponding to ω̃ (see Prop. 7.5 [6]). Moreover, α is also primitive with respect to all Kähler forms ω̃_{λ}. Assume that there exists a complex structure λ ∈ S^{2}, such that C_{X} ∩ℝ[α]⊕ℝω_{λ} is not contained in K_{X}. This condition can be easily rephrased in terms of (−2)- classes and thus becomes a question on the lattice 3U ⊕2(−E_{8}). It looks rather harmless, but for the time being I do not know a complete proof of it. Under this assumption, we may even assume that in fact λ = I. Since α^{2} is harmonic, in fact β^{2} is harmonic for all β ∈ ℝα ⊕ ℝω̃ ⊂ H^{1,1}(ω). Going back to the proof of Prop. 3.2, we see that the second part of it can be adapted to this situation and shows that ψ^{−1}(K_{X} ∩ ℝ[α] ⊕ ℝω) ⊂ ℝα ⊕ ℝω̃, where ψ : K̃_{X} → K_{X}. The space ψ^{−1}(K_{X} ∩ ℝ[α] ⊕ ℝω) is the space of the distinguished Kähler forms whose classes are linear combinations of [α] and ω. Therefore, all these forms are harmonic and linear combinations of α and ω̃ themselves. To conclude, we imitate the proof of Prop. 3.2 and choose a sequence ω_{t} ∈ K_{X} ∩ ℝ[α] ⊕ ℝω converging towards ω′ ∈ C_{X} \ K_{X}. The corresponding sequence ω̃_{t} ∈ K̃_{X} is contained in ℝα ⊕ ℝω̃ and converges towards a form ω̃′. As in the proof of Prop. 3.2 this leads to a contradiction.

5 Hyperkähler manifolds

We will try to improve upon Proposition 3.2 in the case of hyperkähler manifolds. In particular, we will replace the question whether the top exterior power α^{N} of an harmonic form α is harmonic by the corresponding question for the square of α. The motivation for doing so stems from the general philosophy that hyperkähler manifolds should be treated in almost complete analogy to K3 surfaces, whereby the top intersection pairing should be replaced by the Beauville-Bogomolov form [2], which is the higher dimensional analogue of the intersection pairing on a K3 surface. Let us begin by recalling some notations and basic facts. By a compact hyperkähler manifold X we understand a simply-connected compact Kähler manifold, such that H^{0}(X, Ω^{2}) = ℂσ, where σ is an everywhere non-degenerate holomorphic two-form. A Ricci-flat Kähler form ω̃ turns out to be a hyperkähler form (cf. [2]), i.e. there exists a metric g and three complex structures I, J, and K := IJ, such that the corresponding Kähler forms ω̃_{aI+bJ+cK} are closed for all (a, b, c) ∈ S^{2}, such that I is the complex structure defining X, and such that ω̃ = ω̃_{I}. One may renormalize σ, such that σ = ω̃_{J} + iω̃_{K}. In particular, multiplying with σ maps harmonic forms to harmonic forms, for this holds true for the Kähler forms ω̃_{J} and ω̃_{K}. The positive cone C_{X} ⊂ H^{1,1}(X, ℝ) is a connected component of {α ∈ H^{1,1}(X, ℝ) | q_{X}(α) > 0}, where q_{X} is the Beauville-Bogomolov form (cf. [2]).

Proposition 5.1 — Let X be a 2n-dimensional compact hyperkähler manifold with a fixed hyperkähler form ω̃_{0} and the unique holomorphic two-form σ. Then, α^{2}(σσ̄)^{n−1} is harmonic for all α ∈ H^{1,1}(ω_{0}) if and only if the linear subspace H^{1,1}(ω) ⊂ A^{1,1}(X) does not depend on ω ∈ K_{X}.

Proof. Assume that for all α ∈ H^{1,1}(ω_{0}) also α^{2}(σσ̄)^{n−1} is harmonic. If α is in addition strictly positive definite and ω̃ ∈ K̃_{X} with [α] = ω, then α^{2}(σσ̄)^{n−1} = ω̃^{2}(σσ̄)^{n−1}. We adapt Calabi’s classical argument to deduce that in this case α = ω̃: If α^{2}(σσ̄)^{n−1} = ω̃^{2}(σσ̄)^{n−1}, then (α−ω̃)(α+ω̃)(σσ̄)^{n−1} = 0. Since α and ω̃ are strictly positive definite, also (α + ω̃) is strictly positive definite. It can be shown that also (α + ω̃)(σσ̄)^{n−1} is strictly positive. As [α] = ω = [ω̃], the difference α−ω̃ can be written as dd^{c}φ for some real function φ. But by the maximum principle the equation (α+ω̃)(σσ̄)^{n−1}dd^{c}φ = 0 implies φ ≡ const. Hence, α = ω̃. As in the proof of Proposition 3.2 this shows that the intersection of the closed subset H^{1,1}(ω_{0})_{ℝ} with the open cone of strictly positive definite forms in A^{1,1}(X)_{ℝ} is contained in K̃_{X} and one concludes that K̃_{X} ⊂ H^{1,1}(ω_{0})_{ℝ}. Hence, K̃_{X} spans a linear subspace of the same dimension and, by Lemma 2.1 this shows that H^{1,1}(ω) is independent of ω ∈ K_{X}. Conversely, let H^{1,1}(ω) be independent of ω ∈ K_{X}. Then K̃_{X} ⊂ H^{1,1}(ω)_{ℝ} for any ω ∈ K_{X}. Therefore, α^{2}(σσ̄)^{n−1} = c(σσ̄)^{n} with c ∈ ℝ for α in the Zariski-dense open subset K̃_{X} ⊂ H^{1,1}(ω)_{ℝ}. Hence, α^{2}(σσ̄)^{n−1} is harmonic for any α ∈ H^{1,1}(ω) (cf. the proof of Prop. 2.3). □

Similar to Proposition 3.2 one has

Corollary 5.2 — Let X be a 2n-dimensional compact hyperkähler manifold. If the positive cone C_{X} is strictly smaller than the Kähler cone, then for any hyperkähler form ω̃_{0} there exists a harmonic form α ∈ H^{1,1}(ω̃_{0}), such that α^{2}(σσ̄)^{n−1} is not harmonic. □

Of course, one expects that H^{1,1}(ω) does in fact depend on ω, as it is the case for K3 surfaces. This would again follow from the existence of rational curves in nearby hyperkähler manifolds in the twistor space. In fact, for the two main series of examples of higher dimensional hyperkähler manifolds, i.e. Hilbert schemes of points on K3 surfaces and generalized Kummer varieties, this trivially holds true, since in these cases C_{X} is strictly bigger that K_{X} and so Corollary 5.2 applies. But already for global deformations of these examples the situation is not clear. One might speculate that Hilbert schemes respectively Kummer varieties are dense in their deformation spaces, so that arguments similar to those in the proof of Corollary 4.2 could be applied. But an understanding of the global deformations of Hilbert schemes respectively Kummer varieties seems difficult. Actually, it would be more interesting to reverse the argument: Assume that X is a hyperkähler manifold, such that for any small deformation X′ of X the Kähler cone K_{X′} equals C_{X′}. I expect that this is equivalent to saying that H^{1,1}(ω) does not depend on ω. If for some other reason than the existence of rational curves as used in the K3 surface case this can be excluded, then one could conclude that there always is a nearby deformation X′ for which K_{X′} is strictly smaller than C_{X′}. The latter implies the existence of rational curves on X′ (cf. [6]). Along these lines one could try to attack the Kobayashi conjecture, as the existence of rational curves on nearby deformations would say that X itself cannot be hyperbolic. Unfortunately, I cannot carry this through even for K3 surface.

6 Various other examples

Here we collect a few examples where algebraic geometry predicts the failure of harmonicity of the top exterior power of harmonic (1, 1)-forms. In all examples this is linked to the existence of rational curves.

Varieties of general type. Let X be a non-minimal smooth variety of general type. As I learned from Keiji Oguiso this immediately implies that the Kähler cone is strictly smaller than the positive cone. His proof goes as follows: By definition the canonical divisor K_{X} is big and by the Kodaira Lemma (cf. [7]) it can therefore be written as the sum K_{X} = H + E of an ample divisor H and an effective divisor E (with rational coefficients). Consider the segment H_{t} := H + tE with t ∈ [0, 1). If all H_{t} were contained in the positive cone C_{X}, then K_{X} would be in the closure of C_{X}. If the Kähler cone coincided with the positive cone C_{X}, then K_{X} would be nef, contradicting the hypothesis that X is not minimal. Hence t_{0} := sup{t|H_{t} ∈ C_{X}} ∈ (0, 1). If H_{t_{0}} is not nef, then K_{X} is strictly smaller than C_{X}. Thus, it suffices to show that H_{t_{0}} is not nef. If H_{t_{0}} were nef then all expressions of the form H_{t_{0}}^{N−i}.H^{i−1}.E would be non-negative. Then 0 = H_{t_{0}}^{N} = H_{t_{0}}^{N−1}(H + t_{0}E) = H_{t_{0}}^{N−1}.H + t_{0}H_{t_{0}}^{N−1}.E, so both summands must vanish. In particular, 0 = H_{t_{0}}^{N−1}.H = H^{2}.H_{t_{0}}^{N−2} + t_{0}H.H_{t_{0}}^{N−2}.E. Again this yields the vanishing of both terms and in particular 0 = H^{2}.H_{t_{0}}^{N−2}. By induction we eventually obtain 0 = H^{N−1}.H_{t_{0}} and, furthermore, 0 = H^{N−1}.H_{t_{0}} = H^{N} + t_{0}H^{N−1}.E. But this time H^{N} > 0 yields the contradiction. Therefore, for a non-minimal variety of general type one has K_{X} ≠ C_{X} and hence there exist harmonic (with respect to any Kähler metric) (1, 1)-forms with non-harmonic top exterior power. Note that a non-minimal variety contains rational curves. As the reader will notice, the above proof goes through on any manifold X that admits a big, but not nef line bundle L (replacing the canonical divisor). Also in this case the positive cone and the Kähler cone differ.

For a Calabi-Yau manifold X the following proposition shows that if X admits a ‘special’ Kähler form in the sense that the top power of any harmonic (1, 1)-form is harmonic, then X is a unique birational model.

Proposition 6.1 — If φ : X − − → X′ is a birational map between two Calabi-Yau manifolds, then either φ can be extended to an isomorphism or K_{X} is strictly smaller than the positive cone C_{X}. In the latter case, there exists for any Kähler form ω̃ a ω̃-harmonic (1, 1)-form α with α^{N} not harmonic.

Proof. The arguments are very similar to the one in the previous example. Let H^{1,1}(X, ℝ) ≅ H^{1,1}(X′, ℝ) be the natural isomorphism induced by the birational map. Let ω′ ∈ H^{1,1}(X, ℝ) correspond to a Kähler class on X′. By [5] the class ω′ can be represented by a closed positive current. Furthermore, the birational map φ extends to an isomorphism if and only if ω′ ∈ K_{X}. Assume this is not the case. Then ω′ ∉ K_{X}. If ω′ ∈ C_{X} one can apply Prop. 3.2 and we are done. If ω′ ∉ C_{X} we may assume that it is also not in the boundary of C_{X}, as we can change ω′ slightly in the open cone K_{X′}. If ω is a small enough Kähler class on X, then the difference α := ω′ − ω can still be represented by a positive current. Let t_{0} := sup{t|ω_{t} := ω + tα ∈ C_{X}}. Then t_{0} ∈ (0, 1). If K_{X} = C_{X} then ω_{t_{0}}^{N−i}ω^{i−1}α ≥ 0 for all i. Then the above induction argument goes through and we eventually get ω^{N−1}α = 0. Since α is a positive current, this is only possible for α = 0. Hence K_{X} ≠ C_{X}. Note that for hyperkähler manifolds one knows that K_{X′} ⊂ C_{X}. This simplifies the argument. □

Remark 6.2 — i) Again, a non-trivial birational correspondence produces rational curves. We thus have another instance, where the special geometry of the variety is related to the non-harmonicity of products of harmonic forms. ii) Most likely, finer information is encoded by Ricci-flat metrics. Those probably ‘feel’ contractible curves in small deformations. So, as for hyperkähler manifolds I would expect that H^{1,1}(ω) depends on the Ricci-flat Kähler form representing ω. iii) The arguments of the proof of 6.1 can be applied to the case of different birational minimal models (minimal models are not unique!). This shows that in the previous example the Kähler cone could be strictly smaller than the positive cone, even when K_{X} is nef or ample.

Blow-ups. This example is very much in the spirit of the previous two. Let f : X → Y be a non-trivial blow-up of a projective variety Y. Then K_{X} is strictly smaller than C_{X} and, therefore, for any Kähler structure on X there exist harmonic (1, 1)-forms with non-harmonic maximal exterior power. Indeed, if L is an ample line bundle on Y then f^{∗}(L) is nef, but not ample, and it is contained in the positive cone. Hence, f^{∗}(L) ∈ C_{X} \ K_{X}. Note that also the first example could be proved along these lines. By evoking the contraction theorem one shows that any non-minimal projective variety X admits a nontrivial contraction to a projective variety Y. The above argument then yields that K_{X} and C_{X} are different.

7 Chern forms

Let X be a compact Kähler manifold with a Ricci-flat Kähler form ω̃. If F denotes the curvature of the Levi-Cevita connection ∇, then the Bianchi identity reads ∇F = 0. The Kähler-Einstein condition implies Λ_{ω}F = 0. The last equation can be expressed by saying that F is ω̃-primitive. Analogously to the fact that any closed primitive (1, 1)-form is in fact harmonic, one has that for F with ∇F = 0 the primitivity condition Λω̃F = 0 is equivalent to the harmonicity condition ∇ ∗ F = 0. As for untwisted harmonic (1, 1)-forms one might ask for the harmonicity of the product F^{m}. Slightly less ambitious, one could ask whether the trace of this expression, an honest differential form, is harmonic. This trace is, in fact, a scalar multiple of the Chern character ch_{m}(X, ω̃) ∈ A^{m,m}(X)_{ℝ}.

Question. — Let (X, ω̃) be a Ricci-flat Kähler manifold. Are the Chern forms ch_{m}(X, ω̃) harmonic with respect to ω̃ ?

By what was said about K3 surface we shall expect a negative answer to this question at least in this case:

Problem. — Let X be a K3 surface with a hyperkähler form ω̃. Let c_{2} ∈ A^{2,2}(X) be the associated Chern form. Show that c_{2} is not harmonic with respect to ω̃ !

So, this should be seen in analogy to the fact that α^{2} is not harmonic for any primitive harmonic (1, 1)-form α. Here, α is replaced by the curvature F and α^{2} by trF^{2}. It is likely that the non-harmonicity of c_{2} can be shown by standard methods in differential geometry, in particular by using the fact that c_{2} is essentially ∥F ∥ · ω̃^{2} (see [3]), but I do not know how to do this. Furthermore, it is not clear to me what the relation between the above question and the one treated in the previous sections is. I could imagine that the nonharmonicity of ch_{m} in fact implies the existence of harmonic (1, 1)-forms with non-harmonic top exterior power.

Acknowledgements. I wish to thank U. Semmelmann for his interest in this work and G. Hein, M. Lehn, and D. Kaledin for making valuable comments on a first version of it. I am most grateful to Keiji Oguiso for his enthusiastic help with several arguments.

References

[1] Géométrie des surfaces K3: modules et périodes. Séminaires Palaiseau. ed A. Beauville, J.-P. Bourguignon, M. Demazure. Astérisque 126 (1985).

[2] Beauville, A. Variétés Kählériennes dont la première classe de Chern est nulle. J. Diff. Geom. 18 (1983), 755-782.

[3] Besse, A. Einstein Manifolds. Erg. 3. Folge Band 10 (1987).

[4] Demailly, J.-P. Théorie de Hodge des variétés kählériennes complètes. in Introduction à la théorie de Hodge. Panoramas et Synthèses 3 (1996).

[5] Fujiki, A. A Theorem on Bimeromorphic Maps of Kähler Manifolds and Its Applications. Publ. RIMS Kyoto Univ. 17 (1981), 735-754.

[6] Huybrechts, D. The Kähler cone of a compact hyperkähler manifold. Preprint (1999) math.AG/9909109

[7] Kawamata, Y., Matsuda, K., Matsuki, K. Introduction to the Minimal Model Problem. Adv. Stud. Pure Math. 10 (1987), 283-360.

[8] Merkulov, S. Strong homotopy algebras of a Kähler manifold. math.AG/9809172, Int. Math. Res. Notices 3 (1999), 153-164.

[9] Polishchuk, A. Homological mirror symmetry with higher products. Preprint (1999) math.AG/9901025

Daniel Huybrechts

Mathematisches Institut Universität zu Köln Weyertal 86-90 50931 Köln Germany huybrech@mi.uni-koeln.de 240

Documenta Mathematica 6 (2001)