Here’s a perhaps lesser known fact about special 2-forms on Kähler manifolds:

If the Ricci curvature is negative, there are no¹ parallel antihermitian 2-forms.
If the Ricci curvature is positive, there are neither parallel nor harmonic antihermitian 2-forms.
If the manifold is Ricci-flat, then parallel and harmonic antihermitian 2-forms happily coexist because they are the same.

This post will explain how this is proved using a bit of “flipping triangles” magic, and how this helps correct a mistake in a famous article² by N. Koiso from 1983, published in Inventiones Mathematicae.

I was inspired to write this post after my doctoral advisor Uwe Semmelmann told me of a conversation he had with Hajo Hein about this very article, who pointed out that there was at least one serious error in it. We stumbled across a solution when Uwe recently visited me in Paris.

Koiso’s article

In the aforementioned article², Koiso studies among other things the infinitesimal deformations of Einstein metrics and of complex structures on a closed smooth manifold \(M\). Let me first explain what that means.

Einstein deformations

Let \(g\) be an Einstein metric on \(M\) – that is, a Riemannian metric such that its Ricci tensor is proportional to \(g\) itself: \[\Ric_g=Eg,\qquad E\in\RR.\] When looking for interesting Riemannian metrics on a given manifold, Einstein metrics are kind of the sweet spot, because the Einstein condition sits comfortably inbetween the very rigid condition of constant of sectional curvature on the stronger side and the quite lax condition of constant scalar curvature on the weaker side.³ So naturally people want to study the following question:

Question 1: What are the Einstein metrics on a given manifold \(M\)? In particular, how many are there?

The Einstein equation is a nonlinear partial differential equation⁴, and solving it is notoriously difficult. So the first question is a tad too hard in general. Okay, maybe it gets a little bit easier if we look at the solution space locally:

Question 2: Given an Einstein metric \(g\) on \(M\), are there any other Einstein metrics nearby or is \(g\) isolated in the solution set?⁵

Koiso managed to show that the solution set it not too pathological in the sense that if an Einstein metric is not isolated, there is actually a curve of Einstein metrics through it, i.e. a continous non-constant family of Einstein metrics \((g_t)\) where \(t\in\RR\) and \(g_0=g\). We shall call such a family an Einstein deformation.

As it stands, Question 2 is not too interesting because the answer is always “yes”. Why? Because if we take an Einstein metric \(g\) and scale it by some factor \(c>0\), the result \(cg\) will again be Einstein. What’s more, if we take a curve of diffeomorphisms \(\varphi_t: M\to M\) through the identity (\(\varphi_0=\id\)), then the pulled-back metrics \(g_t:=\varphi_t^\ast g\) will be isometric to \(g\) and thus also Einstein. We would thus like to exclude these “trivial deformations” from our considerations:⁶

Question 3: Given an Einstein metric \(g\), does there exist a non-trivial Einstein deformation \((g_t)\) in the sense that \(g_t\) is not related to \(g\) by scaling or diffeomorphism for all \(t\)?

How do we test whether this can be the case? One possibility is to linearize the problem. Recall that a Riemannian metric is nothing but a symmetric 2-tensor field that is positive-definite at each point. The set of symmetric 2-tensor fields \(\Gamma(\Sym^2T^\ast M)\) is a vector space. Thus, given a smooth curve of metrics \((g_t)\), we may differentiate⁷ it to obtain the first order jet \[h:=\frac{\dd}{\dd t}\Big|_{t=0}g_t\in\Gamma(\Sym^2T^\ast M).\]

If \((g_t)\) is an Einstein deformation⁸, then the tensor field \(h\) satisfies the linearized Einstein equation. Assuming that \((g_t)\) is a nontrivial deformation, we may even arrange for \(h\) to preserve the volume form of \(g\) to first order (i.e. fixing the scaling issue) and to be transverse/orthogonal to the orbits of the diffeomorphism group (i.e. fixing the isometry issue).⁹ Such an \(h\) is called an (essential) infinitesimal Einstein deformation (IED), and it satisfies \[\LL h=2E h,\qquad\tr_gh=0,\qquad\delta_gh=0.\] Here \(\tr_g\) denotes the trace and \(\delta_g\) the divergence operator, while \(\LL\) is the Lichnerowicz Laplacian¹⁰.

The linear system of equations above may now be studied in order to get to grips with Question 3. Clearly, if it has no solutions, then there cannot be a nontrivial Einstein deformation. In the other direction, this goes wrong: there may exist IED that cannot be “integrated” into an actual curve. In fact, there exist obstructions against integrability – and these are still kind of mysterious. A concise summary of the topic is contained in Chapter XII of Besse’s famous monograph on Einstein manifolds.¹¹

Complex deformations

Let us now turn to the deformation theory of complex structures, which had originally been developed by Kodaira and Spencer¹² in a more algebro-geometric context, but recast by Koiso into differential-geometric terms. First, recall that an even-dimensional manifold is complex if it has an atlas of charts to \(\CC^n\) (instead of \(\RR^{2n}\)) such that all transition maps are holomorphic. This induces a so-called complex structure \(J\in\Gamma(\End TM)\) on \(M\), an endomorphism of the tangent bundle that is defined by multiplication with \(\i\) in a chart and thus squares to minus the identity, \[J^2=-\id_{TM}.\] By the famous Newlander–Nirenberg theorem, any \(J\in\Gamma(\End TM)\) with \(J^2=-\id_{TM}\) is a complex structure if and only if its so-called Nijenhuis tensor \(N_J\) vanishes.¹³

We may now ask the same questions as above: Given a complex structure on \(M\), are there any other non-isomorphic complex structures nearby? The deformation theory is a little bit different than for Einstein metrics, because there may be complex structures “near” \(J\) that are not connected to it by a curve of complex structures. Nevertheless, we may investigate smooth curves \((J_t)\) of complex structures and their first order jets \[I:=\frac{\dd}{\dd t}\Big|_{t=0}J_t\in\Gamma(\End TM).\]

In order to define what it means to be orthogonal to orbits of the diffeomorphism group, we however need a metric. So let \(g\) be a Riemannian metric on \(M\) that is compatible with \(J\) – this just means that the endomorphism \(J\) is orthogonal with respect to \(g\), i.e. \[g(JX,JY)=g(X,Y)\] for all vector fields \(X,Y\). Differentiating the conditions \(J_t^2=-\id_{TM}\) and \(N_{J_t}=0\) together with the tranversality condition yields a first order jet \(I\in\Gamma(\End TM)\) satisfying \[ IJ+JI=0,\qquad \bar\partial^\nabla I=0,\qquad(\bar\partial^\nabla)^\ast I=0.\] We call such an \(I\) an (essential) infinitesimal complex deformation (ICD). The operator \(\bar\partial^\nabla\) is the coboundary operator for the Dolbeault complex¹⁴ \[ 0\to\Gamma(T^{1,0}M)\stackrel{\bar\partial^\nabla}{\to}\Omega^{0,1}(M,T^{1,0}M)\stackrel{\bar\partial^\nabla}{\to}\Omega^{0,2}(M,T^{1,0}M)\stackrel{\bar\partial^\nabla}{\to}\ldots, \] in which \(I\) is identified with an element of \(\Omega^{0,1}(M,T^{1,0}M)\). Thus \(I\) corresponds to an element of the cohomology \(H^{0,1}(M,T^{1,0}M)\) of this complex, which is canonically isomorphic to the sheaf cohomology \(H^1(M,\Theta)\) of the sheaf \(\Theta\) of holomorphic vector fields on \(M\). This establishes the connection to the Kodaira–Spencer deformation theory!

In constract to the deformation theory of Einstein metrics, the integrability obstructions are much better understood: they manifest as elements of the second cohomology group \(H^2(M,\Theta)\) – in particular, if \(H^2(M,\Theta)=0\), then every ICD is integrable into a curve of complex structures.

Kähler relations

Assume that \((M,g,J)\) is a Kähler manifold, that is a complex manifold with compatible metric \(g\) such that the complex structure \(J\) is parallel with respect to the Levi-Civita connection, i.e. \[\nabla J=0.\] Given an infinitesimal complex deformation \(I\), we may identify it via the metric with a 2-tensor field \(I\in\Gamma(T^\ast M\otimes T^\ast M)\) that is anti-hermitian (which just means that the endomorphism \(I\) anticommutes with \(J\)). Curiously, on a Kähler–Einstein manifold¹⁵, the 2-tensor field \(I\) also satisfies \[\LL I=2EI.\] Decomposing \(I\) into a symmetric and an anti-symmetric part, we obtain a symmetric 2-tensor field \(I_S\) and a 2-form \(I_A\), and since all the involved operators preserve this decomposition, the tensor field \(I_S\) is an infinitesimal Einstein deformation!¹⁶ In fact, the space of symmetric ICD of a Kähler manifold is isomorphic to the space of anti-hermitian IED.

Okay, now what about the 2-form part \(I_A\)?

The problem

In Proposition 8.2 of his article², Koiso claims that on a Kähler manifold, an anti-symmetric anti-hermitian 2-tensor field \(I\in\Gamma(T^\ast M\otimes T^\ast M)\) is an ICD if and only if it is parallel.

This is a mouthful, so let’s break it down: Anti-symmetric just means that \(I\) is a 2-form, i.e. \(I\in\Omega^2(M)\). Generally, on a complex manifold \((M,J)\), the complexified tangent bundle \(TM^\CC\) splits into eigenbundles of \(J\) to the eigenvalues \(\pm\i\), denotes by \(T^{1,0}M\) and \(T^{0,1}M\), respectively.¹⁷ Going to the exterior power, we may split \[\Lambda^2T^\ast M^\CC=\Lambda^{2,0}\oplus\Lambda^{1,1}\oplus\Lambda^{2,0}\] according to the action of \(J\). The (complexified) bundles of hermitian/anti-hermitian 2-forms are then just \(\Lambda^{1,1}\) and \(\Lambda^{(2,0)\oplus(0,2)}:=\Lambda^{2,0}\oplus\Lambda^{0,2}\), respectively.

So the statement above may be reformulated as: any real section \(I\in\Omega^{(2,0)\oplus(0,2)}(M)\) is an ICD if and only if \(\nabla I=0\). Moreover, Koiso claims that this is the case precisely if \(I\) is harmonic, i.e. \(\Delta I=0\). And since we’re on a Kähler manifold, where the Hodge–deRham Laplacian \(\Delta=d^\ast d+dd^\ast\) coincides with twice the \(\bar\partial\)-Laplacian \(\Delta_{\bar\partial}=\bar\partial^\ast\bar\partial+\bar\partial\bar\partial^\ast\), this means that (the (2,0)-part of) \(I\) is holomorphic and the space of anti-symmetric ICD is isomorphic to the Dolbeault cohomology group \(H^{2,0}(M,\CC)\).

It took some time for people to realize that this cannot be true. As pointed out by Dai–Wang–Wei¹⁸ in 2007, there exist examples (complex hyperbolic surfaces) with holomorphic \(2\)-forms (i.e. \(H^{2,0}(M,\CC)\neq0\)) but without ICD!

The solution

In order to see what went wrong, let us revisit the proof of Proposition 8.2.

Why anti-symmetric ICD are parallel

Let \(I\Omega^{(2,0)\oplus(0,2)}(M)\) be an ICD, that is \(\bar\partial^\nabla I=0\) and \((\bar\partial^\nabla)^\ast I=0\). A Weitzenböck formula¹⁹ by Koiso then implies that \[\nabla^\ast\nabla I+2\firstkind I=0,\] where \(\firstkind: \Lambda^2\to\Lambda^2\) is the curvature operator of the first kind defined from the Riemannian curvature \(R\) by \[g(\firstkind(X\wedge Y),Z\wedge W):=g(R(X,Y)Z,W).\] By yet another Weitzenböck formula we can write the Hodge–deRham Laplacian on 2-forms as \[\Delta=\nabla^\ast\nabla+2\firstkind+2E.\] Moreover, we have \(\firstkind I=0\) for anti-hermitian two-forms \(I\). Why? Because \(\nabla J=0\), so \(R(X,Y)J=0\) for all vector fields \(X,Y\) – but this can be rearranged to \(\firstkind(X\wedge Y)\) being hermitian! And since \(\firstkind\) is self-adjoint with respect to \(g\), it follows that \(\firstkind\) annihilates all anti-hermitian 2-forms.

Or for a more high-level explanation: the image of the curvature operator is contained in the holonomy algebra, which in turn is contained in \(\mathfrak{u}(n)\cong\Lambda^{1,1}\) for Kähler manifolds. Thus \(\firstkind\big\vert_{\Lambda^{(2,0)\oplus(0,2)}}=0\).

As a consequence, the ICD condition for our antihermitian 2-form \(I\) reduces to \(\nabla^\ast\nabla I=0\) and thus \(\nabla I=0\) since we’re on a compact manifold.

Parallel and harmonic forms

Koiso goes on to claim that being parallel and being harmonic are equivalent. But this is simply not true. Okay, for \(E=0\) it kind of is – indeed, by the above Weitzenböck formula, \(\Delta I=\nabla^\ast\nabla I\) for \(I\in\Lambda^{(2,0)\oplus(0,2)}\).

For \(E\neq0\), we still have \[\Delta I=\nabla^\ast\nabla I+2EI,\] so parallel and harmonic anti-hermitian 2-forms lie in different eigenspaces of \(\nabla^\ast\nabla\). This is a bit awkward because parallel forms are necessarily closed and coclosed, so in particular harmonic. Hence for \(E\neq0\) there cannot be any parallel antihermitian 2-forms.

What’s more, note that \(\nabla^\ast\nabla\) is a positive-semidefinite operator. If \(E>0\), then any harmonic anti-hermitian 2-form \(I\) would be an eigenform to \(\nabla^\ast\nabla\) to the eigenvalue \(-2E<0\), a contradiction! So there cannot even be harmonic anti-hermitian 2-forms for \(E>0\).

So that settles it: If \(E=0\), then anti-symmetric ICD are indeed harmonic, i.e. indeed the space of anti-symmetric ICD is isomorphic to \(H^{2,0}(M,\CC)\). And if \(E\neq0\), then there are no anti-symmetric ICD at all, irrespective of what \(H^{2,0}(M,\CC)\) does.

Aftermath

What’s weird is that Koiso’s Corollary 9.4 about the dimension of the space of IED on a Kähler–Einstein manifold seems to be correct, even if it relies on the faulty claim about holomorphic 2-forms. This is because he also noticed that there are no parallel anti-hermitian two-forms if \(E\neq0\), and because for \(E=0\) the claim is actually true. So fortunately, nothing else in his paper breaks down. Mathematics is saved once again!

My job here is done. - But you didn't do anthing!

Footnotes

With “there are no \(X\)” (in the context of some vector space) I of course always mean “there are no \(X\) except when \(X=0\)”. ↩
N. Koiso: Einstein Metrics and Complex Structures, Invent. Math. 73, 71-106, 1983. ↩ ↩² ↩³
Also the Einstein equation is kind of important in general relativity, but this is a Riemannian household, so no further talk of that! ↩
No, I am not going to write it out in a local chart so you can see the derivatives. ↩
The keywords “nearby” and “isolated” indicate that there is some topology going on. To be precise, we consider the set of Einstein metrics as a subset of the space \(\mathcal{M}\) of all Riemannian metrics on \(M.\) If you really want to know, \(\mathcal{M}\) is an infinite-dimensional manifold modeled on an inverse limit of Hilbert spaces and thus carries a natural topology – and even a Riemannian metric itself! Don’t worry, this won’t be on the test. ↩
This exclusion of trivial deformations may also be achieved by taking the set of Einstein metrics in \(\mathcal{M}\) and quotienting out the action of the group \(\RR_+\times\mathrm{Diff}(M)\) (i.e. scaling + diffeomorphisms). The result is called the moduli space of Einstein structures, and Question 3 may now be succinctly stated as: “Is the equivalence class \([g]\) isolated in the moduli space?” ↩
The space \(\Gamma(VM)\) of smooth sections of a vector bundle \(VM\) is an infinite-dimensional vector space – but luckily, it is a Fréchet space, so the derivative is well-defined. ↩
Koiso also showed that without restriction, Einstein deformations may be assumed to be smooth, and even analytic. Phew! ↩
This is what is known in physics as the tt gauge. The two t stand for traceless and transverse. ↩
The Lichnerowicz Laplacian is a certain operator on tensor fields of any kind that generalizes the well-known Hodge–deRham Laplacian on differential forms. It is really cool and I am going to write something about it someday. ↩
Arthur L. Besse: Einstein manifolds, Springer-Verlag Berlin, 1987. ↩
K. Kodaira, D.C. Spencer: On the variation of almost-complex structure, Algebraic Geometry and Topology, Princeton, 139-150, 1957. ↩
If you really want to know, the Nijenhuis tensor of \(J\) is a tensor field of type \((1,2)\) defined as \[N_J(X,Y):=[X,Y]-[JX,JY]+J[JX,Y]+J[X,JY]\] for vector fields \(X,Y\). Are you happy now? ↩
I shall not dwell on the origin and meaning of these differential operators, Dolbeault complexes and sheaf cohomology here – instead, I will return to this topic on some later day. ↩
That is, a Kähler manifold \((M,g,J)\) where the metric \(g\) is Einstein. ↩
The tracelessness is an automatic consequence of being anti-hermitian. ↩
If you’re uneasy with bundle manipulation, consider the following vector space analogue: A linear endomorphism \(J\) with \(J^2=-\id\) of a real vector space \(V\) is never diagonalizable – but if \(V\) is complexified, then the complex-linearly extended \(J\) has eigenvalues \(\pm\i\), and the eigenspaces are conjugate. ↩
X. Dai, X. Wang, G. Wei: On the variational stability of Kähler-Einstein metrics, Comm. Anal. Geom. 15 (4), 669-693, 2007. ↩
A Weitzenböck formula is a nontrivial relation on sections of some vector bundle of the form \[D_1^\ast D_1+\ldots D_k^\ast D_k=C,\] where the \(D_i\) are certain first order differential operators called partial gradients and \(C\) is a fibrewise term (i.e. a bundle morphism), usually some sort of curvature operator. See for example:

U. Semmelmann, G. Weingart: The Weitzenböck machine, Compositio Math. 146, 507-540, 2007. ↩