\( \newcommand{\re}{\mathop{\rm Re}\nolimits} \newcommand{\im}{\mathop{\rm Im}\nolimits} \newcommand{\coker}{\mathop{\rm coker}\nolimits} \newcommand{\supp}{\mathop{\rm supp}\nolimits} \newcommand{\ord}{\mathop{\rm ord}\nolimits} \newcommand{\Spec}{\mathop{\rm Spec}\nolimits} \newcommand{\vol}{\mathop{\rm vol}\nolimits} \newcommand\restr[2]{{\left.#1\right|_{#2}}} \newcommand{\transp}{\mathop{\rm \,^t}\nolimits} \newcommand{\sff}{\mathop{\rm I\!I}\nolimits} \newcommand{\tr}{\mathop{\rm Tr}\nolimits} \newcommand{\const}{\mathop{\rm const }\nolimits} \newcommand{\lcm}{\mathop{\rm lcm}\nolimits} \newcommand{\gcd}{\mathop{\rm gcd}\nolimits} \newcommand{\Ric}{\mathop{\rm Ric}\nolimits} \newcommand{\Riem}{\mathop{\rm Riem}\nolimits} \newcommand{\Enorm}{\mathop{\mathcal{E}_{\rm norm}}\nolimits} \newcommand{\Anorm}{\mathop{\mathcal{A}_{\rm norm}}\nolimits} \newcommand{\Cl}{\mathop{\rm Cl}\nolimits} \newcommand{\Spin}{\mathop{\rm Spin}\nolimits} \newcommand{\Pin}{\mathop{\rm Pin}\nolimits} \newcommand{\Hom}{\mathop{\rm Hom}\nolimits} \newcommand{\End}{\mathop{\rm End}\nolimits} \)

Bogomolov-Beauville classification

Table of Contents

The PDF version of this page can be downloaded by replacing html in the its address by pdf. For example /html/sheaf-cohomology.html should become /pdf/sheaf-cohomology.pdf.

This post is a part of the memoire of my M1 internship at I2M. The memoire contains, needless to say, less errors than this page.

1 From the Riemannian results of de Rham and Berger

We will first prove a (conceptually) straightforward result of de Rham decomposition and Berger classification. The following theorem is taken from Beauville's article

Let \(X\) be a compact Kähler manifold with flat Ricci curvature, then

  1. The universal covering space \(\tilde X\) of \(X\) decomposes isometrically and holomorphically as \[\tilde X = E \times\prod_i V_i\times \prod_j X_j\] where \(E = \mathbb{C}^k\), \(V_i\) and \(X_j\) are simply-connected compact manifolds of real dimension \(2m_i\) and \(4r_j\) with irreducible homonomy \(SU(m_i)\) for \(V_i\) and \(Sp(r_j)\) for \(X_j\). One also has uniqueness in the strong sense as in de Rham decomposition.
  2. There exists a finite covering space \(X'\) of \(X\) such that \[ X' = T\times \prod_i V_i \times \prod_j X_j\] where \(T\) is a complex torus.

Note that the first point is obtained directly from Cheeger-Gromoll splitting and de Rham decomposition: The one-dimensional parallel subspaces (of trivial holonomy) are regrouped to \(E\). By Cheeger-Gromoll splitting, \(\tilde X = E\times M\) where \(M\) contains no line and is compact (note that we use compactness of \(X\) here). The irreducible factors in \(M\) are not symmetric spaces as Ricci curvature of symmetric spaces is non-degenerate. Holonomy of these factors are \(SU(m_i)\) and \(Sp(r_j)\) according to Berger list since they are Kähler manifolds and Ricci-flat. It remains to prove the second point.

We will regard each element of \(\pi_1(X)\) by its isometric, free, proper action on \(\tilde X\). As pointed out the arguments in our discussion of uniqueness of de Rham decomposition, every isometry of \(\tilde X\) to itself preserves the components \(T_{x_0}E\), \(T_{x_i}V_i\) and \(T_{x_j}X_j\) of \(T_x \tilde X\), each isometry \(\phi\) of \(\tilde X\) is of form \((\phi_1,\phi_2)\) where \(\phi_1\in Isom(E)\) and \(\phi_2\in Isom(M)\).

We will use here the fact that if \(M\) is a Kähler manifold, compact and Ricci-flat then \(Isom(M)\) equipped with compact-open topology is discrete, therefore finite, which will be proved later (Lemma 1). We note \(\Gamma := \{\phi = (\phi_1,\phi_2)\in \pi_1(X),\ \phi_2 = Id_M\}\) and sometime abusively regard \(\Gamma\) as a subgroup of \(Isom(E)\). Note that \(\Gamma\) is a normal subgroup of \(\pi_1(X)\) with finite index since the quotient is isomorphic to a subgroup of \(Isom(M)\). Therefore \(\tilde X/\Gamma = E/\Gamma\times M\) is compact as a finite cover of \(X\).

We apply the following theorem of Bieberbach.

Let \(E = \mathbb{R}^n\) be an Euclidean space and \(\Gamma\) be a subgroup of \(Isom(E)\) that satisfies

  1. \(\Gamma\) is discrete under compact-open topology.
  2. \(E/\Gamma\) is compact.

Then the subgroup \(\Gamma'\) of translations in \(\Gamma\) is of finite index.

Suppose that the two conditions are satisfied then the theorem gives: \(\tilde X/\Gamma' = E/\Gamma'\times M = T\times \prod_i V_i\times \prod_j X_j\) is a finite cover of \(\tilde X/\Gamma\) as \(\Gamma'\) is a normal subgroup of \(\Gamma\):

Fact. The subgroup of translations in \(Isom(E)\), where \(E = \mathbb{R}^n\) is an Euclidean space, is normal.

Therefore \(X' = \tilde X/\Gamma'\) is a finite cover of \(X\) that we want to find.

It remains to prove that \(\Gamma\) is discrete, which is a consequence of

  1. \(\pi_1(X)\) is discrete, without limit point in \(Isom(E)\times Isom(M)\) (obvious).
  2. \(Isom(M)\) is compact.

In fact given any \(\phi = (\phi_1,\phi_2) \in Isom(E)\times Isom(M)\), there exists by (1.) a neighborhood \( \mathcal{U}_1(\phi_1,\phi_2)\times \mathcal{U}_2(\phi_1,\phi_2)\) of \(\phi\) in \(Isom(E)\times Isom(M)\) such that all points of \(\pi_1(X)\) lying in this region project to \(\phi_1\). By (2.) we can find a neighborhood \( \mathcal{U}_1\) of \(\phi_1\) in \(Isom(E)\) small enough that \( \mathcal{U}_1(\phi_1)\times Isom(M) \subset \cup_{\phi_2\in Isom(M)} \mathcal{U}_1(\phi_1,\phi_2)\times \mathcal{U}_2(\phi_1,\phi_2)\). Therefore the projection of \(\pi_1(X)\) to \(Isom(E)\) is discrete, by consequence \(\Gamma\) is discrete.

Let \(M\) be is a compact, simply-connected, Ricci-flat, Kähler manifold, then the group \(Aut(M)\) of automorphism of \(M\) equipped with compact-open topology is discrete, therefore \(Isom(M)\) is discrete, hence finite.

The idea is that since \(Aut(M)\) is a Lie group, it suffices to prove that its Lie algebra is of dimension 0. This is done using these facts.

  1. The Lie algebra of \(Aut(M)\) can be identified with the vector space of holomorphic vector fields on \(M\).
  2. Bochner's principle: All holomorphic tensor fields on a compact, Ricci-flat Kähler manifold are parallel.
  3. The only invariant vector of the holonomy representation of \(M\) is \(0\) (obvious).

Bochner principle for holomorphic vector fields comes from the following identity (called Weitzenbock formula): \[ \Delta (\frac{1}{2}\|X\|^2) = \| \Delta X\|^2 + g(X, \nabla \text{div} X) + Ric(X,X) \] for every vector field \(X\). If \(X\) is holomorphic then it is harmonic and has \(\text{div} X = 0\). The fact that \(M\) is Ricci-flat gives \(\Delta (\frac{1}{2}\|X\|^2) = \| \nabla X\|^2\) and the function \(\| X\|^2\) is subharmonic, therefore constant since \(M\) is compact. We then have \(\nabla X = 0\),i.e. \(X\) is parallel. The method of Bochner also works for tensor fields of any type in a Ricci-flat Kähler manifold and one also has \(\Delta(\|\tau \|^2) = \|\nabla\tau \|^2\) and that every holomorphic tensor field is parallel. See P. Petersen, Riemannian geometry and A. Besse, Einstein Manifolds for more detail.

2 Towards a classification for complex manifold

To obtain a translation of Theorem 1 in a context of complex manifolds (without any preferred metric a priori), we study the 2 building blocks: manifolds with holonomy \(SU(m)\) and \(Sp(r)\). To be clear, recall that a complex manifold \(X\) is called of Kähler type if one can equip \(X\) with an Hermitian structure whose fundamental form \(\omega\) satisfies \(d\omega= 0\). When we say \(X\) is of Kähler type, we refer to \(X\) as a complex manifold without fixing a metric on \(X\).

2.1 Special unitary manifolds (proper Calabi-Yau manifolds)

Let \(X\) be a compact Kähler manifold with holonomy \(SU(m)\) and complex dimension \(m\geq 3\) then:

  1. \(H^0(X, \Omega_X^p)=0\) for all \(0 < p < m\), by consequence \(\chi( \mathcal{O}_X ) = 1 + (-1)^m\).
  2. \(X\) is projective, that is \(X\) can be embedded into \(\mathbb{P}^N\) as zero-locus of some (finitely) homogeneous polynomials.
  3. \(\pi_1(X)\) is finite and if \(m\) is even, \(X\) is simply connected.

The first point is in fact algebraic in nature: it comes from the fact that the representation of \(SU(m)\) over \(\bigwedge^pT^*_xM\) is irreducible for all \(p\) et non-trivial for \(0

The second point follows the following facts:

  1. (Kodaira's theorem) A compact Kähler manifold with \(H^{2,0}=0\) can be embedded in \(\mathbb{P}^N\).
  2. (Chow's theorem) A compact complex manifold embedded in \( \mathbb{P}^N\) is algebraic, i.e. defined by a finite number of homogeneous polynomials.

The third point is a direct consequence of Riemann-Hurwitz formula. In fact, the universal cover \(\tilde X\) of \(X\) is of holonomy \(SU(m)\). This is due to the following remarks: \(Hol(X)\supset Hol(X')\supset Hol_0(X') = Hol_0(X)\) and \(Hol_0(X) = Hol(X) = SU(m)\) as \(SU(n)\) is connected.

By Theorem 1, \(\tilde X\) is compact by Lemma 1 a finite covering of \(X\) as \(\pi_1(X)\) is finite. As \(\chi(\mathcal{O}_X) = \chi(\mathcal{O}_{\tilde X}) = 2\), one has \(X = \tilde X\), hence \(X\) is simply-connected.

Given a compact manifold \(X\) of Kähler type and complex dimension \(m\), the following properties are equivalent

  1. There exists a compatible metric \(g\) over \(X\) such that \(Hol(X,g) = SU(m)\).
  2. \(K_X\) is trivial and \(H^0(X', \Omega_{X'}^p) =0\) for every \(0 < p < m\) and \(X'\) a finite covering of \(X\).

(1) implies (2) as a finite covering space \(X'\) of a special unitary manifold \(X\) is still a special unitary.

For the implication (2) \(\implies\) (1): by Yau's theorem we equip \(X\) with a Ricci-flat metric, by Theorem 1, there exists a finite cover \(X' = T\times \prod_i V_i\times \prod_j X_j\) where \(T\) is a complex torus, \(Hol(V_i) = SU(m_i), Hol(X_j) = Sp(r_j)\). But \(H^0(X',\Omega^p_{X'})=0\) for \(0

Theorem 2.1 allows us to check if a manifold \(X\) is special unitary by looking at the \(h^{0,p} (0

  1. Elliptic curves over \( \mathbb{C}\) are special unitary, as any statement starting with "for every \(0
  2. A K3 surface (simply-connected surface with trivial canonical bundle) is special unitary, its Hodge diamond is given below.
  3. A quintic threefold (hypersurface of degree 5 in 4-dimensional projective space) is a special unitary manifold, the Hodge diamond of which is given is given below. In particular, the Fermat quintic defined by \[\{(z_0:z_1:z_2:z_3:z_4) \in \mathbb{C}\mathbb{P}^{4}:\ \sum z_i^5 =0 \}\]
  4. In general, any smooth hypersurface \(X\) of \(\mathbb{C}\mathbb{P}^{m+1}\) of degree \(m+2\) satisfies \(h^{0,p}=0\) for all \(0
Table 1: Hodge diamond of a K3 surface.
    1    
  0   0  
1   20   1
  0   0  
    1    
Table 2: Hodge diamond of a quintic threefold.
      1      
    0   0    
  0   1   0  
1   101   101   1
  0   1   0  
    0   0    
      1      

2.2 Irreducible symplectic and hyperkähler manifolds

Let \(X\) be a compact Kähler manifold with holonomy \(Sp(r)\) and complex dimension \(2r\) then:

  1. There exists a holomorphic 2-form \(\varphi\) non-degenerate at every points.
  2. \(H^0(X,\Omega_X^{2l+1}) = 0, H^0(X,\Omega_X^{2l})=\mathbb{C}\varphi^l\) for all \(0\leq l\leq r\). By consequence \(\chi(\mathcal{O}_X)=r+1\).
  3. \(X\) is simply-connected.

The first point of the remark follows directly from our discussion of Berger classification.

The second point is algebraic in nature: The representation of \(Sp(r)\) on \(\bigwedge^p T^*_xM\) splits into

\begin{equation} \label{eq:decomp-varphi} \bigwedge^p T^*_xM = P_p \oplus P_{p-2}\varphi(x) \oplus P_{p-4}\varphi^2(x)\oplus \dots \end{equation}

where \(P_k, 0\leq k\leq r\) are irreducible, non-trivial for \(k>0\) and \(\varphi(x)\in\bigwedge^2 T^*_xM\) uniquely defined up to a constant. Therefore the only invariant elements are \(c\varphi^{p/2}\) where \(c\) is a scalar.

For the last point, one uses the same arguments as Remark 2.1.

Given a compact manifold \(X\) of Kähler type and complex dimension \(2r\), then:

  1. The following properties are equivalent. \(X\) is called hyperkähler if it satisfies one of them.
    1. There exists a compatible metric \(g\) such that \(Hol(X,g) \subset Sp(r)\).
    2. There exists a compatible symplectic structure: a 2-form that is closed, holomorphic and non-degenerate at every point.
  2. The following properties are equivalent. \(X\) is called irreducible symplectic if it satisfies one of them.
    1. There exists a compatible metric \(g\) such that \(Hol(X,g) = Sp(r)\)
    2. \(X\) is simply-connected and there exists (uniquely up to a constant) a compatible symplectic structure on \(X\).

By "compatible", we mean "compatible with the complex structure".

  1. The fact that (a) implies (b) is obvious. For the other way: since \(K_X\) is trivial (existence of global non-null section) by Yau's theorem we equip \(X\) with a Ricci-flat metric, then the symplectic structure \(\varphi\) of \(X\) is parallel by Bochner's principle. Hence the holonomy is in \(Sp(r)\).
  2. For the implication (a) \(\implies\) (b), it suffices to notice that the invariant elements \(\varphi\) in the decomposition \eqref{eq:decomp-varphi} is unique. For the direction (b) \(\implies\) (a), note that \(X\) can be equipped with a Calabi-Yau metric by the (b) \(\implies\) (a) part of (1.), by Theorem 1, \(X = \prod_{j=1}^m X_j\) where \(X_j\) are irreducible compact Kähler manifolds. The symplectique structure \(\varphi\) on \(X\), restricted on each \(X_j\), gives a symplectique structure \(\varphi_j\) of \(X_j\). But any form \(\sum_j \lambda_j pr_j^*\varphi_j\) is another symplectic structure of \(X\), one must have \(m=1\) by uniqueness of \(\varphi\).
  1. One can notice a trivial example: Every special unitary manifold of 2 complex dimensions is irreducible symplectic because \(SU(2)\) is isomorphic to \(Sp(1)\).
  2. Let \(X\) be a smooth cubic hypersurface in \(\mathbb{C}\mathbb{P}^{n+1}\) and \(F(X)= \{ L \in Gr(1, \mathbb{C}\mathbb{P}^{n+1}) , L \subset X\} \subset Gr(1, \mathbb{C}\mathbb{P}^{n+1})\) the manifold formed by lines in \(X\). \(F(X)\) is non-empty when \(n>1\), smooth if \(X\) is smooth and of dimension \(2n-4\). Beauville and Donagi proved that for \(n=4\), \(F(X)\) is irreducible symplectic, therefore hyperkähler.

2.3 Decomposition for complex manifold with vanishing Chern class

Theorem 1 can be translated to a decomposition for complex manifold in the following way:

Let \(X\) be a compact manifold of Kähler type of vanishing first Chern class.

  1. The universal covering space \(\tilde X\) of \(X\) is isomorphic to a product \(E\times \prod_i V_i\times\prod_j X_j\) where \(E = \mathbb{C}^k\) and

    1. Each \(V_i\) is a projective simply-connected manifold of complex dimension \(m_i\geq 3\), with trivial \(K_{V_i}\) and \(H^0(V_i,\Omega_{V_i}^p) = 0\) for \( 0 < p < m_i\)
    2. Each \(X_j\) is an hyperkähler manifold.

    This decomposition is unique up to an order of \(i\) and \(j\).

  2. There exists a finite cover \(X'\) of \(X\) isomorphic to the product \(T\times\prod_i V_i\times\prod_j X_j\).

The theorem follows directly from Theorem 1, the only point that needs proof is the uniqueness, which will be achieved in two steps:

  1. Prove the uniqueness in the case that \(X\) is simply-connected.
  2. Prove that every isomorphism \(\phi:\ \mathbb{C}^k\times Y\longrightarrow \mathbb{C}^h\times Z\) is splitted as \(\phi = (\phi_1,\phi_2)\) where \(\phi_1:\ \mathbb{C}^k\longrightarrow \mathbb{C}^h\) and \(\phi_2:\ Y\longrightarrow Z\) are isomorphisms (by consequence \(h=k\)).

These two steps will be accomplished in the following two lemmas

Let \(Y = \prod_j Y_j\) be a finite product of compact, simply-connected manifold of Kähler type with vanishing Chern class. The Calabi-Yau metrics of \(Y\) are then \(g = \sum_l pr_j^*g_j\) where \(g_j\) are Calabi-Yau metrics of \(Y_j\).

Let \(g\) be a Calabi-Yau metric of \(Y\) and \([\omega]\) its class in \(H^{1,1}(Y)\). Since \(Y_j\) are simply-connected, \([\omega] = \sum_j pr_j^* [\omega_j]\). By Yau's theorem, there exist unique Calabi-Yau metrics \(g_j\) of \(Y_j\) in each class \([\omega_j]\). The metric \(g' = \sum_j pr_j^* g_j\) is in the same class \(\omega\) of \(g\) and is also a Calabi-Yau metric, hence \(g= g' = \sum_j pr_j^*g_j\).

This lemma asserts that when our manifolds \(Y, Y_j\) are equipped with appropriate Calabi-Yau metrics, the decomposition map is also a (Riemannian) isometric, we therefore obtain uniqueness of \(V_i, X_j\) from uniqueness of Theorem 1.

Let \(Y,Z\) be compact, simply-connected manifold of Kähler type, then any isomorphism \(u:\ \mathbb{C}^k\times Y\longrightarrow \mathbb{C}^h\times Z\) is splitted as \(\phi = (\phi_1,\phi_2)\) where \(\phi_1:\ \mathbb{C}^k\longrightarrow \mathbb{C}^h\) and \(\phi_2:\ Y\longrightarrow Z\) are isomorphisms.

It is clear that the composed function \(u_1: \mathbb{C}^k\times Y \longrightarrow \mathbb{C}^h \times Z \longrightarrow \mathbb{C}^h\) is constant in \(Y\), i.e. \(u_1(t,y) = u_1(t)\) as holomorphic functions on \(Y\) are constant, therefore \(u(t,y) = (u_1(t), u_2(t,y))\). As \(u\) is isomorphic, one has \(h\leq k\) then by the same argument for \(u^{-1}\), one has \(h=k\), \(u_1\) is an isomorphism and \(u_2(t,\cdot)\) is an isomorphism from \(Y\) to \(Z\). \(u_2(0,\cdot)^{-1}\circ u_2(t,\cdot)\) is then a curve in \(Aut(Y)\), which is discrete by Lemma 1. Therefore \(u_2(t,\cdot)= u_2(0,\cdot)\) independent of \(t\).

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Last updated 2017-08-31 Thu 02:24.