Calabi-Yau theorem

1. Calabi conjecture
2. Reduction to local charts, Yau theorem
3. A sketch of proof
4. Calabi-Yau manifold

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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 Calabi conjecture

In complex geometry, one usually defines the Ricci curvature to be the real (1,1)-form \(\rho\) with \(\rho(u,v) = Ric(Ju, v) = \text{tr}(w \mapsto R(w,v).Ju)\), as it has the advantage of being an antisymmetric form.

We will call \(\rho\) the Ricci form when it is easy to confuse with the Ricci curvature tensor in Riemannian geometry. We start with the following fact (which is exercise 4.A.3 in Huybrechts, Complex geometry: an introduction).

For our convenience when talking about positivity, we would rather use the anticanonical bundle. Then \(K_{X}^{-1}\) is positive (resp. semi-positive) if and only if \(Ric\) is positive definite (resp. positive semi-definite) as a symmetric form.

We start with the following fact (which is exercise 4.A.3 in Daniel Huybrechts, Complex geometry: an introduction)

Let \((X,g)\) be a compact Kähler manifold. Then \(i\rho(X,g)\) is the curvature of the Chern connection on the canonical bundle \(K_X\). In other words, \(\rho(X,g)\in -2\pi c_1(K_X)\) where \(c_1(K_X)\) is the first Chern class of \(K_X\).

The quadruple \((h, g, \omega, J)\) is said to be compatible if \(g\circ J = g\) and \(\omega(a,b) = g(Ja,b)\) and \(h = g - i\omega\).

When \(J\) is fixed, one of \(h,g,\omega\) that is invariant by \(J\) determines the two others.
For a compatible quadruple, the condition \(\nabla J = 0\) is equivalent to \(d\omega = 0\). The fundamental form \(\omega\) that satisfies \(d\omega = 0\) is called a Kähler form.

The Calabi conjecture asked whether for each form \(R\in c_1(K_X)\) one can find a metric \(g'\) whose new fundamental form \(\omega'\) is in the same class of \(\omega\) and \(Ric(X,g') = R\). We prefer to work with the fundamental form instead of the metric \(g\) as the former is antisymmetric and its derivative is hence easy to define.

2 Reduction to local charts, Yau theorem

2.0.0.1 \(h,g,\omega\) in local coordinates.

We note by \(h_{i\bar j} = h(\partial_{x_i},\partial_{x_j}) = 2g_{\mathbb{C}}(\partial_{z_i},\partial_{z_j})\). By straightforward calculation one has

\begin{align*} \omega & = -\frac{1}{2} Im h_{i\bar j} (dx^i\wedge dx^j + dy^i\wedge dy^j) + Re h_{i\bar j}dx^i\wedge dy^j\\ & = \frac{i}{2}h_{i\bar j}dz^i\wedge d\bar{z^j} \end{align*}

and the condition \(d\omega = 0\) is equivalent to \[ \frac{\partial h_{i\bar j}}{\partial z_k} = \frac{\partial h_{k\bar j}}{\partial z_i} \] We also note by \(h^{i\bar j}\) the inverse transposed of \(h_{i\bar j}\), i.e. \(h^{i\bar j}h_{k\bar j} = \delta_j^k\)

Let \(X\) be an almost complex manifold (manifold with an almost complex structure). Then \(d:\bigwedge^nT^*X\longrightarrow \bigwedge^{n+1}T^*X\) sends \(\bigwedge^{p,q}T^*M\) to \(\bigwedge^{p+1,q}T^*M\oplus \bigwedge^{p,q+1}T^*M\). We denote by \(\partial\) and \(\bar\partial\) the component of \(d\) in \(\bigwedge^{p+1,q}T^*M\) and \(\bigwedge^{p,q+1}T^*M\) respectively.

It would be convenient to define \(d^c =i(\bar\partial - \partial)\) then obviously \(dd^c = 2i\partial\bar\partial\).

2.0.0.2 The Ricci curvature.

The Ricci curvature form is given in local coordinates by \[ Ric_{\omega} = -\frac{1}{2}dd^c\log\det(h_{i\bar j}) \]

2.0.0.3 \(dd^c\) lemma .

We then can state the \(dd^c\) lemma

Let \(\alpha\) be a real, (1,1)-form on a compact Kähler manifold \(M\). Then \(\alpha\) is \(d\) -exact if and only if there exists \(\eta\in C^\infty(M)\) globally defined such that \(\alpha = dd^c\eta\).

2.0.0.4 Yau's theorem.

The \(dd^c\) lemma tells us that every form \(R\in c_1(K_X)\) is of form \(Ric_{\omega} + dd^c\eta\). If one varies the Hermitian product \(h_{i\bar j}\) to \(h_{i\bar j} + \phi_{i\bar j}\) then the new Ricci curvature is \(dd^c\log\det(h_{i\bar j} + \phi_{i\bar j})\). The Calabi conjecture can be restated as the existence of \(\phi\) such that \(h_{i\bar j} + \phi_{i\bar j}\) is definite positive and

\begin{equation} \label{eq:ddc-0} dd^c\left( \log\det(h_{i\bar j} + \phi_{i\bar j}) - \log\det(h_{i\bar j}) -\eta\right) = 0 \end{equation}

The functions \(f\) that satisfies \(dd^cf = 0\) are called pluriharmonic. They also satisfy the maximum principle. By compactness of \(X\), these functions on \(X\) are exactly constant functions. Therefore \eqref{eq:ddc-0} is equivalent to \[ \det(h_{i\bar j} + \phi_{i\bar j}) = e^{c+\eta}\det(h_{i \bar j}) \] or by \(dd^c\) lemma: \[ (\omega + dd^c\phi)^n = e^{c+\eta}\omega^n \] where \(\omega^n\) denotes the repeated wedge product. Note that \((\omega +dd^c\phi)^n - \omega^n\) is exact, one has \(\int_M (\omega +dd^c\phi)^n = V\), the conjecture of Calabi is therefore a consequence of the following theorem.

Given a function \(f\in C^\infty(M), f>0\) such that \(\int_M f\omega^n = V\). There exists, uniquely up to constant, \(\phi\in C^\infty(M)\) such that \(\omega + dd^c\phi >0\) and \[ (\omega + dd^c\phi)^n = f\omega^n \]

3 A sketch of proof

The uniqueness is straightforward. In fact if \(\phi\) and \(\psi\) both satisfy \(\omega + dd^c\phi >0\), \(\omega + dd^c \psi >0\) and \((\omega + dd^c\phi)^n = (\omega + dd^c\psi)^n\) then \(D(\phi - \psi) = 0\) as \[ 0 = \int_M (\phi - \psi)((\omega + dd^c\phi)^n - (\omega + dd^c\psi)^n) = \int_M d(\phi -\psi)\wedge d^c (\phi -\psi) \wedge T \] where \[ T =\sum_{j=0}^{n-1}(\omega + dd^c\phi)^j\wedge (\omega + dd^c\psi)^{n-1-j} \] is a closed (strongly) positive \((n-1,n-1)\) -form.

We will prove the existence of \(\phi\) under the constraint \(\int_M\phi\omega^n = 0\) (which will be useful to prove that \(\mathcal(N)\) is locally diffeomorphism later). We will prove that the set \(S\) of \(t\in [0,1]\) such that there exists \(\phi_t\in C^{k+2,\alpha}(M)\) with \(\int_M \phi_t\omega^n = 0\) that satisfies

\begin{equation} \label{eq:omega-convex-t} (\omega + dd^c\phi_t)^n = (tf + 1-t)\omega^n \end{equation}

is both open and close in \([0,1]\), therefore is the entire interval as \(0\in S\).

To see that \(S\) is open, one only has to prove that the function \(\mathcal{N}\) defined by \[ \phi\mapsto \mathcal{N}(\phi)= \frac{\det(h_{i\bar j} + \phi_{i\bar j})}{\det(h_{i\bar j})} \] or in other words \((\omega + dd^c\phi)^n = \mathcal{N}(\phi)\omega^n\), is a local diffeomorphism. The differential of \( \mathcal{N}\) is given by \[ D \mathcal{N}(\phi).\eta = \mathcal{N}\Delta\eta \] with \(\eta\) varies in \(\{\eta\in C^{k,\alpha}(M):\int_M\eta\omega^n=0\}\). and \(\Delta\) is the Laplace-Beltrami operator which is known to be bijective between \[ \left\{\eta\in C^{k+2,\alpha}(M):\ \int_M\eta = 0\right\} \longrightarrow \left\{f\in C^{k,\alpha}(M):\ \int_M f=0\right\} \] Therefore \( \mathcal{N}\) is a local diffeomorphism and \(S\) is open.

The proof that \(S\) is closed is more technical and is accomplished in 3 steps:

Using Arzela-Ascoli theorem, it suffices to show that \(\{\phi_t:\ t\in S\}\) is bounded in \(C^{k+2,\alpha}\). Then up to a subsequence, one has the uniform convergence of \(\phi_{t_n}\) and all its partial derivatives of order \(\leq k+1\). The \(k+2\) -th order follows from \eqref{eq:omega-convex-t}.
Using Schauder theory, prove that the above bound follows from a priori estimate: \newline There exists \(\alpha\in (0,1)\) and \(C(X,\|f\|_{1,1}, 1/\inf_M f)>0\) such that every \(\phi\in C^4(X)\) satisfying \((\omega +dd^c\phi)^n = f\omega^n\), \(\omega + dd^c\phi >0\) and \(\int_M \phi\omega^n=0\) (we will call such \(\phi\) admissible) has \[\| \phi\|_{2,\alpha} \leq C.\]
Establish the a priori estimate.

To achieve the a priori estimate, one firstly bounds \(\phi\) in \(C^0\), then bound \(\| \Delta\phi\|\) and finally establishs the \(C^{2,\alpha}\) estimate. We will give here some detail of the first step. For more detail, see Z. Blocki, The Calabi-Yau Theorem.

Since \(\phi\) is defined up to an additive constant, what we mean by the \(C^0\) -estimate for \(\phi\) is in fact the estimate of \[ \text{osc}_M \phi := \max_M \phi - \min_M \phi \] by a constant \(C\) that depends only on \(M\) and \(f\). Without losing of generality, one assumes that \(\int_M \omega^n = 1\) and \(\max_M \phi = -1\). Therefore \(\| \phi \|_p \leq \| \phi\|_q\) for \(p\leq q<\infty\).

One has

\begin{align} \int_M (-\phi)^p (f-1)\omega^n &= \int_M (-\phi)^p dd^c\phi \wedge \left( \sum_{j=0}^{n-1} (\omega + dd^c\phi)^j\wedge \omega^{n-1-j}\right) \\ &= p \int_M (-\phi)^{p-1} d\phi \wedge d^c\phi \wedge \left( \omega^{n-1} + \sum_{j=1}^{n-1}(\omega + dd^c\phi)^j\wedge \omega^{n-1-j} \right)\\ &\geq p\int_M (-\phi)^{p-1}d\phi\wedge d^c\phi\wedge \omega^{n-1}\\ &=\frac{4p}{(p+1)^2}\int_M d(-\phi)^{(p+1)/2}\wedge d^c(-\phi)^{(p+1)/2}\wedge \omega^{n-1}\\ &=\frac{c_n p}{(p+1)^2}\| D(-\phi)^{(p+1)/2}\|_2^2 \label{eq:long-edp} \end{align}

where we used the fact that \(\omega + dd^c\phi >0\) in the inequality, and \(c_n\) is a constant depending only on \(n\).

Now we use the following Sobolev inequality on \(M\) (i.e. use Sobolev inequality in each chart as a domain of \( \mathbb{R}^m\) then add up the results): \[ \|v \|_{mq/(m-q)} \leq C(M,q) (\| v\|_q + \|Dv\|_q),\quad \forall v\in W^{1,q}(M), q

Repeatedly apply this inequality (this technique is called Moser's iteration) one has \(\|\phi \|_{p_{k+1}} \leq (Cp_k)^{1/p_k} \|\phi\|_{p_k}\) where the sequence \(p_k\) is defined by \(p_0 = 2\) and \(p_{k+1} = \frac{n}{n-1}p_{k-1} = 2(\frac{n}{n-1})^k\) and \[ \|\phi\|_{\infty} = \lim_{k\to\infty}\|\phi\|_{p_k} \leq \|\phi\|_2 \prod_{j=0}^\infty (Cp_j)^{1/p_j} \] with \(\prod_{j=0}^\infty (Cp_j)^{1/p_j} = (n/(n-1))^{n(n-1)/2} (2C)^{n/2}\)

The fact that \(\|\phi\|_2\) is bounded follows directly from the following lemma.

For any admissible \(\phi\) with \(\max_M \phi = -1\) one has \[ \| \phi\|_p\leq C(M,p),\quad \forall 1\leq p\leq\infty \]

We will prove the lemma with \(p=1\) first. Let \(g\) be the local potential of the Kähler form \(\omega\), i.e. a function defined on each chart (not necessarily agrees on zones where charts are glued together) such that \(\omega = dd^c g = \frac{\sqrt{-1}}{2}g_{i\bar j}dz_i\wedge d\bar z_j\) where \(g_{i\bar j}\) can also be intepreted as \(\frac{\partial^2}{\partial z_i \partial\bar z_j} g\). We also suppose that the function \(g\) is negative on every chart. The fact that \(\omega + dd^c\phi >0\) is rewritten as \((g_{i\bar j} + \phi_{i\bar j}) >0\) in local coordinates.

Note \(u=g + \phi\) the potential of \(\omega + dd^c\phi\) locally defined on each chart, then \(u\) is negative and plurisubharmonic (psh). For every \(x\in B(y,R)\) one has \[ u(x) \leq \frac{1}{\text{vol}(B(x,2R)}\int_{B(x,2R)} u \leq \frac{1}{\text{vol}(B(y,2R))}\int_{B(y,R)}u \] where the first inequality is due to plurisubharmonicity and the second is due to \(u\leq 0\). Therefore \[ \| u\|_{L^1(B(y,R))} \leq \text{vol}(B(y,2R)) \inf_{B(y,R)} |u|, \] hence \[ \|\phi\|_{L^1(B(y,R))} \leq \|u\|_{L^1(B(y,R))} \leq \text{vol}(B(y,2R)) (\inf_{B(y,R)} |\phi| + \max_M |g|) \] To see that \(\|\phi\|_1\) is bounded, we apply the following Lemma 3 to the covering of \(M\) by finitely many ball \(B(y_i,R_i)\), \(c_i = \text{vol}(B(y_i,2R_i))\), \(d_i = c_i \max_M |g|\) and \(r=1\).

The case \(p>1\) follows analoguously using the following estimate: if \(u\) is negative and psh in \(B(y,2R)\) then \[ \|u \|_{L^{p}(B(y,R)} \leq C(n,p,R)\|u\|_{L^1(B(y,2R))} \]

Let \(M\) be a connected compact manifold covered by finitely many local charts \(\{V_i\}_{i=1}^{l}\) and \(r, c_i, d_i>0\). Then for any continuous function \(\phi\) globally defined on \(M\) such that \[ \|\phi\|_{L^1(V_i)} \leq c_i \inf_{V_i} |\phi| + d_i,\quad \min_M |\phi| \leq r, \] one has \(\|\phi\|_1:= \sum_i \|\phi\|_{L^1(V_i)}\leq C(\{V_i\},\{c_i\}, \{d_i\}, r)\)

Let \(p\) be a point in \(M\) where \(|\phi|\) attains its minimum. Since \(M\) is connected, for every \(V_i\), there exists a sequence \(V_{i_k}, 0\leq k\leq l\) such that \[ i_0 = i,\quad V_{i_k}\cap V_{i_{k+1}}\ne \emptyset,\quad p\in V_{i_l} \] One has

\begin{align*} \|\phi\|_{L^1(V_{i_k})} &\leq c_{i_k} \inf_{V_{i_k}} |\phi| + d_{i_k} \leq c_{i_k} \inf_{V_{i_k}\cap V_{i_{k+1}}} |\phi| + d_{i_k}\\ &\leq c_{i_k}\frac{1}{\text{vol}(V_{i_k}\cap V_{i_{k+1}})}\|\phi\|_{L^1(V_{i_{k+1}})} + d_{i_k} \end{align*}

Repeatedly apply this inequality for \(k=0,\dots,l-1\), one has

\begin{align*} \|\phi\|_{L^1(V_i)} &\leq A(i,\{V_j\},\{c_j\}, \{d_j\}) \|\phi\|_{L^1(V_{i_l})} + B(i,\{V_j\},\{c_j\}, \{d_j\})\\ &\leq A(i,\{V_j\},\{c_j\}, \{d_j\}) (c_{i_l} r+ d_{i_l}) + B(i,\{V_j\},\{c_j\}, \{d_j\}) \end{align*}

Take the sum for all \(i=0,\dots, l\) and the result follows.

4 Calabi-Yau manifold

Recall that we defined a Calabi-Yau manifold to be a compact Riemannian manifold of dimension \(2n\) with holonomy contained in \(SU(n)\). We also remark, using parallel transport, the existence of a compatible complex structure (\(U(n)\) suffices) and a holomorphic form non-vanishing at every point. We present here some equivalent definitions of compact Calabi-Yau manifolds.

Let \(X\) be a compact manifold of Kähler type and complex dimension \(n\) then:

The followings are equivalent
1. There exists a Kähler metric such that the global holonomy is in \(SU(n)\).
2. There exists a holomorphic \((n,0)\) form that vanishes nowhere.
3. The canonical bundle \(K_X\) is trivial.
4. The structure group of \(X\) can be reduced to \(SU(n)\).
The following are equivalent. If \(X\) is simply-connected, they are equivalent with the 4 statements above.
1. There exists a Kähler metric such that the local holonomy is in \(SU(n)\).
2. The canonical bundle \(K_X\) is flat.
3. There exists a Kähler metric such that the Ricci curvature vanishes.
4. The first Chern class vanishes.

The proof is straightforward (see Manuscript) with the only non-trivial part is when one needs Calabi-Yau theorem to construct Ricci-flat metric.