Calabi–Yau metric on K3 surface

A hermitian manifold is a manifold \( M^{2n} \) with an almost complex structure \( J\) such and a hermitian metric \( h \), i.e. \( J^2=-1 \), \(h \) is \( i \)-linear, \( h(v,u)=\overline{h(u,v)} \), \( h(u,u) \) is positive and \( h(Ju,v) = ih(u,v) = h(u, -Jv) \) for all \( u,v\in C\otimes TM \). Writing \( h=g + i\omega \), one sees that \( g \) is a Riemannian metric and \(\omega \) is a nondegenerate 2-form of type (1,1), \( g \) and \( \omega \) are \( J \)-invariant, \( J^*g=g, J^*\omega=\omega \) and one can compute the triplet \( (g,\omega,h) \) knowing one of the three.

The following 3 conditions are equivalent and imply integrability of \( J \): (i) \( \omega \) is closed, (ii) \( \omega \) is parallel w.r.t \( g \), (iii) J is parallel w.r.t \( g \). In this case, one calls \( M \) a Kähler manifold, \( \omega \) the Kähler form (or metric) and its cohomology class in \( H^2(M,\mathbb{R}) \) the Kähler class. ¹ It follows from the following lemma that Kähler metrics of the same cohomology class are parameterised by an open cone of \( \Omega^0(M)/\mathbb{R} \).

If we choose a basis \( \{\partial_{x_i}, \partial_{y_i}\}_{i=\overline{1,n}} \) such that \( J \partial_{x_i} = \partial_{y_i} \) then \( dz_i = dx_i + i dy_i \) are \( i \)-eigenvectors of \( J \), i.e. \( J(dz_i) = J^*(dz_i) = i dz_i \) and \( J(d\bar z_i) = -i d\bar z_i \). The hermitian metric can be writen as \( h = \sum h_{ij}dz_i\otimes d\bar z_j \) where \( h_{ij}=h(\partial_{x_i}, \partial_{x_j})=\overline {h_{ji}}\), and Kähler form is \( \omega = \frac{i}{2}h_{ij}dz_i\wedge d\bar z_j \). Because \( J \) is parallel, the Ricci curvature is \( J \)-invariant, and because it is symmetric, we can turn it into a 2-form called Ricci form \( \rho(u,v)=\Ric(Ju,v) \), which we will abusively denote by \( \Ric \) and can be computed in local coordinates as

Let \( (M,J,\omega_0) \) be a compact Kähler manifold, by \( \partial\bar \partial \)-Lemma one can write any (1,1)-form \(\rho\in -2\pi c_1(K_M) \) as \( \rho = \Ric_0-i \partial \bar \partial \eta \) and any Kähler metric cohomologous to \( \omega_0 \) as \( \omega = \omega_0 + i \partial\bar \partial\phi \). By the previous remark, \( \rho \) is the Ricci form of \( \omega \) if and only if

for a constant \( c \). Because \( [\omega]=[\omega_0] \) the two metrics have equal volume, i.e. \( \int_M (f - 1)\omega_0^n=0 \). Yau resolved Calabi conjecture by showing that for any \( f \) with average volume 1, there exists a potential \( \phi\in \Omega^0(M) \) unique up to constant with \( \omega_0 + i \partial\bar \partial\phi >0 \) satisfying \eqref{eq:c-y}.

Since we are interested in Ricci-flat metric, we will suppose that the canonical bundle is holomorphically trivial, i.e. there is a global nowhere-vanishing holomorphic (n,0)-form \( \chi\) uniquely up to constant (due to Maximum Principle), and the Ricci form can be writen as \( \Ric= -i \partial \bar \partial \log \frac{\omega^n}{\chi\wedge\bar\chi}\) by the first point of Remark rem:ric-ddc. Because \( M \) has no non constant (pluri-) harmonic function, the Ricci flat condition is

for a constant \( \lambda \) depending only \( [\omega] \).

One way to generate Kähler metric locally is by using a Kähler potential, i.e. we set \( \omega_0=0 \) and look for \( \omega =: i \partial\bar \partial\phi \). ³ Now if we restrict to rotational potential \( \phi =\phi(\rho)\) where \( \rho=r^2 \) on \( \mathbb{C}^2 \), the Ricci-flat equation becomes \( (\phi')^2 + r^2 \phi'\phi'' = \lambda \). Up to multiplicative constant, one has 2 solutions: the Euclidean potential \( \phi_E =\rho \) and the Eguchi–Hanson potential \( \phi_{EH}(\rho) = -\tanh^{-1}(1 +\rho^2)^{1/2} \) that explodes at 0. Both of them give a positive metric. We will not need the close formula for \( \phi_{EH} \), but it is useful to keep in mind its development at \( \rho \) near \( 0 \) and \( +\infty \).

which will show that the corresponding Kähler metric extends to the blowup and

which shows that when \( \rho \) is big, the metric looks Euclidean.

A complex torus \( T^2 = \mathbb{C}^2/\Lambda \) has a natural involution \( x\mapsto -x \) which admits \( 2^4=16 \) fixed points (there 2 choices, integer or half-integer, for each generator of \( \Lambda \)) and they become 16 singular points in the quotient space \( T^2/\pm \), whose regular set \( X \) is equiped with the induced metric and complex structure of \( \mathbb{C}^2 \). The neighborhood of each singular is modeled by \( \mathbb{C}^2/\pm \) at 0, in term of metric and complex structure. Let \( Z \) be the blowup of \( T^2/\pm \) at its singular points, we will see soon that \( Z \) is Kähler and its canonical bundle is trivial. Our goal is to rediscover Yau's theorem thm:c-y for \( Z \) as a glueing problem.

To see how blowing up change a singular point, we isomorph \( Y:=\mathbb{C}^2/\pm \) with the surface \(\mathcal{C}:= \{(w_1, w_2, w_3): w_3^2=w_1w_2\}\subset \mathbb{C}^3 \) via \( (z_1,z_2)\mapsto (z_1^2, z_2^2, z_1 z_2) \). The blowup \( \hat{\mathbb{C}^3} \) of \( \mathbb{C}^3 \) at 0 is obtained by replacing the origin by complex lines passing by it, i.e. a \( \mathbb{C}P^2 \): \( \hat{\mathbb{C}^3}:=\{((w_1,w_2,w_3),[u_1:u_2:u_3]):\ u_iw_j=u_jw_i \}\subset \mathbb{C}^3\times \mathbb{C}P^2 \). The blowup of \( \mathcal{C} \) at 0 is obtained by taking preimage (the so-called proper transform of \( \mathcal{C} \)) \[ \hat{\mathcal{C}}:= \left\{ ((w_1,w_2,w_3),[u_1:u_2:u_3])\in \mathbb{C}^3\times \mathbb{C}P^2: w_3^2=w_1w_2,\ u_iw_j=u_jw_i,\ u_3^2=u_1u_2 \right\} \]

Now we will use the expansions eq:EH-near-infty to glue Euguchi–Hanson potential \( \phi_{EH} \) to the Euclidean potential outside of a big ball, i.e. let \( \phi_{Y,R} \) be an intepolation of \( \phi_{EH} \) on \( \rho\leq \frac{R}{4} \) and \( \phi_E= \rho\) on \( \rho > R\): \[ \phi_{Y,R}(\rho) = \phi_{EH}(\rho) + \beta (\frac{\rho}{R}).(\phi_E-\phi_{EH})(\rho) \] where \( \beta\in C^\infty(\mathbb{R}) \) taking value in \( [0,1] \), \( \beta=0 \) on \( (-\infty, \frac{1}{4}+\epsilon) \) and \( \beta = 1 \) on \( [1-\epsilon, +\infty) \). Recall that \( \rho = r_Y^2 \) where \( r_Y \) is the Euclidean distance to the origin on \( Y=\widehat{\mathbb{C}^2/\pm 1} \), descended from that of \( \mathbb{C}^2\), and that \( \phi_E-\phi_{EH} = \frac{1}{2}\rho^{-1} + O(\rho^{-3})=\frac{1}{2}r_Y^{-2} + O(r_Y^{-6}) \), so \( |\nabla^k(\phi_{Y,R} - \phi_{EH})|=O(R^{-1-\frac{k}{2}}) \) and if we denote \( \omega_{Y,R} :=i \partial \bar\partial\phi_{Y,R} \) to be the Kähler metric coming from the potential \( \phi_{Y,R} \) then: \[ \omega_{Y,R} = \omega_{EH} + O(R^{-2}),\qquad \omega_{Y,R}^2 = (1+\eta)\omega_{EH}^2 \] where \( \eta \) is supported in \( \frac{1}{2}R^{1/2}\leq r_X\leq R^{1/2} \) and \( |\nabla^k\eta| = O(R^{-2-\frac{k}{2}}) \) . Therefore \( \Ric_{\omega_{Y,R}} = \Ric_{\omega_{EH}} -i \partial\bar \partial\log(1+\eta) = O(R^{-3})\) in the region of interpolation, and is 0 elsewhere.

Now we will equip the Kummer surface \( Z \) with a family of metrics \( \omega_R \) given by scaling down \( Y \) by \( R \) (in length) and glue to \( X \). Concretely, we build a function \( r_X \) on \( X \) which is globally bounded and equal to the Euclidean distance to singular point in \( T^2/\pm 1 \), then near each singular point \( p_i \), we glue the truncated \( X\setminus\{r_X\leq R^{-1/2}\} \) to the region \( \{r_Y\geq R^{1/2}\} \) in \( Y \) and construct the (1,1)-form \( \omega_R \) on \( Z \) that is the flat \(\omega_X \) on the \( X \)-side, i.e. \( X\setminus \{r_X\leq R^{-1/2}\} \), and is \( R^2 \omega_{Y, R} \) on \( Y \)-side, i.e. \(Y\cap\{r_Y\leq R^{1/2}\}\).

A clearer way to to understand the procedure of "scaling down \(Y\) and glueing it to \(X\)" is via tube metric. We first remark that the quantity \( r^-2\omega \) is scale-invariant:

This can be seen via a change of variable \( t=\log r \), so \( g_E = dr^2 + r^2 g_{S^3} = r^2dt^2 + r^2g_{S^3}\). This means \( r^{-2}g_E \) is the product metric \( dt^2 + g_{S^3} \) of \( S^3\times \mathbb{R} \).

Now we come back to the two pictures of \( X \) and \( Y \) and think of \( X \) as a manifold with 16 cylindrical ends at the singular points and think of \( Y \) as having 1 cylindrical end at infinity of \( \mathbb{C}^2/\pm1 \). We mark the \( S^3/\pm1 \) sections in \( X,Y \) that correspond to \( r_X=1\) and \(r_Y=1 \), then as \( R \) increase, we move further to to ends by tube-distance \(\frac{1}{2}\log R \) and glue the two sections \( \{r_X = R^{-1/2}\} \) and \( \{r_Y = R^{1/2}\} \) together. If we choose one, in our case \( X \), and turn the tube metric into the finite one by multiplying it by \( r^2 \), then it becomes the Gromov–Hausdorff limit in Anderson-Bando-Kasue-Nakajima theorem, while the other becomes the "bubble".

Not only does the tube metric provide a clear geometric view of the glueing procedure, but also an analysis toolbox for solving the Ricci-flat equation.

We want to find a Kähler potential \( \phi \) so that \(\tilde\omega_R:= \omega_R + i \partial\bar \partial\phi \) is Ricci-flat, i.e. \( \tilde\omega_R^2 = (\omega_R +i \partial\bar\partial\phi)^2 = \lambda\chi\wedge\bar\chi \) where \( \chi \) is the non-vanishing holomorphic \( (2,0) \)-form. But since \( \omega_R^2 = \frac{1}{1+\eta}\chi\wedge\bar\chi \) where \( \eta \) is as before (in the glueing procedure, we also recaled the holomorphic (2,0)-form on \( Y \)), i.e. it is supported in \( \frac{1}{2}R^{-1/2}\leq r_X\leq R^{1/2} \) and \( |\eta|=O(R^{-2}) \), one has \( ( \omega_R + i \partial\bar \partial\phi)^2 = (1+\eta)\omega_R^2 \).

We then substitute \( \omega_R \) by the tube metric \( \Theta_R :=r_X^{-2}\omega_R \) and rearrange, and divide by \( \Theta_R^2 \) to have \( 2r_X^{-2} i\partial\bar \partial\phi\wedge\Theta_R/\Theta_R^2 + (r_X^{-2} i\partial\bar \partial\phi)^2/\Theta_R^2 = \lambda(1+\eta)-1 \). This is a PDE, with nonlinearity appears in highest order, its linearisation at \( \phi=0 \) is the first term of LHS. We will do a change the variable \( f = r_X\phi \) and multiply the two sides by \( r_X^3 \) to have

The justification for this manoeuvre is that it turns the linear term into the Laplacian with respect to the tube metric.

The linearisation of \eqref{eq:MA-f} is not invertible, since the function \( f=r_X \) is obviously in the kernel. It turns out that this is the only direction in the kernel. So we should modify the RHS of \eqref{eq:MA-f} by adding the term \( \tau r_X \) where \( \tau\in \mathbb{R} \) and look for pair of solution \( (f,\tau) \). The linearised operator, which is self-adjoint, is obviously invertible. And if we can solve \[ 2r_X i\partial\bar \partial(r_X^{-1}f)\wedge\Theta_R/\Theta_R^2 + r_X^{-3}(r_X i\partial\bar \partial(r_X^{-1}f))^2/\Theta_R^2 = (\lambda(1+\eta)-1)r_X^3 + \tau r_X \] Integrate and use the fact that the \( \omega_R \) volume is the same as the \( \tilde \omega_R \) volume, one can see that \( \tau=0 \).

Now suppose that we can solve the linearised equation, it remains to solve the nonlinear one \eqref{eq:MA-f}. To do this, one uses the quantitative Inverse Function theorem.

It is often proved by contraction mapping, but before going to the detail let us explain how the quantity \( \frac{1}{4|F''|F'(a)^{-1}|^2} \) appears using the Taylor expansion of \( F \): \( F(a +\delta) = F(a) + F'(a)\delta + \frac{1}{2}F''(a)\delta^2 +o(\delta^2) \). Infinitesimally to move \( F \) in a direction \( v \) at \( F(a) \), \( x \) should be move in the direction \( F'(a)^{-1}v \), but to go from "infinitesimally" to "locally" we get interference from higher order term. To guarantee that it is negligible, one needs \[ \left|F''(a)\right||\delta|^2\leq \left|F'(a)\delta\right| \] Since \( |F'(a)^{-1}|^{-1}|\delta|\leq|F'(a)\delta| \), one only needs \( |F''||\delta|^2\leq |F'(a)^{-1}|^{-1}|\delta| \), or \( |\delta| \leq \frac{1}{|F'(a)^{-1}||F''|} \). To garuantee this, \( F \) should move less than \( \frac{1}{|F'(a)^{-1}|^2|F''|} \).