## Notes of Gang Tian’s Orsay lecture 28-06-2017

Existence of conic Kaehler-Einstein metrics

Joint work with Feng Wang, Zhejiang university.

A log-Fano manifold is the date of a compact Kaehler manifold ${M}$, a divisor with normal crossings ${D=\sum(1-\beta_i)D_i}$ such that the line bundle

$\displaystyle \begin{array}{rcl} L=K_M^{-1}+\sum(1-\beta_i)D_i \end{array}$

is positive.

A metric ${\omega}$ is a conic Kaehler-Einstein metric if it is smooth Kaehler in ${M\setminus |D|}$ and for every point ${p\in|D|}$ where ${D}$ is defined by ${z_1\cdots z_d=0}$ in some coordinates, ${\omega}$ is equivalent (between two multiplicative constants) to the model cone metric

$\displaystyle \begin{array}{rcl} \omega_{cone}=i(\sum_{i\leq d} \frac{dz_i\wedge d\bar z_i}{|z_i|^{2(1-\beta_i)}}+\sum_{i>d} dz_i\wedge d\bar z_i). \end{array}$

Say that ${\omega}$ is conic Kaehler-Einstein if

$\displaystyle \begin{array}{rcl} Ric(\omega)=\omega+2\pi\sum(1-\beta_i)[D_i]. \end{array}$

1. Necessary conditions

Berman 2016: If ${(M,D)}$ admits a conic KE metric with ${[\omega]=2\pi c_1(L)}$, then ${(M,D)}$ is log-K-stable.

Log-K-stability is defined as follows.

A special degeneration ${(\mathcal{X},\mathcal{D},\mathcal{L})}$ of ${M,D,L)}$ is a 1-parameter family of log-pairs, consisting of

1. A normal log-pair ${(\mathcal{X},\mathcal{D})}$ with a ${{\mathbb C}^*}$-equivariant map ${\pi:(\mathcal{X},\mathcal{D})\rightarrow{\mathbb C}}$,
2. ${\mathcal{L}}$ is an equivariant ${\pi}$-ample ${{\mathbb Q}}$-line bundle.
3. ${\mathcal{X}_t,\mathcal{D}_t,\mathcal{L}_t}$ is isomorphic to ${(M,D,L)}$ for every ${t\not=0}$.

There is a natural compactification ${(\overline{\mathcal{X}},\overline{\mathcal{D}},\overline{\mathcal{L}})}$ of ${(\mathcal{X},\mathcal{D},\mathcal{L})}$ that maps to ${{\mathbb C} P^1}$. Defined number

$\displaystyle \begin{array}{rcl} w(\mathcal{X},\mathcal{D},\mathcal{L})=\frac{n\overline{\mathcal{L}}^{n+1}+(n+1)\overline{\mathcal{L}}^n(K_{\mathcal{X}|{\mathbb C} P^1}+\overline{\mathcal{D}})}{(n+1)L^n}. \end{array}$

If the central fiber is a log-Fano variety ${(M_0,D_0)}$ embedded in ${{\mathbb C} P^N}$ by ${H^0(M_0,L^\ell_{|M_0})}$, then ${w(\mathcal{X},\mathcal{D},\mathcal{L})}$ can be interpreted as a Futaki invariant.

Say that ${(M,D)}$ is log-K-semistable if for any special degeneration ${(\mathcal{X},\mathcal{D},\mathcal{L})}$ has ${w(\mathcal{X},\mathcal{D},\mathcal{L})\geq 0}$. Say that ${(M,D)}$ is log-K-stable if for any special degeneration ${(\mathcal{X},\mathcal{D},\mathcal{L})}$ has ${w(\mathcal{X},\mathcal{D},\mathcal{L})\geq 0}$ and equality holds only for the trivial degeneration ${(\mathcal{\mathcal{X},\mathcal{D},\mathcal{L}})=(M,D,L)\times{\mathbb C}}$.

2. The result

Theorem 1 If ${(M,D)}$ is log-K-stable, the there exists a conic KE metric ${\omega}$ with ${[\omega]=2\pi c_1(L)}$.

Many special cases were known, as consequences of existence of KE metrics on smooth closed manifolds. For instance when ${D}$ is a multiple of ${K_M^{-1}}$.

3. Motivation

We are interested in ${{\mathbb Q}}$-Fano varieties ${X}$. Assume ${X}$ admits a resolution ${\mu:M\rightarrow X}$ such that ${K_M=\mu^* K_X+\sum a_i E_i}$, ${a_i\in(-1,0]}$. For small enough ${\epsilon\in{\mathbb Q}}$, define

$\displaystyle \begin{array}{rcl} L_\epsilon=\pi^*K_X-\sum \epsilon E_i. \end{array}$

If there exists a KE metric ${\omega}$ on ${X}$, then ${\mu^*\omega}$ is a degenerate conic KE metric on ${M}$ with conic angles ${2\pi b_i}$ along ${E_i}$. We expect that there exist conic KE metrics ${\omega_\epsilon}$ on ${(M,\sum (1-b_i-\epsilon)E_i)}$ with ${[\omega_\epsilon]=2\pi c_1(L_\epsilon)}$, which Gromov-Hausdorff converge to ${(X,\omega)}$ as ${\epsilon\rightarrow 0}$.

We think that we are now able to prove the following. If ${X}$ is a K-stable ${{\mathbb Q}}$-Fano variety. Then it admits a generalized KE metric in the above sense.

4. Proof

Many steps are similar to the smooth case. Pick a large integer ${\lambda}$ such that ${\lambda L}$ has a smooth divisor ${E}$. We use a continuity method, solving

$\displaystyle \begin{array}{rcl} Ric(\omega_t)=t\omega_t+\frac{1-t}{\lambda}[E]+\sum(1-\beta_i)[D_i], \end{array}$

${t\in[0,1]}$. The set ${I}$ of ${t}$ such that a solution exists is easily shown to be non-empty (it contains 0) and open. Is it closed? The key point is a ${C^0}$ estimate. It follows from a “partial ${C^0}$-estimate” and log-K-stability. In turn, this follows from an ${L^2}$-estimate and compactness a la Cheeger-Colding-Tian.

4.1. Smoothing conical KE metrics

Say that ${\omega}$ has a K-approximation if there exist Kaehler metrics ${\omega_i=\omega+i\partial\bar\partial\phi_i}$ in the same cohomology class such that

• ${\phi_i\rightarrow 0}$ uniformly on ${M}$ and smoothly outside ${|D|}$,
• ${Ric(\omega_i)\geq K\omega_i}$,
• ${(M,\omega_i)\rightarrow (M,\omega)}$ in Gromov-Hausdorff topology.

We show that if ${Aut^0(M,D)=\{1\}}$ and if for all ${i}$,

$\displaystyle \begin{array}{rcl} (1-K_i)L+(1-\beta_i)D_i\geq 0 \end{array}$

for some ${K_i\leq 1}$, then ${\omega}$ has a K-approximation where ${K=\sum(K_i-1)+1}$.

We solve a modified equation with an extra term involving ${K_i}$‘s. For this, we use the variational approach by Boucksom-Eyssidieux-Guedj-Zeriahi and results of Darwan-Robinstein, Guenancia-Paun.

4.2. Extend B. Wang-Tian’s results to conic case

5. Work in progress

To handle ${{\mathbb Q}}$-Fano varieties, we need to extend Cheeger-Colding to conic cases.