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.


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