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 , a divisor with normal crossings such that the line bundle
A metric is a conic Kaehler-Einstein metric if it is smooth Kaehler in and for every point where is defined by in some coordinates, is equivalent (between two multiplicative constants) to the model cone metric
Say that is conic Kaehler-Einstein if
1. Necessary conditions
Berman 2016: If admits a conic KE metric with , then is log-K-stable.
Log-K-stability is defined as follows.
A special degeneration of is a 1-parameter family of log-pairs, consisting of
- A normal log-pair with a -equivariant map ,
- is an equivariant -ample -line bundle.
- is isomorphic to for every .
There is a natural compactification of that maps to . Defined number
If the central fiber is a log-Fano variety embedded in by , then can be interpreted as a Futaki invariant.
Say that is log-K-semistable if for any special degeneration has . Say that is log-K-stable if for any special degeneration has and equality holds only for the trivial degeneration .
2. The result
Theorem 1 If is log-K-stable, the there exists a conic KE metric with .
Many special cases were known, as consequences of existence of KE metrics on smooth closed manifolds. For instance when is a multiple of .
We are interested in -Fano varieties . Assume admits a resolution such that , . For small enough , define
If there exists a KE metric on , then is a degenerate conic KE metric on with conic angles along . We expect that there exist conic KE metrics on with , which Gromov-Hausdorff converge to as .
We think that we are now able to prove the following. If is a K-stable -Fano variety. Then it admits a generalized KE metric in the above sense.
Many steps are similar to the smooth case. Pick a large integer such that has a smooth divisor . We use a continuity method, solving
. The set of 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 estimate. It follows from a “partial -estimate” and log-K-stability. In turn, this follows from an -estimate and compactness a la Cheeger-Colding-Tian.
4.1. Smoothing conical KE metrics
Say that has a K-approximation if there exist Kaehler metrics in the same cohomology class such that
- uniformly on and smoothly outside ,
- in Gromov-Hausdorff topology.
We show that if and if for all ,
for some , then has a K-approximation where .
We solve a modified equation with an extra term involving ‘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 -Fano varieties, we need to extend Cheeger-Colding to conic cases.