## Project Details

### Description

The PI will work on fundamental problems of Fano K-moduli spaces and local stability theory from both theoretical and concrete aspects. Speciﬁcally, the PI plans to work on the following general problems.

1. Properness of Fano K-moduli spaces. This is one piece of the general conjecture on constructing nicely behaved moduli spaces for Fano varieties using K-stability. The PI plans to reduce this conjecture to a problem on ﬁnite generation of certain minimizing valuation where tools from the Minimal Model Program can be utilized. A related conjecture is the Stability Degeneration Conjecture for normalized volumes over klt singularities. This is a joint work in progress with H. Blum and C. Xu.

2. Explicit K-moduli spaces of log Fano pairs. In recent joint work with K. Ascher and K. DeVleming, the PI established a wall-crossing framework for K-moduli spaces of log Fano pairs when varying the coeﬃcients. He plans to relate these K-moduli spaces in speciﬁc cases to well-studied moduli spaces of curves and K3 surfaces. He plans to construct a log Calabi-Yau moduli space to complete the wall-crossing framework from GIT moduli spaces to KSBA moduli spaces. This is a joint work in progress with K. Ascher and K. DeVleming.

3. Distribution of local volumes. In recent joint work with C. Xu, the PI proved that smooth points have the largest local volume, and the ordinary double point has the second largest local volume in dimension 3. The PI will continue studying the distribution of local volumes for singularities which aims to show that 0 is the only accumulation point. This is a joint work in progress with J. Han and L. Qi.

For broader impact the PI is working with undergraduate and graduate students, helping prepare them for their qualifying exams. He is jointly organizing a conferences series between Yale and Brown Universities.

1. Properness of Fano K-moduli spaces. This is one piece of the general conjecture on constructing nicely behaved moduli spaces for Fano varieties using K-stability. The PI plans to reduce this conjecture to a problem on ﬁnite generation of certain minimizing valuation where tools from the Minimal Model Program can be utilized. A related conjecture is the Stability Degeneration Conjecture for normalized volumes over klt singularities. This is a joint work in progress with H. Blum and C. Xu.

2. Explicit K-moduli spaces of log Fano pairs. In recent joint work with K. Ascher and K. DeVleming, the PI established a wall-crossing framework for K-moduli spaces of log Fano pairs when varying the coeﬃcients. He plans to relate these K-moduli spaces in speciﬁc cases to well-studied moduli spaces of curves and K3 surfaces. He plans to construct a log Calabi-Yau moduli space to complete the wall-crossing framework from GIT moduli spaces to KSBA moduli spaces. This is a joint work in progress with K. Ascher and K. DeVleming.

3. Distribution of local volumes. In recent joint work with C. Xu, the PI proved that smooth points have the largest local volume, and the ordinary double point has the second largest local volume in dimension 3. The PI will continue studying the distribution of local volumes for singularities which aims to show that 0 is the only accumulation point. This is a joint work in progress with J. Han and L. Qi.

For broader impact the PI is working with undergraduate and graduate students, helping prepare them for their qualifying exams. He is jointly organizing a conferences series between Yale and Brown Universities.

Status | Active |
---|---|

Effective start/end date | 7/1/21 → 4/30/23 |

### Funding

- National Science Foundation (DMS 2148266)

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