Birational Geometry and K-stability in Moduli Spaces

The proposed three-day workshop will focus on interrelated mathematical advances in each of the following highly active areas of current research: complex birational geometry (led by MMP), K-stability, and moduli spaces. These domains are of fundamental importance for the (birational) geometry and classification of complex projective varieties as well as for the understanding of their moduli spaces. The workshop will focus on some of the current frontiers that are of particular importance to each of these areas. For logistical reasons, kindly complete the registration form if you intend to participate.

In addition to encouraging and stimulating interactions between relevant experts, the workshop will provide a valuable opportunity to explain to interested graduate students and young researchers the details of the techniques used through mini-courses and specialized lectures. Many speakers are renowned speakers in addition to being leading experts in their field. In particular, the workshop plans to have three mini-lectures, one by Sándor Kovács on the now classical and important KSBA (Kollár-Shepherd-Baron and Alexeev) compactification of moduli spaces, one by Chenyang Xu on the recent roles of the MMP on K-stability and one by Kenneth Ascher on K-moduli and the role of moduli space theory. In addition to our guest speakers, we intend to involve graduate students and postdocs.

One unifying aspect that brought these fields closer together comes from the main geometric invariants of the positivity and stability of the polarized varieties, starting from the classical Seshadri constant to the recently discovered δ invariant that is extracted from the birational geometric study of K-stability to guarantee the necessary and sufficient condition of the existence and uniqueness of the extremal metrics coming from the geometric analysis, and from the use and exploration of test configurations in K-stability, with its intermediate α, β invariants and of various logarithmic thresholds in birational geometry. These have been explored by various means and by various groups of researchers, including S. Boucksom and his collaborators (such as Hisamoto, Jonsson, etc.) via non-Archimedean geometry, the Kollár school, represented in the workshop by senior speakers Sándor Kovács and Chenyang Xu, and specialists in moduli spaces and stacks, represented in the workshop by speakers Kenneth Ascher, Dori Bejleri, with their refinements, generalizations and alternatives to the geometric invariant theory (GIT) of Mumford.

The proposed workshop will also touch on the influence of complex analytic and algebraic geometry on the study of the moduli space of polarized manifolds. In particular, extremal metrics, nonlinear PDE solutions, and the behavior of Weil-Petersson metrics induced on moduli spaces will be considered.