Around Complex Geometry (April 21-23 2023)

Xi Sisi Shen: Canonical metrics in non-Kähler Geometry

Abstract: We will discuss how the existence problem for Kähler-Einstein and constant scalar curvature Kähler (cscK) metrics can be framed in terms of a PDE problem which can be solved using a continuity method approach. From there, we move into extending this existence theory to classes of non-Kahler metrics. 

Gábor Székelyhidi:  Regularity for singular Kahler-Einstein metrics

Abstract: Singular Kahler-Einstein metrics arise naturally when studying limits of sequence of smooth Kahler-Einstein manifolds. Through the work of Donaldson-Sun and Li-Wang-Xu we know that the tangent cones of such singular KE metrics are determined by the underlying complex variety, however it is important to have more refined geometric information. For certain classes of isolated singularities, such as ordinary double points, Hein-Sun provided a precise asymptotic description. I will discuss results extending this work to singularities with more general tangent cones, including those with non-isolated singular sets, as well as "unstable" examples where the tangent cone is not locally biholomorphic to the original complex variety. This is joint work with Shih-Kai Chiu.

Junsheng Zhang: Complete Calabi-Yau manifolds asymptotic to cones

Abstract: We eliminate the possible appearance of an intermediate K-semistable cone in the 2-step degeneration theory developed by Donaldson-Sun. As a byproduct of the proof, we show that every complete Calabi-Yau manifold with maximal volume growth and quadratic curvature decay is polynomially asymptotic to its unique tangent cone at infinity. This is a joint work with Song Sun.

Jacob Keller: The Birational Geometry of K-Moduli Spaces

Abstract: K-stability is a rapidly developing theory that allows one to construct moduli spaces for Fano varieties. In all known examples, K-moduli spaces are uniruled, so their Kodaira dimensions are negative infinity. In this talk we will describe components of K-Moduli spaces which are birational to M_g, in particular they have maximal Kodaira dimension when g is sufficiently large. This component parameterizes certain moduli spaces of vector bundles on smooth curves, and the main difficulty is to show that these moduli spaces are K-stable. To establish this we require good understanding of their toric degenerations.

Yuchen Liu: Moduli of log Calabi-Yau pairs

Abstract: While the theories of KSBA stability and K-stability have been successful in constructing compact moduli spaces of canonically polarized varieties and Fano varieties, respectively, the case of Calabi-Yau varieties remains less well understood. I will discuss a new approach to this problem in the case of log Calabi-Yau pairs (X,D), where X is a Fano variety and D is an anticanonical Q-divisor, in which we consider all semi-log-canonical degenerations. One challenge of this approach is that the moduli stack can be unbounded. Nevertheless, if we consider log Calabi-Yau pairs as degenerations of P^2 with plane curves, we show that there exists a projective good moduli space despite the unboundedness. This is joint work with K. Ascher, D. Bejleri, H. Blum, K. DeVleming, G. Inchiostro, and X. Wang.

Carlo Scarpa: Kähler forms and B-fields

Abstract:  In this talk, we consider the problem of finding a canonical representative of a complexified Kähler class on a compact complex manifold, motivated by fundamental constructions in mirror symmetry. I will describe a new geometric PDE, obtained by coupling the deformed Hermitian Yang-Mills and the constant scalar curvature equations, whose solutions will give the required representative of the class. We will then consider a natural variational framework in which to study the equation, and prove the existence of solutions in some special cases. Based on joint work with Jacopo Stoppa.

Yueqiao Wu: A non-Archimedean characterization of local K-stability

Abstract: Log Fano cone singularities are generalizations of cones over log Fano varieties, and have a local K-stability theory extending the one for log Fano varieties. In this talk, we aim to give a characterization for local K-stability from a non-Archimedean point of view. This characterization will in particular allow us to deal with a more general class of test configurations.

Rémi Reboulet: The birational geometry of GIT quotients

Abstract: If X is a projective variety with an action of a reductive group G, Geometric Invariant Theory (GIT) yields a projective quotient of X by G. This construction depends on the choice of a G-linearised ample line bundle L on X.  Varying L, it is known that we obtain birational GIT quotients, but only finitely many varieties arise in this way. Borrowing from the theory of Zariski--Riemann spaces, we explain how to construct a "universal" GIT quotient capturing all possible varieties birational to a given GIT quotient. This is based on joint work with Ruadhaí Dervan.

Annamaria Ortu: Kähler geometry of holomorphic submersions

Abstract: Proper holomorphic submersions of Kähler manifolds can be thought of as both a generalisation of holomorphic vector bundles and as a way of studying the behaviour of Kähler manifolds in families. On such fibrations, we will define a stability condition for the fibres in terms of K-stability and we will describe a generalisation of a Hermite-Einstein connection, called an optimal symplectic connection. Using this condition we will discuss a deformation theory for fibrations that admit an optimal symplectic connection and we will construct their moduli space.

Sean Paul: Hilbert-Mumford (semi)stability, K-(semi)stability, & (semi)stable pairs

Abstract​: This talk is aimed at early career researchers as well as Phd students in complex differential geometry. The speaker will first discuss each of the stability conditions and introduce the questions that gave rise to them. The second order of business is to compare & contrast each of them (the first and the last are virtually identical). The last piece of business is to state a recent result which says that stability condition 3 (when applied to a polarized manifold (X,L)) is equivalent to the existence of a metric constant scalar curvature in c_1(L) provided that the reduced automorphism group of (X,L) is finite.