Informal Geometric Analysis Seminar

I will discuss a recent joint work with Olivier Biquard about conic Kähler-Einstein metric with cone angle going to zero. We study two situations, one in negative curvature (toroidal compactifications of ball quotients) and one in positive curvature (on Fano manifolds endowed with a smooth anticanonical divisor) leading up to the resolution of a question asked by Donaldson in 2011. 

The notion of pluricomplex energy was introduced by U. Cegrell in 1988. Since then it has played an important role in complex Monge-Ampere equations. I present a recent joint work with Do Duc Thai in which we characterize the class of probability measures on a compact Kahler manifold such that the associated Monge-Ampere equation has a solution of finite energy. Such a characterization was previously only known for particular cases. 

I will discuss some recent progress on the metric version of the SYZ conjecture, which says that for polarized Calabi-Yau manifolds near the large complex structure limit, then on 99% of the manifold there exists a special lagrangian torus fibration. This is unconditionally proved in the case of Fermat hypersurface families, and conditionally proved in general assuming a conjecture in non-archimedean geometry. Time permitting I will try to say a few words about the relative merits of the two methods.

We begin by explaining the correspondence between convex functions on integral polytopes and plurisubharmonic (i.e. "generalized convex") metrics on polarized toric varieties. Under this correspondence, geodesics in the space of toric psh metrics are transformed into affine segments of convex functions. We then show how this result can be extended to more general geodesics of plurisubharmonic metrics in the non-toric case, using a construction of Witt Nyström. If time permits, we will also look into some non-Archimedean aspects of this generalized result, applied to geodesic rays.

We construct Kahler-Einstein metrics with negative scalar curvature near an isolated log canonical (non-log terminal) singularity and we continue to describe the geometry of Kahler-Einstein metric with focus on complex hyperbolic cusp. This is based on joint work with Ved Datar and Jian Song.

We study the Kähler-Ricci flow on varieties of general type. We show that the normalized Kähler-Ricci flow exists at all times in the sense of viscosity, is continuous in an open Zariski set and converges to the singular Kähler-Einstein metric. This gives a partial answer to a question of Feldman-Ilmanen-Knopf on defining and constructing weak solutions of the Kähler-Ricci flow.

Let L be a line bundle with positive singular Hermitian metric he^{-u}, on an n-dimensional compact Kähler manifold X. Let h_k be the dimension of the space of global sections  that are L^2 integrable with respect to the weight e^{-ku}. We show that the limit of h_k/k^n exists, and equals the non-pluripolar volume of the I-model potential associated to u. Joint work with Mingchen Xia.

We introduce a notion of "symplectic complexity" for open symplectic manifolds. This captures purely symplectic features which are different from classical topological invariants such as homology, and it also goes beyond the standard usage of Floer theory. As our main application, we study symplectic embeddings between divisor complements in complex projective space, giving a nearly complete characterization.

Recent developments in complex geometry have highlighted the importance of real Monge-Ampère equations on singular affine manifolds, in particular for the well known SYZ conjecture concerning collapsing families of Calabi-Yau manifolds. We show that for close to symmetric data, the real Monge-Ampère equation on the unit simplex admits a unique solution. This is the first general existence result for Monge-Ampère equations on a singular affine manifold and as a corollary we strengthen a recent result by Y. Li which confirms the SYZ conjecture in special cases. Time permitting, I will talk about a built in phenomena reminiscent of free boundary problems and the proof, which uses tools from optimal transport. 

Extremal Kähler metrics were introduced by Calabi in the 80’s as a type of canonical Kähler metric on a Kähler manifold, and are a generalisation of constant scalar curvature Kähler metrics in the case when the manifold admits automorphisms. A natural question is when the blowup of a manifold in a point admits an extremal Kähler metric. We completely settle the question in terms of a finite dimensional moment map/GIT condition, generalising work of Arezzo-Pacard, Arezzo-Pacard-Singer and Székelyhidi. Our methods also allow us to deal with a certain semistable case that has not been considered before, where the original manifold does not admit an extremal metric, but is infinitesimally close to doing so. As a consequence of this, we solve the first non-trivial special case of a conjecture of Donaldson. This is joint work with Ruadhaí Dervan.

A filling is an oriented surface bounding a link. Classifications of Legendrian knots and their exact Lagrangian fillings are central questions in low-dimensional contact and symplectic topology. Lagrangian fillings can be constructed via local moves in finite steps. In this talk, I will show that most Legendrian torus links have infinitely many exact Lagrangian fillings. These fillings are constructed using Legendrian loops, and proven to be distinct using the microlocal theory of sheaves and the theory of cluster algebras. This is a joint work with Roger Casals.

Totally geodesic submanifolds, when they exist, have been essential to the resolution of many open problems for finite-volume hyperbolic manifolds. For instance, Gromov–Piatetski-Shapiro famously used them to construct examples of non-arithmetic hyperbolic manifolds in every dimension and Millson used them to show that in all dimensions there is always an arithmetic hyperbolic manifold with positive first Betti number. In this talk, we will show that geodesic submanifolds of these two types of hyperbolic manifolds, "arithmetic" and "non-arithmetic", behave very differently. More specifically, we will first recall the definition of arithmetic and non-arithmetic manifolds and then show that the presence of infinitely many geodesic submanifolds characterizes the arithmeticity of a hyperbolic manifold. Along the way we will also discuss how this relates to rigidity of certain representations of the fundamental group. This is joint work with Bader, Fisher, and Stover.

In this talk, we introduce the notion of (singular) K ähler-Einstein metrics on mildly singular varieties. Extending Di Nezza-Lu’s approach to the setting of big cohomology classes, we show that singlar Kähler-Einstein metrics on log canonical varieties of general type have continuous potentials on a Zariski open set.

Filtered vector spaces arise as natural invariants in symplectic geometry, algebraic geometry, and topology. For instance, in symplectic geometry, infinite dimensional Floer theoretic invariants such as symplectic cohomology are filtered by action, whereas in algebraic geometry, the ring of algebraic functions on a smooth affine variety is filtered by the order of pole at infinity. In this talk, we are going to discuss how to relate these filtrations, and their growth. The main tool for this is the categorification of these filtrations, which also allows us to use the techniques of homological algebra to study the growth. This is joint work with Laurent Cote.

Let M be a complete, connected Riemannian surface and suppose that S is a discrete subset of M. What can we learn about M from the knowledge of all distances in the surface between pairs of points of S? We prove that if the distances in S correspond to the distances in a 2-dimensional lattice, or more generally in an arbitrary net in R^2, then M is isometric to the Euclidean plane. We thus find that Riemannian embeddings of certain discrete metric spaces are rather rigid. A corollary is that a subset of Z^3 that strictly contains a two-dimensional lattice cannot be isometrically embedded in any complete Riemannian surface. This is a joint work with B. Klartag.