Geometry

In 1999, M. Gromov asked the following question. Let two algebraic varieties X and Y admit embeddings to a third one Z such that the open complements Z - X and Z - Y are isomorphic. How far are X and Y from being birationally equivalent? This question is closely related to the structure of the Grothendieck ring of varieties, and in particular to the new notion of L-equivalence. I will report on the progress in this area in the last 20 years. We have situations when X and Y are birational: this is the case when X, Y, Z are smooth and projective (Larsen--Lunts 2001, Kontsevich--Tschinkel 2018). We also have examples when they are not birational, which come from L-equivalence (Borisov 2014, Hassett--Lai 2018 and others). Finally, we have a filtration on the Grothendieck ring of varieties giving a full control over such questions (Zakharevich 2014, Lin--Shinder 2022).

I will report on joint work in progress with Luca Giovenzana.  I will briefly describe some developments in the abstract theory of quasi del Pezzo pseudolattices, before showing how this theory arises naturally in the contexts of type II degenerations of K3 surfaces and elliptically fibred K3 surfaces. This can be thought of as a manifestation of mirror symmetry; I will discuss what it could tell us about mirror symmetry for K3 surfaces and the 2-dimensional Fano/LG correspondence.

The global dimension of a Bridgeland stability condition is a real number that measures the distance between interacting semistable objects. I will explain how a low global dimension of families of stability conditions is relevant in establishing wall-crossing formulas and I will show that geometric stability conditions on weak Fano surfaces have global dimension 2.

Geometric flows give a useful analytic tool to study important problems in geometry and topology.  A key example is the Ricci flow, which (after possibly rescaling) has Einstein metrics as its critical points.  In 7 dimensions, 3-Sasakian geometry leads to two natural Einstein metrics with positive scalar curvature, both of which are induced by special structures known as nearly parallel G2-structures.  The nearly parallel G2-structures are critical points (again up to scaling) for two different flows in G2 geometry: the Laplacian flow and the Laplacian coflow.  In this talk, I will describe how the behaviour is very different for all these geometric flows on 3-Sasakian 7-manifolds, particularly in terms of the stability of the critical points. This is joint work with A. Kennon.

In relation with the problem of looking for canonical metrics on a compact Kähler manifold, the geometry of the space of Kähler potentials has been intensively studied in the last 20 years. The latter space can be endowed of a family of Finsler metrics d_p. As shown by Darvas, the (d_p) distance between two potentials with bounded laplacian can be expressed with a very nice formula. In a joint paper with Chinh Lu we prove that this formula still holds for more singular potentials having only finite entropy.

If G is a compact Lie group, then T^*G^c, the complex cotangent bundle of its complexification, has a natural hyperKähler metric. This surprising fact was proved by Kronheimer in 1988 by showing that a moduli space of solutions of Nahm’s equations on a finite real interval is diffeomorphic to T^*G^c.  (The story was further elucidated by Dancer and Swann).   Little is known explicitly about this metric except in the simplest cases.  In joint work with Richard Melrose and Raphael Tsiamis we have been studying the asymptotic behaviour of this metric, which turns out to be quite intricate.   The goal of my talk is to explain some of this work.

When searching for examples satisfying certain geometric properties, it is often convenient to examine manifolds constructed by gluing simple pieces together.  One common example of such a construction involves gluing disk bundles together along their common boundary.  On the other hand, many geometric phenomena impose strong topological conditions on the underlying manifold, such as the existence of a decomposition into a union of disk bundles (glued along a common boundary).

Given that they arise frequently from these two different viewpoints, it thus makes sense to study manifolds which decompose as a union of disk bundles in their own right.  In this talk, I will report on joint work with J. DeVito in this direction.