The Sheffield Algebraic Geometry Seminar

I will discuss recent joint work with Siddarth Kannan (MIT), in which we develop a calculus for S_n-equivariant Euler characteristics of moduli spaces of stable curves and maps, based on graph enumeration. Our formulas pass through the theory of wreath symmetric functions and their relationship with automorphisms of graphs.

Mirror symmetry predicts a relation between the Gromov-Witten invariants of a symplectic manifold X and the BCOV theory of the mirror space Y. Kontsevich proposed that this is a consequence of homological mirror symmetry. In particular, one should be able to extract the Gromov-Witten invariants of X from its Fukaya category. I will report on recent work with Amanda Hirschi, in which we construct an (open-closed) Deligne-Mumford field theory associated to a single Lagrangian. I will explain how this relates to Costello's proposed method for proving Kontsevich's conjecture.

The Fulton–MacPherson configuration space is a well-known compactification of the ordered configuration space of a projective variety. We describe a construction of its logarithmic analogue: it is a simple normal crossings compactification of the configuration space of points on X\D, where X is a smooth projective variety and D is a simple normal crossings divisor, and is constructed via logarithmic and tropical geometry. Moreover, given a semistable degeneration of X, we construct a logarithmically smooth degeneration of the Fulton–MacPherson space of X. Both constructions parametrise point configurations on certain target degenerations, arising from both logarithmic geometry and the original Fulton–MacPherson construction. The degeneration satisfies a “degeneration formula” — each irreducible component of its special fibre can be described as a proper birational modification of a product of logarithmic Fulton–MacPherson configuration spaces. Time permitting, we explore some potential applications to enumerative geometry.

Understanding when stable vector bundles with fixed topological invariants exist – or fail to exist – is a longstanding and difficult problem. While there are many topological obstructions, algebraic ones are rarer: beyond the classical Bogomolov inequality, which gives a quadratic bound in terms of the rank and the first two Chern characters, few general tools are available. In higher dimensions, one expects new inequalities involving higher Chern characters – the so-called higher Bogomolov inequalities. For threefolds, a conjectural inequality of Bayer–Macrì–Toda provides a candidate obstruction, but the conjecture remains open in many cases, including most Calabi–Yau threefolds. A proof would have applications to the existence of stability conditions, enumerative geometry, and birational geometry.

In this talk, I will explain how a weaker form of the Bayer–Macrì–Toda conjecture on Calabi–Yau threefolds can be reduced to bounding the higher rank Brill–Noether theory of a sectional curve in the threefold. This reduction allows us to verify the conjecture in many new cases and to construct stability conditions on numerous Calabi–Yau threefolds (including all previously known cases). This is joint work with Feyzbakhsh, Koseki, and Liu.

The Hopf index theorem counts solutions of equations via a winding number. I’ll review this and give 8 equivalent formulations. Then I’ll describe 8 analogues for equations which form “isotropic sections” of “bundles with quadratic form”. There are applications to “cosection localised virtual cycles” and to “DT^4 virtual cycles”, but mainly I’ll focus on more elementary topics. This is joint work with Martijn Kool, Jeongseok Oh and Jørgen Rennemo.