Geometry and Analysis Seminar

The quadratic Euler characteristic of an algebraic variety is a (virtual) symmetric bilinear form which refines the topological Euler characteristic and contains interesting arithmetic information when the base field is not algebraically closed. For smooth projective varieties, it has a quite concrete expression in terms of the cup product and Serre duality for Hodge cohomology. However, for singular varieties, it is defined abstractly (using either cut and paste relations or motivic homotopy theory) and is still rather mysterious. I will first introduce this invariant and place it in the broader context of quadratic enumerative geometry. I will then explain some progress on concrete computations, first for symmetric powers (joint with Lenny Taelman) and second for conductor formulas for hypersurface singularities (older results with Marc Levine and Vasudevan Srinivas on the one hand, and joint work in progress with Ran Azouri, Niels Feld, Yonathan Harpaz and Tasos Moulinos on the other).

The relationship between the topological or homotopy-invariant properties of a symplectic manifold X and the set of possible immersed or embedded Lagrangian submanifolds of X is rich and mostly mysterious.  In 2020, D. Alvarez-Gavela, Y. Eliashberg, and D. Nadler proved that any Weinstein manifold (e.g. an affine variety) admitting a Lagrangian plane field retracts onto a Lagrangian submanifold with arboreal singularities (a certain class of singularities which can be described combinatorially). I will discuss work in progress with D. Alvarez-Gavela and T. Large investigating the other direction, in which we prove a partial converse to the AGEN result and show that most Weinstein manifolds do not admit such skeleta. This suggests that the Floer-theoretic invariants of some well-known open symplectic manifolds may be more complicated than expected.

Let $S$ be a smooth projective surface with $p_g>0$ and $H^1(S,{\mathbb Z})=0$. 
We consider the moduli spaces $M=M_S^H(r,c_1,c_2)$ of $H$-semistable sheaves on $S$ of rank $r$ and 
with Chern classes $c_1,c_2$. Associated a suitable class $v$ the Grothendieck group of vector bundles
on $S$ there is a deteminant line bundle $\lambda(v)\in Pic(M)$, and also a tautological sheaf $\tau(v)$ on $M$.

In this talk we derive a conjectural generating function for the virtual Verlinde numbers, i.e. the virtual holomorphic 
Euler characteristics of all determinant bundles $\lambda(v)$ on M, and for Segre invariants associated to $\tau(v)$ . 
The argument is based on conjectural blowup formulas and a virtual version of Le Potier's strange duality. 
Time permitting we also sketch a common refinement of these two conjectures, and their proof for Hilbert schemes of points.
 

The moduli space of torsion-free $G_2$-structures on a compact $7$-manifold $M$ is a smooth manifold, locally diffeomorphic to an open subset of $H^3(M)$. It is endowed with a natural metric which arises as the Hessian of a potential, the properties of which are still poorly understood. In this talk, we will review what is known of the geometry of $G_2$-moduli spaces and present new formulae for the fourth derivative of the potential and the curvatures of the associated metric. We explain some interesting consequences for the simplest examples of $G_2$-manifolds, when the universal cover of $M$ is $\mathbb{R}^7$ or $\mathbb{R}^3 \times K3$. If time permits, we also make some comments on the general case.

In this joint work with Filip Zivanovic, we construct symplectic cohomology for a class of symplectic manifolds that admit ${\mathbb C}^*$-actions and which project equivariantly and properly to a convex symplectic manifold. The motivation for studying these is a large class of examples known as Conical Symplectic Resolutions, which includes quiver varieties, resolutions of Slodowy varieties, and hypertoric varieties. These spaces are highly non-exact at infinity, so along the way we develop foundational results to be able to apply Floer theory. Motivated by joint work with Mark McLean on the Cohomological McKay Correspondence, our goal is to describe the ordinary cohomology of the resolution in terms of a Morse-Bott spectral sequence for positive symplectic cohomology. These spectral sequences turn out to be quite computable in many examples. We obtain a filtration on ordinary cohomology by cup-product ideals, and interestingly the filtration can be dependent on the choice of circle action.

For a compact Lie group $G$, its massless Coulomb branch algebra is the $G$-equivariant Borel-Moore homology of its based loop space. This algebra is the same as the algebra of regular functions on the BFM space. In this talk, we will explain how this algebra acts on the equivariant symplectic cohomology of Hamiltonian $G$-manifolds when the symplectic manifolds are open and convex. This is a generalization of the closed case where symplectic cohomology is replaced with quantum cohomology. Following Teleman, we also explain how it relates to the Coulomb branch algebra of cotangent-type representations. This is joint work with Eduardo González and Dan Pomerleano.

Let $(M, g)$ be a Riemannian manifold equipped with a calibration $k$-form $\alpha$. In earlier work with Cheng and Madnick (AJM 2021), we studied the analytic properties of a special class of $k$-harmonic maps into $M$ satisfying a first order nonlinear PDE, whose images (away from a critical set) are $\alpha$-calibrated submanifolds of $M$. We call these maps Smith immersions, as they were originally introduced in an unpublished preprint of Aaron Smith. They have nice properties related to conformal geometry, and are higher-dimensional analogues of the $J$-holomorphic map equation. In new joint work (arXiv:2311.14074) with my PhD student Anton Iliashenko, we have obtained analogous results for maps out of $M$. Slightly more precisely, we define a special class of $k$-harmonic maps out of $M$, satisfying a first order nonlinear PDE, whose fibres (away from a critical set) are $\alpha$-calibrated submanifolds of $M$. We call these maps Smith submersions. I will give an introduction to both of these sets of equations, and discuss many future questions.