Algebraic and Symplectic Geometry Seminar

Let $X_0 $ be a rational surface with a cyclic quotient singularity $(1,a)/r$.  Kawamata constructed a remarkable vector bundle  $F_0$  on $X_0$ such that the finite-dimensional algebra End$(F_0)$ "absorbs'' the singularity of $X_0$ in a categorical sense. If we deform over an irreducible component of the versal deformation space of $X_0$ (as described by Kollár and Shepherd-Barron), the vector bundle $F_0$ also deforms to a vector bundle $F$. These results were established using abstract methods of birational geometry, making the explicit computation of the family of algebras challenging. We will utilise homological mirror symmetry to compute End$(F)$ explicitly in a certain bulk-deformed Fukaya category. In the case of a $Q$-Gorenstein smoothing, this algebra End$(F)$ is a matrix order over $k[t]$ and "absorbs" the singularity of the corresponding terminal 3-fold singularity. This is based on joint work with Jenia Tevelev.

The Bogomolov-Miyaoka-Yau inequality for minimal compact complex surfaces of general type was proved in 1977 independently by Miyaoka, using methods of algebraic geometry, and by Yau, as an outgrowth of his proof of the Calabi conjectures. In this talk, we outline our program to prove the conjecture that symplectic 4-manifolds with $b^+>1$ obey the Bogomolov-Miyaoka-Yau inequality. Our method uses Morse theory on the gauge theoretic moduli space of non-Abelian monopoles, where the Morse function is a Hamiltonian for a natural circle action and natural two-form.  We shall describe generalizations of Donaldson’s symplectic subspace criterion (1996) from finite to infinite dimensions. These generalized symplectic subspace criteria can be used to show that the natural two-form is non-degenerate and thus an almost symplectic form on the moduli space of non-Abelian monopoles. This talk is based on joint work with Tom Leness and the monographs https://arxiv.org/abs/2010.15789  (to appear in AMS Mathematical Surveys and Monographs), https://arxiv.org/abs/2206.14710 and https://arxiv.org/abs/2410.13809.