Johannes Krah showed that the blowup of $\mathbf{P}^{2}$ in 10 general points admits a phantom subcategory. We construct three types of objects in such a phantom: projections of skyscraper sheaves, a strong generator, and a family of objects with two nonzero cohomology sheaves. We study the deformation theory of these objects to show that the phantom contains rich geometry, such as encoding the blowdown map to $\mathbf{P}^{2}$. We also show that there exists a co-connective dg-algebra whose derived category is a phantom.
Product-quotient surfaces form an interesting class of surfaces of general type. We explore the stability manifold of these surfaces. In cases where the group action is not free, we construct non-geometric stability conditions.
I will discuss recent joint work with T. Kelly (Queen Mary University of London) on Chern characters of coherent sheaves on projective hypersurfaces. The formulas we get will be explicitly in terms of both the terms and differentials in the infinite free resolution of the corresponding graded module. We will also discuss some examples like complete intersections and applications to special cases of the Hodge conjecture.
Given a symplectic manifold together with a mirror representing Lagrangian submanifold, we can construct mirror correpondences such as an A-infinity functor or a ring isomorphism in a geometric way. We discuss how we can generalize the construction of mirror isomorphisms when we are given noncompact manifolds or finite abelian group actions.
The classical Chern character can be viewed as a ring homomorphism from the Grothendieck group to the cohomology ring. Building on earlier work, Shklyarov generalized this and defined a categorical Chern character for dg categories, which assigns to each object a class in Hochschild homology. He used it to prove a Hirzebruch–Riemann–Roch theorem for dg algebras. In this talk, I will first introduce the category of matrix factorizations over curved category algebras. Then, following Shklyarov, I will compute the Chern character of these matrix factorizations and show that it recovers the Kapustin–Li formula and its equivariant versions. Time permitting, I will explain how this might be related to recent work of Favero and Kelly, “The Chern character of a coherent sheaf on a smooth projective hypersurface.” This is based on my joint work with David Favero.
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Abstract