We construct a full exceptional collection of vector bundles in the bounded derived category of coherent sheaves on the Grassmannian IGr(3,8) of isotropic 3-dimensional subspaces in an 8-dimensional symplectic vector space.
The purpose of this article is to develop techniques for estimating basis log canonical thresholds on logarithmic surfaces. To that end, we develop new local intersection estimates that imply log canonicity. Our main motivation and application is to show the existence of Kähler–Einstein edge metrics on all but finitely many families of asymptotically log del Pezzo surfaces, partially confirming a conjecture of two of us. In an appendix we show that the basis log canonical threshold of Fujita–Odaka coincides with the greatest lower Ricci bound invariant of Tian.
We study two rational Fano threefolds with an action of the icosahedral group 𝔄5. The first one is the famous Burkhardt quartic threefold, and the second one is the double cover of the projective space branched in the Barth sextic surface. We prove that both of them are 𝔄5-Fano varieties that are 𝔄5-birationally superrigid. This gives two new embeddings of the group 𝔄5 into the space Cremona group.
We discuss Calabi–Yau and fractional Calabi–Yau semiorthogonal components of derived categories of coherent sheaves on smooth projective varieties. The main result is a general construction of a fractional Calabi–Yau category from a rectangular Lefschetz decomposition and a spherical functor. We give many examples of applications of this construction and discuss some general properties of Calabi–Yau categories.
These are (somewhat informal) lecture notes for the CIME summer school “Geometric Representation Theory and Gauge Theory” in June 2018. In these notes we review the constructions and results of Braverman et al. (Adv Theor Math Phys 22(5):1017–1147, 2018; Adv Theor Math Phys 23(1):75–166, 2019; Adv Theor Math Phys 23(2):253–344, 2019) where a mathematical definition of Coulomb branches of 3d N = 4 quantum gauge theories (of cotangent type) is given, and also present a framework for studying some further mathematical structures (e.g. categories of line operators in the corresponding topologically twisted theories) related to these theories.
This is a companion paper of [Part II]. We study Coulomb branches of unframed and framed quiver gauge theories of type ADE. In the unframed case they are isomorphic to the moduli space of based ra- tional maps from P^1 to the flag variety. In the framed case they are slices in the affine Grassmannian and their generalization. In the appendix, written jointly with Joel Kamnitzer, Ryosuke Kodera, Ben Webster, and Alex Weekes, we identify the quantized Coulomb branch with the truncated shifted Yangian.
We propose a conjectural construction of various slices for double affine Grass- mannians as Coulomb branches of 3-dimensional N = 4 supersymmetric affine quiver gauge theories. It generalizes the known construction for the usual affine Grassman- nians, and makes sense for arbitrary symmetric Kac-Mody algebras.