arXiv subjects
Publications and source records attributed to Kazushi Ueda.
We introduce a class of noncommutative surfaces called Artin--Schelter surfaces of del Pezzo types, which contains del Pezzo surfaces as special cases. We show that the moduli stacks of marked Artin--Schelter surfaces of del Pezzo types are birational to the moduli stacks of tuples consisting of a smooth projective curve of genus 1, two line bundles of degree 3, and a collection of line bundles of degree 1 on the curve.
We prove homological mirror symmetry for projective hypersurfaces of sufficiently high degree using a functor from the wrapped Fukaya category of an affine hypersurface to the Fukaya category of its boundary at infinity.
We discuss homological mirror symmetry for Rabinowitz Fukaya categories of Milnor fibers of double suspensions of invertible polynomials, and prove it for Brieskorn--Pham polynomials which are not of Calabi--Yau type. This allows a calculation of the Rabinowitz Floer homology of the Milnor fiber as the Hochschild homology of the dg category of equivariant matrix factorizations.
We introduce a compact moduli scheme of marked noncommutative cubic surfaces as the GIT moduli scheme of relations of a quiver associated with a full strong exceptional collection on a cubic surface. It is a toric variety containing the configuration space of six points on a plane in general position as a locally closed subvariety, and birationally parametrizes a class of AS-regular $\mathbb{Z}$-algebras.
We show that the Satake-Baily-Borel compactification of the moduli space of lattice polarized K3 surfaces parametrizing K3 surfaces of degree 2 with four rational double points of type $D_4$ is the projective 3-space. We also show that the corresponding graded ring of automorphic forms is generated by four elements of weight 2 and one element of weight 11 with one relation of weight 22.
We introduce the notion of stable orbifold projective curves, and show that the moduli stack of stable orbifold projective curves is isomorphic to the moduli stack of weighted pointed stable curves in the sense of Hassett with respect to the weights determined by the automorphism groups of the stacky points.
Boundary points on the moduli space of pointed curves corresponding to collisions of marked points have modular interpretations as degenerate curves. In this paper, we study degenerations of orbifold projective curves corresponding to collisions of stacky points from the point of view of noncommutative algebraic geometry.
We construct a consistent dimer model having the same symmetry as its characteristic polygon. This produces examples of non-commutative crepant resolutions of non-toric non-quotient Gorenstein singularities in dimension 3.
We show that the base spaces of the semiuniversal unfoldings of some weighted homogeneous singularities can be identified with moduli spaces of $A_\infty$-structures on the trivial extension algebras of the endomorphism algebras of the tilting objects. The same algebras also appear in the Fukaya categories of their mirrors. Based on these identifications, we discuss applications to homological mirror symmetry for Milnor fibers, and give a proof of homological mirror symmetry for an $n$-dimensional affine hypersurface of degree $n + 2$ and the double cover of the $n$-dimensional affine space branched along a degree $2n + 2$ hypersurface. Along the way, we also give a proof of a conjecture of Seidel from math/0206155 which may be of independent interest.
We introduce a conjecture on homological mirror symmetry relating the symplectic topology of the complement of a smooth ample divisor in a K3 surface to algebraic geometry of type III degenerations, and prove it when the degree of the divisor is either 2 or 4.
We show that the ring of modular forms with characters for the even unimodular lattice of signature (2,18) is obtained from the invariant ring of $\mathrm{Sym}(\mathrm{Sym}^8(V) \oplus \mathrm{Sym}^{12}(V))$ with respect to the action of $\mathrm{SL}(V)$ by adding a Borcherds product of weight 132 with one relation of weight 264, where $V$ is a 2-dimensional $\mathbb{C}$-vector space. The proof is based on the study of the moduli space of elliptic K3 surfaces with a section.
We prove homological mirror symmetry for Milnor fibers of simple singularities in dimensions greater than one, which are among the log Fano cases of Conjecture 1.5 in arXiv:1806.04345. The proof is based on a relation between matrix factorizations and Calabi--Yau completions. As an application, we give an explicit computation of the Hochschild cohomology group of the derived $n$-preprojective algebra of a Dynkin quiver for any $n \geq 1$, and the symplectic cohomology group of the Milnor fiber of any simple singularity in any dimension greater than one.
We show that the ring of modular forms with characters for $\mathrm{O}(2,4;\mathbb{Z})$ is generated by forms of weights 4, 4, 6, 8, 10, 10, 12, and 30 with three relations of weights 8, 20, and 60. The proof is based on the study of a moduli space of K3 surfaces.
We study a correspondence between 3d $\mathcal{N}=2$ topologically twisted Chern-Simons-matter theories on $S^1 \times Σ_g$ and quantum $K$-theory of Grassmannians. Our starting point is a Frobenius algebra depending on a parameter $β$ associated with an algebraic Bethe ansatz introduced by Gorbounov-Korff. They showed that the Frobenius algebra with $β=-1$ is isomorphic to the (equivariant) small quantum $K$-ring of the Grassmannian, and the Frobenius algebra with $β=0$ is isomorphic to the equivariant small quantum cohomology of the Grassmannian. We apply supersymmetric localization formulas to the correlation functions of supersymmetric Wilson loops in the Chern-Simons-matter theory and show that the algebra of Wilson loops is isomorphic to the Frobenius algebra with $β=-1$. This allows us to identify the algebra of Wilson loops with the quantum $K$-ring of the Grassmannian. We also show that correlation functions of Wilson loops on $S^1 \times Σ_g$ satisfy the axiom of 2d TQFT. For $β=0$, we show the correspondence between an A-twisted GLSM, the Frobenius algebra for $β=0$, and the quantum cohomology of the Grassmannian. We also discuss deformations of Verlinde algebras, omega-deformations, and the $K$-theoretic $I$-functions of Grassmannians with level structures.
We discuss a construction of AS-regular algebras from acyclic spherical helices, generalizing works of Bondal and Polishchuk [BP93] and Van den Bergh [VdB11].
We show that the ring of modular forms with characters for the even unimodular lattice of signature (2,10) is generated by forms of weights 4, 10, 12, 16, 18, 22, 24, 28, 30, 36, 42, and 252 with one relation of weight 504. The proof is based on the comparison of the orbifold quotient of the symmetric domain with the root stack of the coarse moduli space.
Motivated by recent works on localizations in A-twisted gauged linear sigma models, we discuss a generalization of toric residue mirror symmetry to complete intersections in Grassmannians.
We discuss the geometry of rational maps from a projective space of an arbitrary dimension to the product of projective spaces of lower dimensions induced by linear projections. In particular, we give an algebro-geometric variant of the projective reconstruction theorem by Hartley and Schaffalitzky [HS09].