Non-commutative Grassmann variety as a moduli space
We construct a non-commutative version of the Grassmann variety $G(2,4)$ as a non-commutative moduli space of linear subspaces in a projective space.
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arXiv subjects
Yujiro Kawamata
Publications and source records attributed to Yujiro Kawamata.
Non-commutative deformations of tilting bundles and the derived McKay correspondence
We prove that the tilting bundle and the derived McKay correspondence extends under formal non-commutative deformations by using Cech cohomology of non-commutative schemes.
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Semi-orthogonal decomposition and smoothing
We investigate the behavior of semi-orthogonal decompositions of bounded derived categories of singular varieties under flat deformations to smooth varieties. We consider a Q-Gorenstein smoothing of a surface with a quotient singularity, and prove that a pretilting sheaf, which is constructed from a non-commutative deformation of a divisorial sheaf and generates a semi-orthogonal component of a bounded derived category, deforms to a direct sum of exceptional vector bundles which are mutually totally orthogonal.
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Non-Commutative Deformations of Derived McKay Correspondence for A(n) singularities
The derived McKay correspondence conjecture says that there is an equivalence of triangulated categories between the bounded derived categories of commutative and non-commutative crepant resolutions of a Gorenstein singularity. We will prove that this derived equivalence extends between the semi-universal non-commutative deformations of the commutative and the non-commutative crepant resolutions of a toric surface singularity.
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On formal non-commutative deformations of smooth varieties
We will develop a formal non-commutative (NC) deformation theory of smooth algebraic varieties $X$ defined over a field $k$, and describe a semi-universal deformation where the tangent space $T^1$ and the obstruction space $T^2$ are given as subspaces of Hochschild cohomology groups.
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Deformations over non-commutative base
We make some remarks on deformations over non-commutative base. We describe the base algebra of versal deformations using $T^1$ and $T^2$.
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Non-commutative deformations of perverse coherent sheaves and rational curves
We consider non-commutative deformations of sheaves on algebraic varieties. We develop some tools to determine parameter algebras of versal non-commutative deformations for partial simple collections and the structure sheaves of smooth rational curves. We apply them to universal flopping contractions of length 2 and higher. We confirm Donovan-Wemyss conjecture in the case of deformations of Laufer's flops.
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On the derived category of a weighted projective threefold
We calculate a semi-orthogonal decomposition of the bounded derived category of coherent sheaves on P(1,1,1,3) using a tilting bundle.
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On Fujita's semi-ampleness in the rank one case
We prove that a conjecture of Fujita on the semi-ampleness is true in the case of rank one direct summand, though it is wrong in higher rank case by Catanese and Dettweiler.
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On the extension problem of pluricanonical forms
We review some recent development on the extension problem of pluricanonical forms from a divisor to the ambient space in [Si], [K5] and [N3] with simplified proofs. A section for a correction is added.
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On non-commutative formal deformations of coherent sheaves on an algebraic variety
We review the theory of non-commutative deformations of sheaves and describe a versal deformation by using an A-infinity algebra and the change of differentials of an injective resolution. We give some explicit non-trivial examples.
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Semi-orthogonal decomposition of a derived category of a 3-fold with an ordinary double point
We consider semi-orthogonal decompositions of derived categories for 3-dimensional projective varieties in the case when the varieties have ordinary double points.
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Non-commutative deformations of simple objects in a category of perverse coherent sheaves
We determine versal non-commutative deformations of some simple collections in the categories of perverse coherent sheaves arising from tilting generators for projective morphisms.
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On multi-pointed non-commutative deformations and Calabi-Yau threefolds
We shall develop a theory of multi-pointed non-commutative deformations of a simple collection in an abelian category, and construct relative exceptional objects and relative spherical objects in some cases. This is inspired by a work by Donovan and Wemyss.
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Birational geometry and derived categories
This paper is based on a talk at a conference "JDG 2017: Conference on Geometry and Topology". We survey recent progress on the DK hypothesis connecting the birational geometry and the derived categories stating that the K-equivalence of smooth projective varieties should correspond to the equivalence of their derived categories, and the K-inequality to the fully faithful embedding.
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Fujita decomposition over higher dimensional base
We generalize a result of Fujita, on the decomposition of Hodge bundles over curves, to the case of a higher dimensional base.
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Derived McKay correspondence for GL(3,C)
We prove that the equivariant derived category for a finite subgroup of GL(3,C) has a semi-orthogonal decomposition into the derived category of a certain partial resolution, called a maximal Q-factorial terminalization, of the corresponding quotient singularity and a relative exceptional collection. This is a generalization of a result of Bridgeland, King and Reid.
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Derived categories of toric varieties III
We prove derived McKay correspondence in special cases and the decomposition of toric K-equivalence into flops.
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