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Kotaro Kawatani

Publications and source records attributed to Kotaro Kawatani.

13 recordsLinked to original sources
arXiv2026

Stability conditions and infinitesimal deformation of curves

Let $\mathcal X$ be an infinitesimal deformation of a smooth projective curve $X_0$ over a field. We study stability conditions under such deformations and show that the derived push-forward functor associated with the inclusion $X_0 \to \mathcal X$ induces an isomorphism between the space of stability conditions on $\mathcal X$ and that on $X_0$. This yields a direct comparison between the deformed and undeformed settings. As an application, we prove that the autoequivalence group $\mathrm{Aut}{\mathbf D^b(\mathcal X)}$ naturally acts on $\mathbf D^b(X_0)$, providing a link between derived symmetries and the deformation structure.

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arXiv2023

$R$-linear triangulated categories and stability conditions

Let $R$ be a commutative ring. We introduce the notion of support of objects in an $R$-linear triangulated category. As an application, we study the non-existence of Bridgeland stability conditions on $R$-linear triangulated categories.

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arXiv2021

On a deformation of gluing stability conditions

On a triangulated category $\mathbf D$ equipped with a semiorthogonal decomposition $\mathbf D=\langle{\mathbf D_{1}},{\mathbf D_{2}}\rangle$, Collins and Polishchuk develop a gluing construction of stability condition on $\mathbf D$. The gluing construction gives a stability condition on $\mathbf D$ from these on $\mathbf D_{1}$ and $\mathbf D_{2}$. We study a deformation of gluing stability conditions on for a nice semiorthogonal decomposition. As a consequence, we construct a continuous family of stability conditions by showing a deformation property introduced by Bridgeland's original paper. Here the deformation property is weaker than the support property which is the standard solution for the continuousness. After proving the continuousness of the family, we show that each stability condition in the family satisfies the support property via specialization. More precisely we find a stability condition with support property at the boundary of the family. Finally applying these results, we study the space of stability conditions on the category of morphisms in a triangulated category.

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arXiv2021

Stability conditions on affine Noetherian schemes

We show that the existence of locally finite stability conditions on the bounded derived category $\mathbf{D}^{b}(X)$ of coherent sheaves on an affine Noetherian scheme $X$ is equivalent to $\dim X=0$. We also study the spaces of stability conditions on the category of morphisms $\mathbf M_{X}$ in the derived category of the scheme $X$ and show that the spaces of stability conditions on $\mathbf{D}^{b}(X)$ and $\mathbf{M}_{X}$ are homotopy equivalent to each other.

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arXiv2020

Stability conditions on morphisms in a category

Let $\mathrm{h}\mathscr{C}$ be the homotopy category of a stable infinity category $\mathscr{C}$. Then the homotopy category $\mathrm{h}\mathscr{C}^{Δ^{1}}$ of morphisms in the stable infinity category $\mathscr{C}$ is also triangulated. Hence the space $\mathsf{Stab}\,{ \mathrm{h}\mathscr{C}^{Δ^{1}}}$ of stability conditions on $\mathrm{h}\mathscr{C}^{Δ^{1}}$ is well-defined though the non-emptiness of $\mathsf{Stab}\,{ \mathrm{h}\mathscr{C}^{Δ^{1}}}$ is not obvious. Our basic motivation is a comparison of the homotopy type of $\mathsf{Stab}{\mathrm{h}\mathscr{C}}$ and that of $\mathsf{Stab}{\mathrm{h}\mathscr{C}^{Δ^{1}}}$. Under the motivation we show that functors $d_{0}$ and $d_{1} \colon \mathscr{C}^{Δ^{1}} \rightrightarrows \mathscr{C}$ induce continuous maps from $\mathsf{Stab} {\mathrm{h}\mathscr{C}}$ to $\mathsf{Stab}{\mathrm{h}\mathscr{C}^{Δ^{1}}}$ contravariantly where $d_{0}$ (resp. $d_{1}$) takes a morphism to the target (resp. source) of the morphism. As a consequence, if $\mathsf{Stab}{\mathrm{h}\mathscr{C}}$ is nonempty then so is $\mathsf{Stab}{\mathrm{h}\mathscr{C}^{Δ^{1}}}$. Assuming $\mathscr{C}$ is the derived infinity category of the projective line over a field, we further study basic properties of $d_{0}^{*} $ and $d_{1}^{*}$. In addition, we give an example of a derived category which does not have any stability condition.

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arXiv2018

Nonexistence of semiorthogonal decompositions and sections of the canonical bundle

For any admissible subcategory of the bounded derived category of coherent sheaves on a smooth proper variety, we prove that sections of the canonical bundle impose a strong constraint on the supports of the objects of the subcategory or its semiorthogonal complement. We also show that admissible subcategories are rigid under the actions of topologically trivial autoequivalences. As applications of these results, we prove that the derived category of various minimal varieties in the sense of the minimal model program admit no non-trivial semiorthogonal decompositions, generalizing the result for curves due to the second author to higher dimensions. The case of minimal surfaces is further investigated in detail.

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arXiv2018

Pure sheaves and Kleinian singularities

Grothendieck proved that any locally free sheaf on a projective line over a field (uniquely) decomposes into a direct sum of line bundles. Ishii and Uehara construct an analogue of Grothendieck's theorem for pure sheaves on the fundamental cycle of the Kleinian singularity $A_n$. We first study the analogue for the other Kleinian singularities except for $A_n$. We also study the classification of rigid pure sheaves on the reduced scheme of the fundamental cycles. The classification is related to the classification of spherical objects in a certain Calabi-Yau $2$-dimensional category.

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arXiv2013

A hyperbolic metric and stability conditions on K3 surfaces with ρ=1

In this article we introduce a hyperbolic metric on the (normalized) space of stability conditions on projective K3 surfaces $X$ with Picard rank $ρ(X) =1$. And we show that all walls are geodesic in the normalized space with respect to the hyperbolic metric. Furthermore we demonstrate how the hyperbolic metric is helpful for us by discussing mainly three topics. We first make a study of so called Bridgeland's conjecture. In the second topic we prove a famous Orlov's theorem without the global Torelli theorem. In the third topic we give an explicit example of stable complexes in large volume limits by using the hyperbolic metric. Though Bridgeland's conjecture may be well-known for algebraic geometers, we would like to start from the review of it.

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arXiv2011

Stability of Gieseker stable sheaves on K3 surfaces in the sense of Bridgeland and some applications

We show that some Gieseker stable sheaves on a projective K3 surface $X$ are stable with respect to a stability condition of Bridgeland on the derived category of $X$ if the stability condition is in explicit subsets of the space of stability conditions depending on the sheaves. Furthermore we shall give two applications of the result. As a part of these applications, we show that the fine moduli space of Gieseker stable torsion free sheaves on a K3 surface with Picard number one is the moduli space of $μ$-stable locally free sheaves if the rank of the sheaves is not a square number.

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arXiv2011

Stability conditions and $μ$-stable sheaves on K3 surfaces with Picard number one

In this article, we show that some semi-rigid $μ$-stable sheaves on a projective K3 surface $X$ with Picard number 1 are stable in the sense of Bridgeland's stability condition. As a consequence of our work, we show that the special set $U(X) \subset \Stab (X)$ reconstructs $X$ itself. This gives a sharp contrast to the case of an abelian surface.

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