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Yuuji Tanaka

Publications and source records attributed to Yuuji Tanaka.

26 records · Page 2Linked to original sources
arXiv2017

Some boundedness properties of solutions to the Vafa-Witten equations on closed four-manifolds

We consider a set of gauge-theoretic equations on closed oriented four-manifolds, which was introduced by Vafa and Witten. The equations involve a triple consisting of a connection and extra fields associated to a principal bundle over a closed oriented four-manifold. They are similar to Hitchin's equations over compact Riemann surfaces, and as part of the resemblance, there is no $L^2$-bound on the curvature without an $L^2$-bound on the extra fields. In this article, however, we observe that under the particular circumstance where the curvature does not become concentrated and the limiting connection is not locally reducible, one obtains an $L^2$-bound on the extra fields.

math.DG↗
arXiv2016

On the moduli space of Donaldson-Thomas instantons

In alignment with a programme by Donaldson and Thomas [DT], Thomas [Th] constructed a deformation invariant for smooth projective Calabi-Yau threefolds, which is now called the Donaldson-Thomas invariant, from the moduli space of (semi-)stable sheaves by using algebraic geometry techniques. In the same paper [Th], Thomas noted that certain perturbed Hermitian-Einstein equations might possibly produce an analytic theory of the invariant. This article sets up the equations on symplectic 6-manifolds, and gives the local model and structures of the moduli space coming from the equations. We then describe a Hitchin-Kobayashi style correspondence for the equations on compact Kähler threefolds, which turns out to be a special case of results by Alvarez-Consul and Garcia-Prada [AG].

math.DG↗
arXiv2014

Stable sheaves with twisted sections and the Vafa-Witten equations on smooth projective surfaces

This article describes a Hitchin-Kobayashi style correspondence for the Vafa-Witten equations on smooth projective surfaces. This is an equivalence between a suitable notion of stability for a pair $(\mathcal{E}, φ)$, where $\mathcal{E}$ is a locally-free sheaf over a surface $X$ and $φ$ is a section of $\text{End} (\mathcal{E}) \otimes K_{X}$; and the existence of a solution to certain gauge-theoretic equations, the Vafa-Witten equations, for a Hermitian metric on $\mathcal{E}$. It turns out to be a special case of results obtained by Alvarez-Consul and Garcia-Prada. In this article, we give an alternative proof which uses a Mehta-Ramanathan style argument originally developed by Donaldson for the Hermitian-Einstein problem, as it relates the subject with the Hitchin equations on Riemann surfaces, and surely indicates a similar proof of the existence of a solution under the assumption of stability for the Donaldson-Thomas instanton equations described in arXiv:0805.2192 on smooth projective threefolds; and more broadly that for the quiver vortex equation on higher dimensional smooth projective varieties.

math.DG↗
arXiv2013

A stability of vector bundles with twisted sections and the Donaldson-Thomas instantons on compact Kähler threefolds

We consider a version of Hermitian-Einstein equation but perturbed by a Higgs field with a solution called a Donaldson-Thomas instanton on compact Kähler threefolds. The equation could be thought of as a generalization of the Hitchin equation on Riemann surfaces to Kähler threefolds. In the appendix of arXiv:0805.2192, following an analogy with the Hitchin equation, we introduced a stability condition for a pair consisting of a locally-free sheaf over a compact Kähler threefold and a section of the associated sheaf of the endomorphisms tensored by the canonical bundle of the threefold. In this article, we prove a Hitchin--Kobayashi-type correspondence for this and the Donaldson-Thomas instanton on compact Kähler threefolds.

math.DG↗
arXiv2013

A removal singularity theorem of the Donaldson-Thomas instantons on compact Kähler threefolds

We consider a perturbed Hermitian-Einstein equation, which we call the Donaldson-Thomas equation, on compact Kähler threefolds. In arXiv:0805.2195, we analysed some analytic properties of solutions to the equation, in particular, we proved that a sequence of solutions to the Donaldson-Thomas equation has a subsequence which smoothly converges to a solution to the Donaldson-Thomas equation outside a closed subset of the Hausdorff dimension two. In this article, we prove that some of these singularities can be removed.

math.DG↗
arXiv2013

A weak compactness theorem of the Donaldson-Thomas instantons on compact Kähler threefolds

In arXiv:0805.2192, we set up a gauge-theoretic equation on symplectic 6-manifolds, which is a version of the Hermitian-Einstein equation perturbed by Higgs fields, and call Donaldson-Thomas equation, to analytically approach the Donaldson-Thomas invariants. In this article, we consider the equation on compact Kähler threefolds, and study some of analytic properties of solutions to them, using analytic methods in higher-dimensional Yang-Mills theory developed by Nakajima and Tian with some additional arguments concerning an extra non-linear term coming from the Higgs fields. We prove that a sequence of solutions to the Donaldson-Thomas equation of a unitary vector bundle over a compact Kähler threefold has a converging subsequence outside a closed subset whose real 2-dimensional Hausdorff measure is finite, provided that the L^2-norms of the Higgs fields are uniformly bounded. We also prove an n/2-compactness theorem of solutions to the equations on compact Kähler threefolds.

math.DG↗
arXiv2012

A construction of Spin(7)-instantons

Joyce constructed examples of compact eight-manifolds with holonomy Spin(7), starting with a Calabi-Yau four-orbifold with isolated singular points of a special kind. That construction can be seen as the gluing of ALE Spin(7)-manifolds to each singular point of the Calabi-Yau four-orbifold divided by an anti-holomorphic involution fixing only the singular points. On the other hand, there are higher-dimensional analogues of anti-self-dual instantons in four dimensions on Spin(7)-manifolds, which are called Spin(7)-instantons. They are minimizers of the Yang-Mills action, and the Spin(7)-instanton equation together with a gauge fixing condition forms an elliptic system. In this article, we construct Spin(7)-instantons on the examples of compact Spin(7)-manifolds above, starting with Hermitian-Einstein connections on the Calabi-Yau four-orbifolds and ALE spaces. Under some assumptions on the Hermitian-Einstein connections, we glue them together to obtain Spin(7)-instantons on the compact Spin(7)-manifolds. We also give a simple example of our construction.

math.DG↗