arXiv subjects

Kotaro Kawai

Publications and source records attributed to Kotaro Kawai.

17 recordsLinked to original sources

arXiv2026

Anisotropic calibrations, Fueter maps and mirror symmetry

Let $(M,g)$ be a Riemannian manifold. Choose a pair $(α,H)$, where $α$ is a calibration and $H$ is a calibrated distribution. Using these data, we define a 1-parameter family of forms $α_\varepsilon$ and study its adiabatic limit as $\varepsilon\rightarrow 0$. We show that (i) the limit is a calibration in a generalized sense, (ii) under the usual closedness assumptions, the adiabatic calibrated submanifolds are anisotropic minimal in the classical sense defined in the Calculus of Variations/PDE theory. We apply this construction to $G_2$-manifolds endowed with an associative distribution. Here, one can also define the notion of Fueter maps. We prove that, in the case of isometric immersions, adiabatic calibrated submanifolds coincide with Fueter maps: this is a first-order analogue of the classical relationship between minimal submanifolds and harmonic maps. We provide explicit examples and prove local analytic existence theorems for adiabatic calibrated submanifolds. Applying mirror symmetry as described by the real Fourier-Mukai transform in the standard ``toy model'' situation, the picture is as follows: adiabatic limits correspond to large radius limits, calibrated (associative) submanifolds correspond to deformed Donaldson-Thomas connections, adiabatic calibrated submanifolds correspond to $G_2$-instantons.

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arXiv2025

A monotonicity formula for minimal connections

For Hermitian connections on a Hermitian complex line bundle over a Riemannian manifold $(X,g)$, we can define the ``volume", which can be considered to be the ``mirror" of the standard volume for submanifolds. We call the critical points minimal connections. In this paper, (1) we prove monotonicity formulas for minimal connections with respect to some versions of volume functionals under certain conditions on $\dim X$ and the curvature of $g$. These formulas would be important in bubbling analysis. As a corollary, we obtain the vanishing theorem for minimal connections on the odd dimensional Euclidean space. (2) We see that the formal ``large radius limit" of the defining equation of minimal connections is that of Yang--Mills connections. Then the existence theorem of minimal connections is proved for a ``sufficiently large" metric. (3) We can consider deformed Donaldson--Thomas (dDT) connections on $G_2$-manifolds as ``mirrors" of calibrated (associative) submanifolds. We show that dDT connections are minimal connections, just as calibrated submanifolds are minimal submanifolds. By the argument specific to dDT connections, we obtain the stronger monotonicity formulas and vanishing theorem for dDT connections than in (1).

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arXiv2023

Some observations on deformed Donaldson-Thomas connections

A deformed Donaldson-Thomas (dDT) connection is a Hermitian connection of a Hermitian line bundle over a $G_2$-manifold $X$ satisfying a certain nonlinear PDE. This is considered to be the mirror of a (co)associative cycle in the context of mirror symmetry. It can also be considered as an analogue of a $G_2$-instanton. In this paper, we see that some important observations that appear in other geometric problems are also found in the dDT case as follows. (1) A dDT connection exists if a 7-manifold has full holonomy $G_2$ and the $G_2$-structure is ``sufficiently large". (2) The dDT equation is described as the zero of a certain multi-moment map. (3) The gradient flow equation of a Chern-Simons type functional of Karigiannis and Leung, whose critical points are dDT connections, agrees with the ${\rm Spin}(7)$ version of the dDT equation on a cylinder with respect to a certain metric on a certain space. This can be considered as an analogue of the observation in instanton Floer homology for 3-manifolds.

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arXiv2023

Almost formality of manifolds of low dimension

In this paper we introduce the notion of Poincaré DGCAs of Hodge type, which is a subclass of Poincaré DGCAs encompassing the de Rham algebras of closed orientable manifolds. Then we introduce the notion of the small algebra and the small quotient algebra of a Poincaré DGCA of Hodge type. Using these concepts, we investigate the equivalence class of $(r-1)$ connected $(r>1)$ Poincaré DGCAs of Hodge type. In particular, we show that a $(r-1)$ connected Poincaré DGCA of Hodge type ${\mathcal A}^\ast$ of dimension $n \le 5r-3$ is $A_\infty$-quasi-isomorphic to an $A_3$-algebra and prove that the only obstruction to the formality of ${\mathcal A}^\ast$ is a distinguished Harrison cohomology class $[μ_3] \in {\mathsf{Harr}}^{3,-1} (H^*({\mathcal A}^\ast), H^*({\mathcal A}^\ast))$. Moreover, the cohomology class $[μ_3]$ and the DGCA isomorphism class of $H^*({\mathcal A}^\ast)$ determine the $A_\infty$-quasi-isomorphism class of ${\mathcal A}^\ast$. This can be seen as a Harrison cohomology version of the Crowley-Nordström results [D. Crowley, J. Nordström, The rational homotopy type of $(n-1)$-connected manifolds of dimension up to $5n-3$, arXiv:1505.04184v2] on rational homotopy type of $(r-1)$-connected $(r>1)$ closed manifolds of dimension up to $5r-3$. We also derive the almost formality of closed $G_2$-manifolds, which have been discovered recently by Chan-Karigiannis-Tsang in [K.F. Chan, S. Karigiannis and C.C. Tsang, The ${\mathcal L}_B$-cohomology on compact torsion-free ${\rm G}_2$ manifolds and an application to `almost' formality, arXiv:1801.06410, to appear in Ann. Global Anal. Geom.], from our results and the Cheeger-Gromoll splitting theorem.

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arXiv2022

Deformation theory of deformed Hermitian Yang-Mills connections and deformed Donaldson-Thomas connections

A deformed Donaldson-Thomas (dDT) connection is a Hermitian connection of a Hermitian line bundle over a $G_2$-manifold $X$ satisfying a certain nonlinear PDE. This is considered to be the mirror of a (co)associative cycle in the context of mirror symmetry. The dDT connection is an analogue of a deformed Hermitian Yang-Mills (dHYM) connection which is extensively studied recently. In this paper, we study the moduli spaces of dDT and dHYM connections. In the former half, we prove that the deformation of dDT connections is controlled by a subcomplex of the canonical complex, an elliptic complex defined by Reyes Carrión, by introducing a new coclosed $G_2$-structure. If the deformation is unobstructed, we also show that the connected component is a $b^{1}$-dimensional torus, where $b^{1}$ is the first Betti number of $X$. A canonical orientation on the moduli space is also given. We also prove that the obstruction of the deformation vanishes if we perturb the $G_2$-structure generically under some mild assumptions. In the latter half, we prove that the moduli space of dHYM connections, if it is nonempty, is a $b^{1}$-dimensional torus, especially, it is connected and orientable. We also prove the existence of a family of moduli spaces along a deformation of underlying structures if some necessary conditions are satisfied.

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arXiv2022

Mirror of volume functionals on manifolds with special holonomy

We can define the ``volume'' $V$ for Hermitian connections on a Hermitian complex line bundle over a Riemannian manifold $X$, which can be considered to be the ``mirror'' of the standard volume for submanifolds. This is called the Dirac-Born-Infeld (DBI) action in physics. In this paper, (1) we introduce the negative gradient flow of $V$, which we call the line bundle mean curvature flow. Then, we show the short-time existence and uniqueness of this flow. When $X$ is Kähler, we relate the negative gradient of $V$ to the angle function and deduce the mean curvature for Hermitian metrics on a holomorphic line bundle defined by Jacob and Yau. (2) We relate the functional $V$ to a deformed Hermitian Yang--Mills (dHYM) connection, a deformed Donaldson--Thomas connection for a $G_2$-manifold (a $G_2$-dDT connection), a deformed Donaldson--Thomas connection for a ${\rm Spin}(7)$-manifold (a ${\rm Spin}(7)$-dDT connection), which are considered to be the ``mirror'' of special Lagrangian, (co)associative and Cayley submanifolds, respectively. When $X$ is a compact ${\rm Spin}(7)$-manifold, we prove the ``mirror'' of the Cayley equality, which implies the following. (a) Any ${\rm Spin}(7)$-dDT connection is a global minimizer of $V$ and its value is topological. (b) Any ${\rm Spin}(7)$-dDT connection is flat on a flat line bundle. (c) If $X$ is a product of $S^1$ and a compact $G_2$-manifold $Y$, any ${\rm Spin}(7)$-dDT connection on the pullback of the Hermitian complex line bundle over $Y$ is the pullback of a $G_2$-dDT connection modulo closed 1-forms. We also prove analogous statements for $G_2$-manifolds and Kähler manifolds of dimension 3 or 4.

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arXiv2021

The real Fourier-Mukai transform of Cayley cycles

The real Fourier-Mukai transform sends a section of a torus fibration to a connection over the total space of the dual torus fibration. By this method, Leung, Yau and Zaslow introduced deformed Hermitian Yang-Mills (dHYM) connections for Kähler manifolds and Lee and Leung introduced deformed Donaldson-Thomas (dDT) connections for $G_2$- and ${\rm Spin}(7)$-manifolds. In this paper, we suggest an alternative definition of a dDT connection for a manifold with a ${\rm Spin}(7)$-structure which seems to be more appropriate by carefully computing the real Fourier-Mukai transform again. We also post some evidences showing that the definition we suggest is compatible with dDT connections for a $G_2$-manifold and dHYM connections of a Calabi-Yau 4-manifold. Another importance of this paper is that it motivates our study in our other papers. That is, based on the computations in this paper, we develop the theories of deformations of dDT connections for a manifold with a ${\rm Spin}(7)$-structure and the "mirror" of the volume functional, which is called the Dirac-Born-Infeld (DBI) action in physics.

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arXiv2021

Deformation theory of deformed Donaldson-Thomas connections for ${\rm Spin}(7)$-manifolds

A deformed Donaldson-Thomas connection for a manifold with a ${\rm Spin}(7)$-structure, which we call a ${\rm Spin}(7)$-dDT connection, is a Hermitian connection on a Hermitian line bundle $L$ over a manifold with a ${\rm Spin}(7)$-structure defined by fully nonlinear PDEs. It was first introduced by Lee and Leung as a mirror object of a Cayley cycle obtained by the real Fourier-Mukai transform and its alternative definition was suggested in our other paper. As the name indicates, a ${\rm Spin}(7)$-dDT connection can also be considered as an analogue of a Donaldson-Thomas connection (${\rm Spin}(7)$-instanton). In this paper, using our definition, we show that the moduli space $\mathcal{M}_{{\rm Spin}(7)}$ of ${\rm Spin}(7)$-dDT connections has similar properties to these objects. That is, we show the following for an open subset $\mathcal{M}'_{{\rm Spin}(7)} \subset \mathcal{M}_{{\rm Spin}(7)}$. (1) Deformations of elements of $\mathcal{M}'_{{\rm Spin}(7)}$ are controlled by a subcomplex of the canonical complex introduced by Reyes Carrión by introducing a new ${\rm Spin}(7)$-structure from the initial ${\rm Spin}(7)$-structure and a ${\rm Spin}(7)$-dDT connection. (2) The expected dimension of $\mathcal{M}'_{{\rm Spin}(7)}$ is finite. It is $b^1$, the first Betti number of the base manifold, if the initial ${\rm Spin}(7)$-structure is torsion-free. (3) Under some mild assumptions, $\mathcal{M}'_{{\rm Spin}(7)}$ is smooth if we perturb the initial ${\rm Spin}(7)$-structure generically. (4) The space $\mathcal{M}'_{{\rm Spin}(7)}$ admits a canonical orientation if all deformations are unobstructed.

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arXiv2020

Conformal transformations of the pseudo-Riemannian metric of a homogeneous pair

We introduce a new notion of a homogeneous pair for a pseudo-Riemannian metric $g$ and a positive function $f$ on a manifold $M$ admitting a free $\mathbb{R}_{>0}$-action. There are many examples admitting this structure. For example, (a) a class of pseudo-Hessian manifolds admitting a free $\mathbb{R}_{>0}$-action and a homogeneous potential function such as the moduli space of torsion-free $G_2$-structures, (b) the space of Riemannian metrics on a compact manifold, and (c) many moduli spaces of geometric structures such as torsion-free ${\rm Spin}(7)$-structures admit this structure. Hence we provide the unified method for the study of these geometric structures. We consider conformal transformations of the pseudo-Riemannian metric $g$ of a homogeneous pair $(g, f)$. Showing that the pseudo-Riemannian manifold $(M, (v \circ f) g)$, where $v: \mathbb{R}_{>0} \rightarrow \mathbb{R}_{>0}$ is a smooth function, has the structure of a warped product, we study the geometric structures such as the sectional curvature, geodesics and the metric completion (if $g$ is positive definite) w.r.t. $(v \circ f) g$ in terms of those on the level set of $f$. In particular, (1) we can generalize the result of Clarke and Rubinstein about the metric completion of the space of Riemannian metrics w.r.t. the conformal transformations of the Ebin metric, and (2) two canonical Riemannian metrics on the $G_2$ moduli space have different metric completions.

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arXiv2019

Frölicher-Nijenhuis cohomology on $G_2$- and ${\rm Spin}(7)$-manifolds

In this paper we show that a parallel differential form $Ψ$ of even degree on a Riemannian manifold allows to define a natural differential both on $Ω^\ast(M)$ and $Ω^\ast(M, TM)$, defined via the Frölicher-Nijenhuis bracket. For instance, on a Kähler manifold, these operators are the complex differential and the Dolbeault differential, respectively. We investigate this construction when taking the differential w.r.t. the canonical parallel $4$-form on a $G_2$- and ${\rm Spin}(7)$-manifold, respectively. We calculate the cohomology groups of $Ω^\ast(M)$ and give a partial description of the cohomology of $Ω^\ast(M, TM)$.

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arXiv2018

Frölicher-Nijenhuis bracket on manifolds with special holonomy

In this article, we summarize our recent results on the study of manifolds with special holonomy via the Frölicher-Nijenhuis bracket. This bracket enables us to define the Frölicher-Nijenhuis cohomologies which are analogues of the $d^c$ and the Dolbeault cohomologies in Kähler geometry, and assigns an $L_\infty$-algebra to each associative submanifold. We provide several concrete computations of the Frölicher-Nijenhuis cohomology.

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arXiv2018

Cohomogeneity One Coassociative Submanifolds in the Bundle of Anti-self-dual 2-forms over the 4-sphere

Coassociative submanifolds are 4-dimensional calibrated submanifolds in $G_{2}$-manifolds. In this paper, we construct explicit examples of coassociative submanifolds in $Λ^{2}_{-} S^{4}$, which is the complete $G_{2}$-manifold constructed by Bryant and Salamon. Classifying the Lie groups which have 3- or 4-dimensional orbits, we show that the only homogeneous coassociative submanifold is the zero section of $Λ^{2}_{-} S^{4}$ up to the automorphisms and construct many cohomogeneity one examples explicitly. In particular, we obtain examples of non-compact coassociative submanifolds with conical singularities and their desingularizations.

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arXiv2017

Second order deformations of associative submanifolds in nearly parallel $G_2$-manifolds

Associative submanifolds $A$ in nearly parallel $G_2$-manifolds $Y$ are minimal 3-submanifolds in spin 7-manifolds with a real Killing spinor. The Riemannian cone over $Y$ has the holonomy group contained in ${\rm Spin(7)}$ and the Riemannian cone over $A$ is a Cayley submanifold. Infinitesimal deformations of associative submanifolds were considered by the author. This paper is a continuation of the work. We give a necessary and sufficient condition for an infinitesimal associative deformation to be integrable (unobstructed) to second order explicitly. As an application, we show that the infinitesimal deformations of a homogeneous associative submanifold in the 7-sphere given by Lotay, which he called $A_3$, are unobstructed to second order.

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arXiv2017

Frölicher-Nijenhuis bracket and geometry of $G_2$-and ${\rm Spin}(7)$-manifolds

We extend the characterization of the integrability of an almost complex structure $J$ on differentiable manifolds via the vanishing of the Frölicher-Nijenhuis bracket $[J, J] ^{FN}$ to an analogous characterization of torsion-free $G_2$-structures and torsion-free Spin(7)-structures. We also explain the Fernández-Gray classification of $G_2$-structures and the Fernández classification of Spin(7)-structures in terms of the Frölicher-Nijenhuis bracket.

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arXiv2017

Deformations of homogeneous associative submanifolds in nearly parallel $G_{2}$-manifolds

A nearly parallel $G_{2}$-manifold $Y$ is a Riemannian 7-manifold whose cone $C(Y) = \mathbb{R}_{>0} \times Y$ has the holonomy group contained in ${\rm Spin(7)}$. In other words, it is a spin 7-manifold with a real Killing spinor. We have a special class of calibrated submanifolds called Cayley submanifolds in $C(Y)$. An associative submanifold in $Y$ is a minimal 3-submanifold whose cone is Cayley. We study its deformations, namely, Cayley cone deformations, explicitly when it is homogeneous in the 7-sphere $S^{7}$.

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arXiv2016

Stabilities of affine Legendrian submanifolds and their moduli spaces

We introduce the notion of affine Legendrian submanifolds in Sasakian manifolds and define a canonical volume called the $ϕ$-volume as odd dimensional analogues of affine Lagrangian (totally real or purely real) geometry. Then we derive the second variation formula of the $ϕ$-volume to obtain the stability result in some $η$-Einstein Sasakian manifolds. It also implies the convexity of the $ϕ$-volume functional on the space of affine Legendrian submanifolds. Next, we introduce the notion of special affine Legendrian submanifolds in Sasaki-Einstein manifolds as a generalization of that of special Legendrian submanifolds. Then we show that the moduli space of compact connected special affine Legendrian submanifolds is a smooth Fréchet manifold.

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arXiv2015

Some associative submanifolds of the squashed 7-sphere

The squashed 7-sphere $S^{7}$ is a 7-sphere with an Einstein metric given by the canonical variation and its cone $\mathbb{R}^{8} - \{ 0 \}$ has full holonomy ${\rm Spin}(7)$. There is a canonical calibrating 4-form $Φ$ on $\mathbb{R}^{8} - \{ 0 \}$. A minimal 3-submanifold in $S^{7}$ is called associative if its cone is calibrated by $Φ$. In this paper, we classify two types of fundamental associative submanifolds in the squashed $S^{7}$. One is obtained by the intersection with a 4-plane and the other is homogeneous. Then we study their infinitesimal associative deformations and explicitly show that all of them are integrable.

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