arXiv subjects
Publications and source records attributed to Naoto Yotsutani.
We construct two families of projective bundles that admit two-coupled Kähler-Einstein (cKE) metrics but do not ordinary Kähler-Einstein metrics in sufficiently high dimensions. We also prove that no such examples arise among toric Fano threefolds, for any decomposition of $c_1(X)$, when considering the natural projection onto $\mathbb P^1$.
For a polarized toric variety, we provide a sufficient criterion ensuring that a uniformly K-stable polarized toric variety $(X,L)$ is asymptotically Chow polystable, under the assumption that the obstruction to asymptotic Chow semistability (the Futaki-Ono invariant) vanishes. Our approach is based on a detailed study of triangulations of neighborhoods of the vertices of the associated moment polytope $Δ$. As an application, we prove that every uniformly K-stable polarized smooth toric variety $(X,L)$ with vanishing Futaki-Ono invariant is asymptotically Chow polystable.
Let $X$ be a smooth projective toric variety, and let $\widetilde{X}$ denote the blow-up of $X$ at finitely many distinct torus-invariant points. In this paper, we derive an explicit combinatorial formula for the Chow weight of $\widetilde{X}$ in terms of the underlying toric manifold $X$ and the symplectic cuts of its associated Delzant polytope. As an application, we study toric blow-ups of the projective plane and compare their Chow stability with that of blow-ups at general points.
We prove that the only Bott manifolds such that the Futaki invariant vanishes for any Kähler class are isomorphic to the products of the projective lines.
We show that there exists a toric Fano manifold of dimension $10$ that does not admit an extremal Kähler metric in the first Chern class, answering a question of Mabuchi. By taking a product with a suitable toric Fano manifold, one can also produce a toric Fano manifold of dimension $n$ admitting no extremal Kähler metric in the first Chern class for each $n \geq 11$.
Let $X \to \mathbb P^N$ be a smooth linearly normal projective variety. It was proved by Paul that the $K$-energy of $(X, {ω_{FS}}|_{X})$ restricted to the Bergman metrics is bounded from below if and only if the pair of (rescaled) Chow/Hurwitz forms of $X$ is numerically semistable. In this paper, we provide a necessary and sufficient condition for a given smooth toric variety $X_P$ to be numerically semistable with respect to $\mathcal O_{X_P}(i)$ for a positive integer $i$. Applying this result to a smooth polarized toric variety $(X_P, L_P)$, we prove that $(X_P, L_P)$ is asymptotically numerically semistable if and only if it is K-semistable for toric degenerations.
In this paper, we prove that a Gorenstein toric Fano variety $(X, -K_{X})$ is asymptotically Chow semistable then it is Ding polystable with respect to toric test configurations (Theorem 1.3). This extends the known result obtained by others (Theorem 1.2) to the case where $X$ admits Gorenstein singularity. We also show the additivity of the Mabuchi constant for the product toric Fano varieties in Proposition 1.5 based on the author's recent work (Ono, Sano and Yotsutani in arXiv:2305.05924). Applying this formula to certain toric Fano varieties, we construct infinitely many examples that clarify the difference between relative K-stability and relative Ding stability in a systematic way (Proposition 1.4). Finally, we verify relative Chow stability for Gorenstein toric del Pezzo surfaces using the combinatorial criterion developed in (Yotsutani and Zhou in Tohoku Math. J. 71 (2019), 495-524.) and specifying the symmetry of the associated polytopes as well.
In this paper we study the relative Chow and $K$-stability of toric manifolds in the toric sense. First, we give a criterion for relative $K$-stability and instability of toric Fano manifolds in the toric sense. The reduction of relative Chow stability on toric manifolds will be investigated using the Hibert-Mumford criterion in two ways. One is to consider the maximal torus action and its weight polytope. We obtain a reduction by the strategy of Ono [Ono13], which fits into the relative GIT stability detected by Székelyhidi. The other way relies on $\mathbb{C}^*$-actions and Chow weights associated to toric degenerations following Donaldson and Ross-Thomas [D02, RT07]. As applications of our main theorem, we partially determine the relative $K$-stability of toric Fano threefolds and present counter-examples which are relatively $K$-stable in the toric sense but which are asymptotically relatively Chow unstable. In the end, we explain the erroneous parts of the published version of this article (corresponding to Sections 1-5), which provides some inconclusive results for relative $K$-stability in Table 6.
Let $X$ be a simple normal crossing (SNC) compact complex surface with trivial canonical bundle which includes triple intersections. We prove that if $X$ is $d$-semistable, then there exists a family of smoothings in a differential geometric sense. This can be interpreted as a differential geometric analogue of the smoothability results due to Friedman, Kawamata-Namikawa, Felten-Filip-Ruddat, Chan-Leung-Ma, and others in algebraic geometry. The proof is based on an explicit construction of local smoothings around the singular locus of $X$, and the first author's existence result of holomorphic volume forms on global smoothings of $X$. In particular, these volume forms are given as solutions of a nonlinear elliptic partial differential equation. As an application, we provide several examples of $d$-semistable SNC complex surfaces with trivial canonical bundle including double curves, which are smoothable to complex tori, primary Kodaira surfaces and $K3$ surfaces. We also provide several examples of such complex surfaces including triple points, which are smoothable to $K3$ surfaces.
Previously we constructed Calabi-Yau threefolds by a differential-geometric gluing method using Fano threefolds with their smooth anticanonical $K3$ divisors (New York J. Math. 20: 1-33, 2014). In this paper, we further consider the diffeomorphism classes of the resulting Calabi-Yau threefolds (which are called the doubling Calabi-Yau threefolds) starting from different pairs of Fano threefolds with Picard number one. Using the classifications of simply-connected $6$-manifolds in differential topology and the $λ$-invariant introduced by Lee (J. Math. Pures Appl. 141: 195-219, 2020), we prove that any two of the doubling Calabi-Yau threefolds with Picard number two are not diffeomorphic to each other when the underlying Fano threefolds are distinct families.
The purpose of this paper is to clarify all of the uniformly relatively Ding stable toric Fano threefolds and fourfolds as well as unstable ones. The key player in our classification result is the Mabuchi constants, which can be calculated by combinatorial data of the associated moment polytopes due to the work of Yao [33]. In Tables 1-3, we give the list of uniform relative Ding stability of all toric Fano manifolds in dimension up to four with the values of the Mabuchi constants. As an application of our main theorem (Theorem 1.1), we clarify the difference between relative $K$-stability and relative Ding stability by considering some specific toric Fano manifolds (Corollaries 1.6 and 1.9). In the proof of Corollary 1.9, we used Bott tower structure of relatively Ding unstable toric Fano manifolds.
Let $X$ be a normal crossing compact complex surface with triple points. We prove that there exists a family of smoothings of $X$ when $X$ satisfies suitable conditions. Since our differential geometric proof also includes the case where $X$ is neither Kählerian nor $H^1(X, \mathcal O_X)=0$, this generalizes Friedman's result on degenerations of $K3$ surfaces in algebraic geometry.
Chow stability is one notion of Mumford's Geometric Invariant Theory for studying the moduli space of polarized varieties. Kapranov, Sturmfels and Zelevinsky detected that Chow stability of polarized toric varieties is determined by its inherent {\it secondary polytope}, which is a polytope whose vertices correspond to regular triangulations of the associated polytope \cite{KSZ}. In this paper, we give a purely convex-geometrical proof that the Chow form of a projective toric variety is $H$-semistable if and only if it is $H$-polystable with respect to the standard complex torus action $H$. This \emph{essentially} means that Chow semistability is equivalent to Chow polystability for any (not-necessaliry-smooth) projective toric varieties.
We give a differential-geometric construction of compact manifolds with holonomy $\mathrm{Spin}(7)$ which is based on Joyce's second construction of compact $\mathrm{Spin}(7)$-manifolds in \cite{Joyce00} and Kovalev's gluing construction of $G_2$-manifolds in \cite{Kovalev03}. We also give some examples of compact $\mathrm{Spin}(7)$-manifolds, at least one of which is \emph{new}. Ingredients in our construction are \emph{orbifold admissible pairs with} a compatible antiholomorphic involution. Here in this paper we need orbifold admissible pairs $(\overline{X}, D)$ consisting of a four-dimensional compact Kähler orbifold $\overline{X}$ with isolated singular points modelled on $\mathbb{C}^4/\mathbb{Z}_4$, and a smooth anticanonical divisor $D$ on $\overline{X}$. Also, we need a compatible antiholomorphic involution $σ$ on $\overline{X}$ which fixes the singular points in $\overline{X}$ and acts freely on the anticanoncial divisor $D$. If two orbifold admissible pairs $(\overline{X}_1, D_1)$, $(\overline{X}_2, D_2)$ with $\dim_{\mathbb{C}} \overline{X}_i = 4$ and compatible antiholomorphic involutions $σ_i$ on $\overline{X}_i$ satisfy the \emph{gluing condition}, we can glue $(\overline{X}_1 \setminus D_1)/\braket{σ_1}$ and $(\overline{X}_2 \setminus D_2)/\braket{σ_2}$ together to obtain a compact Riemannian $8$-manifold $(M, g)$ whose holonomy group $\mathrm{Hol}(g)$ is contained in $\mathrm{Spin}(7)$. Furthermore, if the $\widehat{A}$-genus of $M$ equals $1$, then $M$ is a $\mathrm{Spin}(7)$-manifold, i.e., a compact Riemannian manifold with holonomy $\mathrm{Spin}(7)$. We shall investigate our gluing construction using $(\overline{X}_i,D_i)$ with $i=1,2$ when $D_1=D_2=D$ and $D$ is a complete intersection in a weighted projective space, as well as when $(\overline{X}_1,D_1)=(\overline{X}_2,D_2)$ and $σ_1=σ_2$ (the \emph{doubling} case).
We give a differential-geometric construction of Calabi-Yau fourfolds by the `doubling' method, which was introduced in \cite{DY14} to construct Calabi-Yau threefolds. We also give examples of Calabi-Yau fourfolds from toric Fano fourfolds. Ingredients in our construction are \emph{admissible pairs}, which were first dealt with by Kovalev in \cite{K03}. Here in this paper an admissible pair $(\overline{X},D)$ consists of a compact Kähler manifold $\overline{X}$ and a smooth anticanonical divisor $D$ on $\overline{X}$. If two admissible pairs $(\overline{X}_1,D_1)$ and $(\overline{X}_2,D_2)$ with $\dim_{\mathbb{C}}\overline{X}_i=4$ satisfy the \emph{gluing condition}, we can glue $\overline{X}_1\setminus D_1$ and $\overline{X}_2\setminus D_2$ together to obtain a compact Riemannian $8$-manifold $(M,g)$ whose holonomy group $\mathrm{Hol}(g)$ is contained in $\mathrm{Spin}(7)$. Furthermore, if the $\widehat{A}$-genus of $M$ equals $2$, then $M$ is a Calabi-Yau fourfold, i.e., a compact Ricci-flat Kähler fourfold with holonomy $\mathrm{SU}(4)$. In particular, if $(\overline{X}_1,D_1)$ and $(\overline{X}_2,D_2)$ are identical to an admissible pair $(\overline{X},D)$, then the gluing condition holds automatically, so that we obtain a compact Riemannian $8$-manifold $M$ with holonomy contained in $\mathrm{Spin}(7)$. Moreover, we show that if the admissible pair is obtained from \emph{any} of the toric Fano fourfolds, then the resulting manifold $M$ is a Calabi-Yau fourfold by computing $\widehat{A}(M)=2$.
We give a differential-geometric construction and examples of Calabi-Yau threefolds, at least one of which is {\it{new}}. Ingredients in our construction are {\it admissible pairs}, which were dealt with by Kovalev in \cite{K03} and further studied by Kovalev and Lee in \cite{KL11}. An admissible pair $(\overline{X},D)$ consists of a three-dimensional compact Kähler manifold $\overline{X}$ and a smooth anticanonical $K3$ divisor $D$ on $\overline{X}$. If two admissible pairs $(\overline{X}_1,D_1)$ and $(\overline{X}_2,D_2)$ satisfy the {\it gluing condition}, we can glue $\overline{X}_1\setminus D_1$ and $\overline{X}_2\setminus D_2$ together to obtain a Calabi-Yau threefold $M$. In particular, if $(\overline{X}_1,D_1)$ and $(\overline{X}_2,D_2)$ are identical to an admissible pair $(\overline{X},D)$, then the gluing condition holds automatically, so that we can {\it always} construct a Calabi-Yau threefold from a {\it single} admissible pair $(\overline{X},D)$ by {\it doubling} it. Furthermore, we can compute all Betti and Hodge numbers of the resulting Calabi-Yau threefolds in the doubling construction.
Donaldson proved that if a polarized manifold $(V,L)$ has constant scalar curvature Kähler metrics in $c_1(L)$ and its automorphism group Aut$(M,L)$ is discrete, $(V,L)$ is asymptotically Chow stable. In this paper, we shall show an example which implies that the above result does not hold in the case when Aut$(V,L)$ is not discrete.