arXiv subjects

Yohsuke Imagi

Publications and source records attributed to Yohsuke Imagi.

11 recordsLinked to original sources

arXiv2026

Deformations of Compact Calabi--Yau and Fano Varieties with Isolated Singularities

Let $X$ be a compact log-canonical Kähler $n$-fold, $n\ge3,$ with trivial canonical sheaf and with isolated singularities. We prove that the generic fibres of a semi-universal deformation of $X$ have Du Bois invariant $b^{1,n-2}=0$ at the singular points. Under a certain topological hypothesis on $X$ the generic fibres have also link invariant $l^{1,n-2}=0.$ If $X$ is a projective log-canonical $n$-fold, $n\ge3,$ with ample anti-canonical sheaf and with isolated singularities then the generic fibres have $b^{1,n-2}=l^{1,n-2}=0$ (without the topological hypothesis). These are generalizations of recent results of Tenie `Global smoothing of singular Fano and Calabi--Yau varieties' and an older result of Namikawa `Deformation theory of Calabi--Yau threefolds and certain invariants of singularities.'

math.AG↗

arXiv2026

Remarks on the Gluing Theorems for Compact Special Lagrangian Submanifolds with Isolated Conical Singularities

Let X be a compact special Lagrangian submanifold with isolated conical singularities, and let there exist local smoothings of these singularities. It is then known that under certain hypotheses there exist global smoothings of X. We prove the same results under weaker hypotheses. We prove also that amongst these hypotheses that concerning the cohomology classes of X and the local smoothings is strictly necessary. We prove further that the global smoothings are $C^\infty$ with respect to the relevant parameters.

math.DG↗

arXiv2025

Nearby Special Lagrangians

Let $X$ be a Calabi--Yau manifold and $Q\subset X$ a closed connected embedded special Lagrangian; closed Lagrangians mean compact Lagrangian submanifolds without boundary. We prove that if the fundamental group $π_1Q$ is abelian then there exists a Weinstein neighbourhood of $Q\subset X$ in which every closed irreducibly immersed special Lagrangian with unobstructed Floer cohomology is $C^1$ close to $Q.$ We prove also that if $π_1Q$ is virtually solvable then for every positive integer $R$ there exists a Weinstein neighbourhood of $Q\subset X$ in which every closed irreducibly immersed special Lagrangian of degree $\le R$ and with unobstructed Floer cohomology is unbranched; that is, the projection $L\to Q$ is a covering map. We prove a stronger statement when $π_1Q$ is finite and a weaker statement when $π_1Q$ has no non-abelian free subgroups. The $π_1Q$ conditions, the Floer cohomology condition and the special Lagrangian condition are all essential as we show by counterexamples.

math.SG↗

arXiv2025

Embedding Theorems for Calabi--Yau Conifolds

We prove that compact Calabi--Yau varieties with certain isolated singularities are projective. In dimension 3 we do this by analysis, supposing given conifold metrics. In higher dimensions it follows more readily from Ohsawa's degenerate spectral sequence.

math.AG↗

arXiv2025

Deformations of Compact Calabi--Yau Conifolds

Let $X$ be a compact normal Kähler space whose canonical sheaf is a rank-one free $\mathcal O_X$ module and whose singularities are isolated, rational and quasi-homogeneous. We prove then that under a topological hypothesis the obstruction to deforming $X$ concentrates upon its singularities, generalizing partially the results of Namikawa--Gross. We prove also that under a certain hypothesis the locally trivial deformations of $X$ are unobstructed.

math.AG↗

arXiv2021

Generalized Thomas-Yau Uniqueness Theorems

We generalize Thomas-Yau's uniqueness theorem in two ways. We prove a stronger statement for special Lagrangians and include minimal Lagrangians in Kähler-Einstein manifold or more generally J-minimal Lagrangians introduced by Lotay and Pacini. In every case the heart of the proof is to make certain Hamiltonian perturbations. For this we use the method by Imagi, Joyce and Oliveira dos Santos.

math.DG↗

arXiv2019

Example of Compact Special Lagrangians with a Stable Singularity

We construct a family of compact almost Calabi--Yau manifolds of complex dimension 3 and therein a corresponding family of compact special Lagrangians with one-point singularities modelled upon that T^2-cone constructed by Harvey--Lawson and characterized by Haskins as a stable T^2-cone in the terminology by Joyce.

math.DG↗

arXiv2016

Surjectivity of a Gluing Construction in Special Lagrangian Geometry

This paper is motivated by a relatively recent work by Joyce in special Lagrangian geometry, but the basic idea of the present paper goes back to an earlier pioneering work of Donaldson in Yang--Mills gauge theory; Donaldson discovered a global structure of a (compactified) moduli space of Yang--Mills instantons, and a key step to that result was the proof of surjectivity of Taubes' gluing construction. In special Lagrangian geometry we have currently no such a global understanding of (compactified) moduli spaces, but in the present paper we determine a neighbourhood of a `boundary' point. It is locally similar to Donaldson's result, and in particular as Donaldson's result implies the surjectivity of Taubes' gluing construction so our result implies the surjectivity of Joyce's gluing construction in a certain simple case.

math.DG↗

arXiv2015

Uniqueness results for special Lagrangians and Lagrangian mean curvature flow expanders in C^m

We prove two main results: (a) Suppose $L$ is a closed, embedded, exact special Lagrangian $m$-fold in ${\mathbb C}^m$ for $m\ge 3$ asymptotic at infinity to the union $Π_1\cupΠ_2$ of two transverse special Lagrangian planes $Π_1,Π_2$ in ${\mathbb C}^m$. Then $L$ is one of the explicit 'Lawlor neck' family of examples found by Lawlor (Invent. math. 95, 1989). (b) Suppose $L$ is a closed, embedded, exact Lagrangian mean curvature flow expander in ${\mathbb C}^m$ for $m\ge 3$ asymptotic at infinity to the union $Π_1\cupΠ_2$ of two transverse Lagrangian planes $Π_1,Π_2$ in ${\mathbb C}^m$. Then $L$ is one of the explicit family of examples found by Joyce, Lee and Tsui, arXiv:0801.3721. If instead $L$ is immersed rather than embedded, the only extra possibility in (a),(b) is $L=Π_1\cupΠ_2$. Our methods, which are new and can probably be used to prove other similar uniqueness theorems, involve $J$-holomorphic curves, Lagrangian Floer cohomology, and Fukaya categories from symplectic topology. When $m=2$, (a) is easy to prove using hyperkahler geometry, and (b) is proved by Lotay and Neves, arXiv:1208.2729.

math.SG↗

arXiv2014

A Uniqueness Theorem for Gluing Calibrated Submanifolds

`Gluing' is a technique of constructing solutions to non-linear (elliptic) partial differential equations such as Yang--Mills equations, minimal surface equations and Einstein equations. Calibrated submanifolds are a certain class of minimal surfaces, and there are various examples of them constructed by the gluing technique. We have existence theorems in that sense, but there seems to have been no uniqueness theory for higher-dimensional ones such as special Lagrangian submanifolds, which we discuss in the present paper.

math.DG↗

arXiv2011

A Uniqueness Theorem for Gluing Special Lagrangian Submanifolds

Butscher, D. Lee, Y. Lee, and Joyce constructed a special Lagrangian submanifold by gluing a Lawlor neck into a transverse intersection point of two special Lagrangian submanifolds. We prove a uniqueness theorem for the gluing of flat special Lagrangian tori of real dimension 3 in a flat complex torus of complex dimension 3.

math.DG↗