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Yuji Odaka

Publications and source records attributed to Yuji Odaka.

At least 19 recordsLinked to original sources
arXiv2025

On Sun-Zhang's theory of Fano fibrations -- weighted volumes, moduli and bubbling Fano fibrations

We revisit the recent theory of Sun-Zhang on general Fano fibration (germs) which emerged from the study of non-compact Kahler-Ricci soliton metrics, primarily from an algebro-geometric perspective. In addition to reviewing the existing framework, we present new results, conjectures, and remarks. These include methods for computing weighted volumes via (restricted) volumes, Laplace transforms, and incomplete Gamma-functions, and a conjectural algebro-geometric construction (``bubbling") of Fano fibration with asymptotically conical base from degenerating Fano fibration.

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arXiv2025

Canonical torus action on symplectic singularities

We show that any symplectic singularity lying on a smoothable projective symplectic variety locally admits a good action of $(\mathbb{C}^*)^r$, which is canonical. Under mild assumptions, we actually prove such singularity germ is the cone vertex over a contact orbifold with weak K\"ahler-Einstein metric, forcing $r=1$. In particular, it admits a (canonical) good $\mathbb{C}^*$-action, which also extends to (canonical) actions of $\mathbb{H}^*\supset SU(2)$. These settle Kaledin's conjecture conditionally but in a substantially stronger form by establishing the canonicity, the extensibility of the action, for instance. Our key idea is to use the Donaldson-Sun theory on local K\"ahler metrics in complex differential geometry to connect with the theory of Poisson deformations of symplectic varieties. For general symplectic singularities, we prove the same assertions, assuming that the Donaldson-Sun theory extends to such singularities along with suitable singular (hyper)K\"ahler metrics. Conversely, our results can also be used to study the local behavior of such metrics around the germ.

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arXiv2024

Algebraic geometry of bubbling Kahler metrics

We give an algebro-geometric or non-archimedean framework to study bubbling phenomena of Kahler metrics with Euclidean volume growth, after [DS17, Sun23, dBS23]. In particular, for any degenerating family to log terminal singularity, we algebraically construct a finite sequence of birational modifications of the family with milder degenerations, and compare with analytic bubbling constructions in loc.cit. We also provide approaches in terms of coordinates and valuations. Our discussion partially depends on the general framework of stability theory in our [Od24b] (arXiv:2406.02489) after [HL14, AHLH23].

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arXiv2024

Stability theory over toroidal or Novikov type base and Canonical modifications

We set up a generalization of ubiquitous one-parameter families in algebraic geometry and their use for stability theories ([GIT, HL, AHLH]) to families over toric varieties and their analytic analogues. The language allows us to reformulate degenerations of ``irrational" direction in the literature as canonical objects in a unified manner. Accordingly, we generalize the (semi)stable reduction-type theorem for $\Theta$-stratification in [AHLH] of Langton type to our higher rank setup. We also establish a complex analytic analogue of the results. As an infinitesimal analogue of toric spectrum, we also use Novikov type rings, as they provide greater canonicity, but their use can be avoided logically for readers who prefer not to use such rings. Our other papers discuss applications of our framework and the main theorems to problems around K-stability and Kahler geometry.

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arXiv2024

Compact moduli of Calabi-Yau cones and Sasaki-Einstein spaces

We construct proper moduli algebraic spaces of K-polystable $\mathbb{Q}$-Fano cones (a.k.a. Calabi-Yau cones) or equivalently their links i.e., Sasaki-Einstein manifolds with singularities. As a byproduct, it gives alternative algebraic construction of proper K-moduli of $\mathbb{Q}$-Fano varieties. In contrast to the previous algebraic proof of its properness ([BHLLX, LXZ]), we do not use the $\delta$-invariants ([FO, BJ]) nor the $L^2$-normalized Donaldson-Futaki invariants. We use the local normalized volume of [Li] and the higher $\Theta$-stable reduction instead.

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arXiv2022

Minimization of Arakelov K-energy for many cases

We prove that for various polarized varieties over $\overline{\mathbb{Q}}$, which broadly includes K-trivial case, K-ample case, Fano case, minimal models, certain classes of fibrations, certain metrized "minimal-like" models minimizes the Arakelov theoretic analogue of the Mabuchi K-energy, as conjectured in [Od15]. This is an Arakelov theoretic analogue of [H22b].

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arXiv2022

Special Lagrangian fibrations, Berkovich retraction, and crystallographic groups

We explicitly construct special Lagrangian fibrations on finite quotients of maximally degenerating abelian varieties, glue with Berkovich retraction in non-Archimedean geometry by using "hybrid" technique. We also study their symmetries explicitly which can be regarded as crystallographic groups. In particular, a conjecture of Kontsevich-Soibelman is solved at an enhanced level for finite quotients of abelian varieties in any dimension.

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arXiv2022

Semi-toric and toroidal compactifications as log minimal models, and applications to weak K-moduli

We give a characterization of toroidal (resp., semi-toric) compactifications due to Ash-Mumford-Rapoport-Tai (resp., Looijenga) as log minimal models and apply it to study weak K-moduli compactifications, giving a different proof to a theorem of Alexeev-Engel. We also discuss towards further generalization, in particular revisit Shah-Sterk compactification of moduli of polarized Enriques surfaces to show compatibility with log K-stability.

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arXiv2021

On log minimality of weak K-moduli compactifications of Calabi-Yau varieties

For moduli of polarized smooth K-trivial a.k.a., Calabi-Yau varieties in a general sense, we revisit a classical problem of constructing its "weak K-moduli" compactifications which parametrizes K-semistable (i.e., semi-log-canonical K-trivial) degenerations. Although weak K-moduli is not unique in general, they always contain a unique partial compactification (K-moduli). Our main theorem is the log minimality of their normalizations, under some conditions. Partially to confirm that known examples satisfy the conditions, we also include an appendix on the algebro-geometric reconstruction of Kulikov models via the MMP, which has been folklore at least but we somewhat strengthen.

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arXiv2021

Fano Shimura varieties with mostly branched cusps

We prove that the Satake-Baily-Borel compactification of certain Shimura varieties are Fano varieties, Calabi-Yau varieties or have ample canonical divisors with mild singularities. We also prove some variants statements, give applications and discuss various examples including new ones, for instance, the moduli spaces of unpolarized (log) Enriques surfaces.

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arXiv2020

Degenerated Calabi-Yau varieties with infinite components, Moduli compactifications, and limit toroidal structures

For any degenerating Calabi-Yau family, we introduce new limit space which we call galaxy, whose dense subspace is the disjoint union of countably infinite open Calabi-Yau varieties, parametrized by the rational points of the Kontsevich-Soibelman's essential skeleton, while dominated by the Huber adification over the Puiseux series field. Other topics include: projective limits of toroidal compactifications, locally modelled on what we call the limit toric varieties, the way to attach tropicalized family to given Calabi-Yau family, which are weakly related to each other.

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arXiv2020

PL density invariant for type II degenerating K3 surfaces, Moduli compactification and hyperKahler metrics

A protagonist here is a new-type invariant for type II degenerations of K3 surfaces, which is explicit PL (piecewise linear) convex function from the interval with at most 18 non-linear points. Forgetting its actual function behaviour, it also classifies the type II degenerations into several combinatorial types, depending on the type of root lattices as appeared in classical examples. From differential geometric viewpoint, the function is obtained as the density function of the limit measure on the collapsing hyperKahler metrics to conjectural segments, as in the work of Honda-Sun-Zhang. On the way, we also reconstruct a moduli compactification of elliptic K3 surfaces in the works of Brunyate, Ascher-Bejleri, Alexeev-Brunyate-Engel in a more elementary manner, analyze the cusps more explicitly. We also interpret the glued hyperKahler fibration of Hein-Sun-Viaclovsky-Zhang as a special case from our viewpoint, discuss other cases, and possible relations with Landau-Ginzburg models in the mirror symmetry context.

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arXiv2020

Polystable log Calabi-Yau varieties and Gravitational instantons

Open Calabi-Yau manifolds and log Calabi-Yau varieties have been broadly studied over decades. Regarding them as "semistable" objects, we propose to consider their good proper subclass, which we regard as certain poly-stable ones, morally corresponding to semistable with closed (minimal) orbits} as the classical analogue of GIT. We partially confirm that the new polystability seems equivalent to the existence of non-compact complete Ricci-flat Kahler metrics with small volume growths, notably many examples of gravitational instantons. Also, we prove some compactness or polystable reduction type results, partially motivated by bubbles of compact Ricci-flat metrics.

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arXiv2018

Collapsing K3 Surfaces and Moduli Compactification

This note is a summary of our work [OO] which provides an explicit and global moduli-theoretic framework for the collapsing of Ricci-flat Kahler metrics and we use it to study especially the K3 surfaces case. For instance, it allows us to discuss their Gromov-Hausdorff limits along any sequences, which are even not necessarily "maximally degenerating". Our results also give a proof of Kontsevich-Soibelman [KS04, Conjecture 1] (cf., [GW00, Conjecture 6.2]) in the case of K3 surfaces as a byproduct.

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arXiv2017

Tropical Geometric Compactification of Moduli, II - $A_g$ case and holomorphic limits -

We compactify the classical moduli variety $A_g$ of principally polarized abelian varieties of complex dimension $g$ by attaching the moduli of flat tori of real dimensions at most $g$ in an explicit manner. Equivalently, we explicitly determine the Gromov-Hausdorff limits of principally polarized abelian varieties. This work is analogous to the first of our series (available at arXiv:1406.7772v2), which compactified the moduli of curves by attaching the moduli of metrized graphs. Then, we also explicitly specify the Gromov-Hausdorff limits along holomorphic family of abelian varieties and show that they form special non-trivial subsets of the whole boundary. We also do it for algebraic curves case and observe a crucial difference with the case of abelian varieties.

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arXiv2016

On the K-stability of Fano varieties and anticanonical divisors

We apply a recent theorem of Li and the first author to give some criteria for the K-stability of Fano varieties in terms of anticanonical Q-divisors. First, we propose a condition in terms of certain anticanonical Q-divisors of given Fano variety, which we conjecture to be equivalent to the K-stability. We prove that it is at least sufficient condition and also relate to the Berman-Gibbs stability. We also give another algebraic proof of the K-stability of Fano varieties which satisfy Tian's alpha invariants condition.

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arXiv2015

Canonical Kahler metrics and Arithmetics -- Generalising Faltings heights

We extend the Faltings modular heights of abelian varieties to general arithmetic varieties and show direct relations with the Kahler-Einstein geometry, the Minimal Model Program, heights of Bost and Zhang, and give some applications. Along the way, we propose arithmetic Yau-Tian-Donaldson conjecture, an equivalence of a purely arithmetic property of variety and its metrical property, and partially confirm it.

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