On the Existence of Good Minimal Models for K\"ahler Varieties with Projective Albanese Map

Yu-Ting Huang

arxiv: 2502.18800 · v2 · submitted 2025-02-26 · 🧮 math.AG

On the Existence of Good Minimal Models for K\"ahler Varieties with Projective Albanese Map

Yu-Ting Huang This is my paper

Pith reviewed 2026-05-23 02:53 UTC · model grok-4.3

classification 🧮 math.AG MSC 14E30

keywords Kähler varietiesminimal modelsAlbanese mapklt pairsbirational geometrygood minimal models

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The pith

Compact Kähler klt pairs admit good minimal models when the Albanese map is projective and the general fiber has one.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves existence of good minimal models for compact Kähler klt pairs under two conditions: the Albanese map must be a projective morphism, and the general fiber must already possess a good minimal model. This extends known results from the projective setting to Kähler varieties by using the Albanese map to reduce the problem on the total space to the fiber. A sympathetic reader cares because good minimal models organize the birational geometry of varieties, and confirming their existence in the Kähler case removes an obstruction to classification beyond the projective world. The argument proceeds by induction on the fiber after fixing the Albanese map.

Core claim

We establish the existence of a good minimal model for a compact Kähler klt pair (X, B) when the Albanese map of X is a projective morphism and the general fiber of (X, B) has a good minimal model.

What carries the argument

The projective Albanese map of X, which reduces the minimal model question for the total space (X, B) to the same question on its general fiber.

Load-bearing premise

The general fiber of the pair must already have a good minimal model.

What would settle it

A compact Kähler klt pair whose Albanese map is projective, whose general fiber has a good minimal model, yet the total space lacks a good minimal model.

read the original abstract

In this article, we establish the existence of a good minimal model for a compact K\"ahler klt pair $(X, B)$ when the Albanese map of $X$ is a projective morphism and the general fiber of $(X, B)$ has a good minimal model.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Huang gives a clean reduction showing that a compact Kähler klt pair has a good minimal model once the Albanese map is projective and the general fiber already does.

read the letter

The paper's main contribution is a reduction theorem: if (X, B) is a compact Kähler klt pair whose Albanese map is a projective morphism, and if the general fiber already has a good minimal model, then the total space does too. This moves some of the algebraic MMP machinery into the Kähler setting without claiming to remove the fiber hypothesis. The reduction itself looks standard, relying on properties of the Albanese map and klt singularities that carry over from the projective case. That part is useful because it organizes the problem and isolates where the remaining work must happen. The soft spot is exactly what the abstract flags: the result is conditional on the fiber. In practice this means the theorem applies only where the fiber case is already settled, which often reduces to known algebraic situations or low-dimensional cases. Without more on when the fiber hypothesis holds in genuinely non-algebraic Kähler examples, the advance stays limited to the reduction step. The citation pattern is light and focused on prior MMP work, which is appropriate. No obvious circularity appears in the stated logic. This is for readers already inside the Kähler MMP literature who need to track conditional results. A specialist working on extending minimal models beyond projective varieties would want to see the details. It deserves peer review because the claim is new, the reduction is explicit, and the area is active enough that referees can check the technical steps against existing Kähler techniques.

Referee Report

0 major / 1 minor

Summary. The manuscript establishes the existence of a good minimal model for a compact Kähler klt pair (X, B) when the Albanese map of X is a projective morphism and the general fiber of (X, B) has a good minimal model.

Significance. If the result holds, it provides a useful reduction in the minimal model program for Kähler varieties, allowing the problem for the total space to be reduced to the fiber case via the projective Albanese morphism. This could facilitate inductive arguments in non-projective settings, building on standard properties of klt pairs and Albanese maps.

minor comments (1)

The abstract states the main theorem clearly but the provided text contains no proof details, error analysis, or verification steps, making it impossible to assess the soundness of the reduction argument from the available material.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for reviewing the manuscript and for the accurate summary of the main result. The referee has not raised any specific major comments or concerns, and we have no revisions to propose at this stage.

Circularity Check

0 steps flagged

No circularity: result is explicitly conditional on fiber hypothesis

full rationale

The paper states a conditional existence result: a good minimal model exists for the total space (X,B) provided the Albanese map is projective and the general fiber already possesses a good minimal model. This is a standard reduction step relying on properties of the Albanese morphism and klt pairs; the fiber hypothesis is openly declared as an input rather than derived or fitted from the conclusion. No equations, self-citations, or ansatzes are shown to reduce the central claim to itself by construction. The derivation chain therefore remains self-contained against external benchmarks in the Kähler minimal model program.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on standard background in birational geometry and the minimal model program for Kähler manifolds; the fiber hypothesis is the main domain assumption.

axioms (2)

domain assumption Standard definition and properties of klt pairs and good minimal models in the Kähler category
Invoked implicitly as the setting for the theorem (abstract).
domain assumption Existence of good minimal models for the general fiber is given as hypothesis
Central reduction step stated in the abstract.

pith-pipeline@v0.9.0 · 5561 in / 1183 out tokens · 50352 ms · 2026-05-23T02:53:44.514238+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages · 1 internal anchor

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[1] [1]

Existence of minimal models for varieties of log general type

Caucher Birkar, Paolo Cascini, Christopher Hacon, and James McKernan. Existence of minimal models for varieties of log general type. Journal of the American Mathematical Society , 23(2):405--468, 2010

work page 2010

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Chen and Christopher D

Jungkai A. Chen and Christopher D. Hacon. On the irregularity of the image of the iitaka fibration. Communications in Algebra , 32(1):203--215, 2004. https://arxiv.org/abs/https://doi.org/10.1081/AGB-120027861 arXiv:https://doi.org/10.1081/AGB-120027861 , https://doi.org/10.1081/AGB-120027861 doi:10.1081/AGB-120027861

work page doi:10.1081/agb-120027861 2004

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o ring. Rational curves on compact k \

Junyan Cao and Andreas H \"o ring. Rational curves on compact k \"a hler manifolds. Journal of Differential Geometry , 114(1):1--39, 2020

work page 2020

[4] [4]

Collins and Valentino Tosatti

Tristan C. Collins and Valentino Tosatti. Kähler currents and null loci. Inventiones mathematicae , 202(3):1167–1198, March 2015. URL: http://dx.doi.org/10.1007/s00222-015-0585-9, https://doi.org/10.1007/s00222-015-0585-9 doi:10.1007/s00222-015-0585-9

work page doi:10.1007/s00222-015-0585-9 2015

[5] [5]

Existence of good minimal models for k \"a hler varieties of maximal albanese dimension

Omprokash Das and Christopher Hacon. Existence of good minimal models for k \"a hler varieties of maximal albanese dimension. Comptes Rendus. Math \'e matique , 362(S1):83--91, 2024

work page 2024

[6] [6]

Transcendental minimal model program for projective varieties, 2024

Omprokash Das and Christopher Hacon. Transcendental minimal model program for projective varieties, 2024. URL: https://arxiv.org/abs/2412.07650, https://arxiv.org/abs/2412.07650 arXiv:2412.07650

work page arXiv 2024

[7] [7]

On the 4-dimensional minimal model program for kähler varieties

Omprokash Das, Christopher Hacon, and Mihai Păun. On the 4-dimensional minimal model program for kähler varieties. Advances in Mathematics , 443:109615, 2024. URL: https://www.sciencedirect.com/science/article/pii/S0001870824001300, https://doi.org/10.1016/j.aim.2024.109615 doi:10.1016/j.aim.2024.109615

work page doi:10.1016/j.aim.2024.109615 2024

[8] [8]

Singularities of theta divisors and the birational geometry of irregular varieties

Lawrence Ein and Robert Lazarsfeld. Singularities of theta divisors and the birational geometry of irregular varieties. Journal of the American Mathematical Society , pages 243--258, 1997

work page 1997

[9] [9]

On maximal Albanese dimensional varieties

Osamu Fujino. On maximal albanese dimensional varieties. arXiv preprint arXiv:0911.2851 , 2009

work page internal anchor Pith review Pith/arXiv arXiv 2009

[10] [10]

Minimal model program for projective morphisms between complex analytic spaces

Osamu Fujino. Minimal model program for projective morphisms between complex analytic spaces. arXiv preprint arXiv:2201.11315 , 2022

work page arXiv 2022

[11] [11]

Higher obstructions to deforming cohomology groups of line bundles

Mark Green and Robert Lazarsfeld. Higher obstructions to deforming cohomology groups of line bundles. Journal of the American Mathematical Society , 4(1):87--103, 1991

work page 1991

[12] [12]

On the canonical bundle formula and adjunction for generalized kaehler pairs, 2024

Christopher Hacon and Mihai Paun. On the canonical bundle formula and adjunction for generalized kaehler pairs, 2024. URL: https://arxiv.org/abs/2404.12007, https://arxiv.org/abs/2404.12007 arXiv:2404.12007

work page arXiv 2024

[13] [13]

Existence of log canonical closures

Christopher D Hacon and Chenyang Xu. Existence of log canonical closures. Inventiones mathematicae , 192:161--195, 2013

work page 2013

[14] [14]

Minimal models and the kodaira dimension of algebraic fiber spaces

Yujiro Kawamata. Minimal models and the kodaira dimension of algebraic fiber spaces. Journal für die reine und angewandte Mathematik , 363:1--46, 1985. URL: http://eudml.org/doc/152778

work page 1985

[15] [15]

, TITLE =

Yujiro Kawamata. Flops connect minimal models. Publ. Res. Inst. Math. Sci , 44:419--423, 2008. https://doi.org/10.2977/PRIMS/1210167332 doi:10.2977/PRIMS/1210167332

work page doi:10.2977/prims/1210167332 2008

[16] [16]

Varieties fibered by good minimal models

Ching-Jui Lai. Varieties fibered by good minimal models. Mathematische Annalen , 350:533--547, 2011

work page 2011

[17] [17]

Torsion points on the cohomology jump loci of compact k \"a hler manifolds

Botong Wang. Torsion points on the cohomology jump loci of compact k \"a hler manifolds. Math. Res. Lett , 23(2):545--563, 2016

work page 2016

[18] [18]

On the Iitaka Conjecture c_ n,m for K\"ahler Fibre Spaces

Juanyong Wang. On the Iitaka Conjecture c_ n,m for K\"ahler Fibre Spaces . Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques , Ser. 6, 30(4):813--897, 2021. URL: http://www.numdam.org/articles/10.5802/afst.1690/, https://doi.org/10.5802/afst.1690 doi:10.5802/afst.1690

work page doi:10.5802/afst.1690/ 2021