Geometry & Topology Volume 29, issue 3 (2025)

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

Submission form

Editorial board

Subscriptions

ISSN (electronic): 1364-0380

ISSN (print): 1465-3060

Author index

To appear

Other MSP journals

On boundedness and moduli spaces of K-stable Calabi–Yau fibrations over curves

Kenta Hashizume and Masafumi Hattori

Geometry & Topology 29 (2025) 1619–1691

DOI: 10.2140/gt.2025.29.1619

Bibliography

1	V Alexeev, Log canonical singularities and complete moduli of stable pairs, preprint (1996) arXiv:alg-geom/9608013
2	V Alexeev, Moduli spaces M_g,n(W) for surfaces, from: "Higher-dimensional complex varieties", de Gruyter (1996) 1 MR1463171
3	J Alper, H Blum, D Halpern-Leistner, C Xu, Reductivity of the automorphism group of K-polystable Fano varieties, Invent. Math. 222 (2020) 995 MR4169054
4	F Ambro, Shokurov’s boundary property, J. Differential Geom. 67 (2004) 229 MR2153078
5	C Birkar, Anti-pluricanonical systems on Fano varieties, Ann. of Math. 190 (2019) 345 MR3997127
6	C Birkar, Boundedness and volume of generalised pairs, preprint (2021) arXiv:2103.14935
7	C Birkar, Singularities of linear systems and boundedness of Fano varieties, Ann. of Math. 193 (2021) 347 MR4224714
8	C Birkar, Moduli of algebraic varieties, preprint (2022) arXiv:2211.11237
9	C Birkar, Geometry of polarised varieties, Publ. Math. Inst. Hautes Études Sci. 137 (2023) 47 MR4588595
10	C Birkar, P Cascini, C D Hacon, J McKernan, Existence of minimal models for varieties of log general type, J. Amer. Math. Soc. 23 (2010) 405 MR2601039
11	H Blum, M Jonsson, Thresholds, valuations, and K-stability, Adv. Math. 365 (2020) 107062 MR4067358
12	H Blum, Y Liu, Openness of uniform K-stability in families of ℚ-Fano varieties, Ann. Sci. École Norm. Sup. 55 (2022) 1 MR4411858
13	H Blum, C Xu, Uniqueness of K-polystable degenerations of Fano varieties, Ann. of Math. 190 (2019) 609 MR3997130
14	S Boucksom, Remarks on minimal models of degenerations, unpublished note (2014)
15	S Boucksom, T Hisamoto, M Jonsson, Uniform K-stability, Duistermaat–Heckman measures and singularities of pairs, Ann. Inst. Fourier (Grenoble) 67 (2017) 743 MR3669511
16	G Chen, The J-equation and the supercritical deformed Hermitian–Yang–Mills equation, Invent. Math. 225 (2021) 529 MR4285141
17	X Chen, J Cheng, On the constant scalar curvature Kähler metrics, II : Existence results, J. Amer. Math. Soc. 34 (2021) 937 MR4301558
18	G Codogni, Z Patakfalvi, Positivity of the CM line bundle for families of K-stable klt Fano varieties, Invent. Math. 223 (2021) 811 MR4213768
19	B Conrad, Grothendieck duality and base change, 1750, Springer (2000) MR1804902
20	V V Datar, V P Pingali, A numerical criterion for generalised Monge–Ampère equations on projective manifolds, Geom. Funct. Anal. 31 (2021) 767 MR4317503
21	P Deligne, D Mumford, The irreducibility of the space of curves of given genus, Inst. Hautes Études Sci. Publ. Math. 36 (1969) 75 MR262240
22	R Dervan, Uniform stability of twisted constant scalar curvature Kähler metrics, Int. Math. Res. Not. 2016 (2016) 4728 MR3564626
23	R Dervan, P Naumann, Moduli of polarised manifolds via canonical Kähler metrics, preprint (2018) arXiv:1810.02576
24	R Dervan, J Ross, Stable maps in higher dimensions, Math. Ann. 374 (2019) 1033 MR3985107
25	R Dervan, L M Sektnan, Moduli theory, stability of fibrations and optimal symplectic connections, Geom. Topol. 25 (2021) 2643 MR4310897
26	R Dervan, L M Sektnan, Optimal symplectic connections on holomorphic submersions, Comm. Pure Appl. Math. 74 (2021) 2132 MR4303016
27	S K Donaldson, Scalar curvature and projective embeddings, I, J. Differential Geom. 59 (2001) 479 MR1916953
28	S K Donaldson, Scalar curvature and stability of toric varieties, J. Differential Geom. 62 (2002) 289 MR1988506
29	B Fantechi, L Göttsche, L Illusie, S L Kleiman, N Nitsure, A Vistoli, Fundamental algebraic geometry : Grothendieck’s FGA explained, 123, Amer. Math. Soc. (2005) MR2222646
30	J Fine, Constant scalar curvature Kähler metrics on fibred complex surfaces, J. Differential Geom. 68 (2004) 397 MR2144537
31	J Fine, Fibrations with constant scalar curvature Kähler metrics and the CM-line bundle, Math. Res. Lett. 14 (2007) 239 MR2318622
32	A Fujiki, G Schumacher, The moduli space of extremal compact Kähler manifolds and generalized Weil–Petersson metrics, Publ. Res. Inst. Math. Sci. 26 (1990) 101 MR1053910
33	O Fujino, Fundamental theorems for the log minimal model program, Publ. Res. Inst. Math. Sci. 47 (2011) 727 MR2832805
34	O Fujino, Basepoint-free theorems : saturation, b-divisors, and canonical bundle formula, Algebra Number Theory 6 (2012) 797 MR2966720
35	O Fujino, Effective basepoint-free theorem for semi-log canonical surfaces, Publ. Res. Inst. Math. Sci. 53 (2017) 349 MR3669739
36	O Fujino, Semipositivity theorems for moduli problems, Ann. of Math. 187 (2018) 639 MR3779955
37	K Fujita, Uniform K-stability and plt blowups of log Fano pairs, Kyoto J. Math. 59 (2019) 399 MR3960299
38	K Fujita, Y Odaka, On the K-stability of Fano varieties and anticanonical divisors, Tohoku Math. J. 70 (2018) 511 MR3896135
39	D Gieseker, Global moduli for surfaces of general type, Invent. Math. 43 (1977) 233 MR498596
40	U Görtz, T Wedhorn, Algebraic geometry, I : Schemes with examples and exercises, Vieweg + Teubner (2010) MR2675155
41	A Grothendieck, Éléments de géométrie algébrique, IV : Étude locale des schémas et des morphismes de schémas, III, Inst. Hautes Études Sci. Publ. Math. 28 (1966) 5 MR0217086
42	C D Hacon, C Xu, Existence of log canonical closures, Invent. Math. 192 (2013) 161 MR3032329
43	C D Hacon, J McKernan, C Xu, ACC for log canonical thresholds, Ann. of Math. 180 (2014) 523 MR3224718
44	C D Hacon, J McKernan, C Xu, Boundedness of moduli of varieties of general type, J. Eur. Math. Soc. 20 (2018) 865 MR3779687
45	R Hartshorne, Algebraic geometry, 52, Springer (1977) MR463157
46	B Hassett, S J Kovács, Reflexive pull-backs and base extension, J. Algebraic Geom. 13 (2004) 233 MR2047697
47	M Hattori, A decomposition formula for J-stability and its applications, (2021) arXiv:2103.04603
48	M Hattori, On K-stability of Calabi–Yau fibrations, preprint (2022) arXiv:2203.11460
49	M Hattori, Minimizing CM degree and specially K-stable varieties, Int. Math. Res. Not. 2024 (2024) 5728 MR4728720
50	M Hattori, On fibration stability after Dervan–Sektnan and singularities, Osaka J. Math. 61 (2024) 195 MR4729041
51	M Hattori, A Zanardini, On the GIT stability of linear systems of hypersurfaces in projective space, preprint (2022) arXiv:2212.09364
52	C Jiang, On birational boundedness of Fano fibrations, Amer. J. Math. 140 (2018) 1253 MR3862064
53	C Jiang, Boundedness of ℚ-Fano varieties with degrees and alpha-invariants bounded from below, Ann. Sci. École Norm. Sup. 53 (2020) 1235 MR4174851
54	M Kawakita, Inversion of adjunction on log canonicity, Invent. Math. 167 (2007) 129 MR2264806
55	S Keel, S Mori, Quotients by groupoids, Ann. of Math. 145 (1997) 193 MR1432041
56	J Kollár, Projectivity of complete moduli, J. Differential Geom. 32 (1990) 235 MR1064874
57	J Kollár, Effective base point freeness, Math. Ann. 296 (1993) 595 MR1233485
58	J Kollár, Hulls and husks, preprint (2008) arXiv:0805.0576
59	J Kollár, Singularities of the minimal model program, 200, Cambridge Univ. Press (2013) MR3057950
60	J Kollár, Families of varieties of general type, 231, Cambridge Univ. Press (2023) MR4566297
61	J Kollár, S Mori, Birational geometry of algebraic varieties, 134, Cambridge Univ. Press (1998) MR1658959
62	J Kollár, N I Shepherd-Barron, Threefolds and deformations of surface singularities, Invent. Math. 91 (1988) 299 MR922803
63	R Lazarsfeld, Positivity in algebraic geometry, I : Classical setting : line bundles and linear series, 48, Springer (2004) MR2095471
64	C Li, C Xu, Special test configuration and K-stability of Fano varieties, Ann. of Math. 180 (2014) 197 MR3194814
65	C Li, X Wang, C Xu, On the proper moduli spaces of smoothable Kähler–Einstein Fano varieties, Duke Math. J. 168 (2019) 1387 MR3959862
66	Y Liu, C Xu, Z Zhuang, Finite generation for valuations computing stability thresholds and applications to K-stability, Ann. of Math. 196 (2022) 507 MR4445441
67	H Matsumura, Commutative algebra, 56, Benjamin Cummings (1980) MR575344
68	R Miranda, On the stability of pencils of cubic curves, Amer. J. Math. 102 (1980) 1177 MR595010
69	R Miranda, The moduli of Weierstrass fibrations over ℙ¹, Math. Ann. 255 (1981) 379 MR615858
70	D Mumford, Lectures on curves on an algebraic surface, 59, Princeton Univ. Press (1966) MR209285
71	D Mumford, Stability of projective varieties, Enseign. Math. 23 (1977) 39 MR450272
72	D Mumford, Abelian varieties, 5, Tata Inst. Fund. Res. (2008) MR2514037
73	D Mumford, J Fogarty, F Kirwan, Geometric invariant theory, 34, Springer (1994) MR1304906
74	N Nakayama, Invariance of the plurigenera of algebraic varieties under minimal model conjectures, Topology 25 (1986) 237 MR837624
75	Y Odaka, On the GIT stability of polarized varieties: a survey, lecture notes (2010)
76	Y Odaka, The Calabi conjecture and K-stability, Int. Math. Res. Not. 2012 (2012) 2272 MR2923166
77	Y Odaka, On log minimality of weak K-moduli compactifications of Calabi–Yau varieties, preprint (2021) arXiv:2108.03832
78	M Olsson, Algebraic spaces and stacks, 62, Amer. Math. Soc. (2016) MR3495343
79	Z Patakfalvi, K Schwede, W Zhang, F-singularities in families, Algebr. Geom. 5 (2018) 264 MR3800355
80	W K Seiler, Global moduli for polarized elliptic surfaces, Compos. Math. 62 (1987) 187 MR898733
81	J Song, Nakai–Moishezon criterions for complex Hessian equations, preprint (2020) arXiv:2012.07956
82	P Belmans, A J de Jong, others, The Stacks project, electronic reference (2005–)
83	G Tian, On Kähler–Einstein metrics on certain Kähler manifolds with C₁(M) > 0, Invent. Math. 89 (1987) 225 MR894378
84	G Tian, On Calabi’s conjecture for complex surfaces with positive first Chern class, Invent. Math. 101 (1990) 101 MR1055713
85	G Tian, Kähler–Einstein metrics with positive scalar curvature, Invent. Math. 130 (1997) 1 MR1471884
86	R Vakil, The rising sea: foundations of algebraic geometry, (2010)
87	E Viehweg, Quasi-projective moduli for polarized manifolds, 30, Springer (1995) MR1368632
88	C Xu, Z Zhuang, On positivity of the CM line bundle on K-moduli spaces, Ann. of Math. 192 (2020) 1005 MR4172625
89	C Xu, Z Zhuang, Uniqueness of the minimizer of the normalized volume function, Camb. J. Math. 9 (2021) 149 MR4325260
90	A Zanardini, Stability of pencils of plane sextics and Halphen pencils of index two, Manuscripta Math. 172 (2023) 353 MR4632931
91	K Zhang, A quantization proof of the uniform Yau–Tian–Donaldson conjecture, J. Eur. Math. Soc. 26 (2024) 4763 MR4780493