---
_id: '13268'
abstract:
- lang: eng
text: We give a simple argument to prove Nagai’s conjecture for type II degenerations
of compact hyperkähler manifolds and cohomology classes of middle degree. Under
an additional assumption, the techniques yield the conjecture in arbitrary degree.
This would complete the proof of Nagai’s conjecture in general, as it was proved
already for type I degenerations by Kollár, Laza, Saccà, and Voisin [10] and independently
by Soldatenkov [18], while it is immediate for type III degenerations. Our arguments
are close in spirit to a recent paper by Harder [8] proving similar results for
the restrictive class of good degenerations.
acknowledgement: The first author is supported by the ERC Synergy Grant HyperK. The
second author is supported by the Max Planck Institute for Mathematics and the Institute
of Science and Technology Austria. This project has received funding from the European
Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie
grant agreement No 101034413.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: D.
full_name: Huybrechts, D.
last_name: Huybrechts
- first_name: Mirko
full_name: Mauri, Mirko
id: 2cf70c34-09c1-11ed-bd8d-c34fac206130
last_name: Mauri
citation:
ama: Huybrechts D, Mauri M. On type II degenerations of hyperkähler manifolds. Mathematical
Research Letters. 2023;30(1):125-141. doi:10.4310/mrl.2023.v30.n1.a6
apa: Huybrechts, D., & Mauri, M. (2023). On type II degenerations of hyperkähler
manifolds. Mathematical Research Letters. International Press. https://doi.org/10.4310/mrl.2023.v30.n1.a6
chicago: Huybrechts, D., and Mirko Mauri. “On Type II Degenerations of Hyperkähler
Manifolds.” Mathematical Research Letters. International Press, 2023. https://doi.org/10.4310/mrl.2023.v30.n1.a6.
ieee: D. Huybrechts and M. Mauri, “On type II degenerations of hyperkähler manifolds,”
Mathematical Research Letters, vol. 30, no. 1. International Press, pp.
125–141, 2023.
ista: Huybrechts D, Mauri M. 2023. On type II degenerations of hyperkähler manifolds.
Mathematical Research Letters. 30(1), 125–141.
mla: Huybrechts, D., and Mirko Mauri. “On Type II Degenerations of Hyperkähler Manifolds.”
Mathematical Research Letters, vol. 30, no. 1, International Press, 2023,
pp. 125–41, doi:10.4310/mrl.2023.v30.n1.a6.
short: D. Huybrechts, M. Mauri, Mathematical Research Letters 30 (2023) 125–141.
date_created: 2023-07-23T22:01:14Z
date_published: 2023-06-21T00:00:00Z
date_updated: 2024-01-16T12:00:47Z
day: '21'
department:
- _id: TaHa
doi: 10.4310/mrl.2023.v30.n1.a6
ec_funded: 1
external_id:
arxiv:
- '2108.01587'
isi:
- '001027656000006'
intvolume: ' 30'
isi: 1
issue: '1'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.48550/arXiv.2108.01587
month: '06'
oa: 1
oa_version: Preprint
page: 125-141
project:
- _id: fc2ed2f7-9c52-11eb-aca3-c01059dda49c
call_identifier: H2020
grant_number: '101034413'
name: 'IST-BRIDGE: International postdoctoral program'
publication: Mathematical Research Letters
publication_identifier:
eissn:
- 1945-001X
issn:
- 1073-2780
publication_status: published
publisher: International Press
quality_controlled: '1'
scopus_import: '1'
status: public
title: On type II degenerations of hyperkähler manifolds
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 30
year: '2023'
...
---
_id: '14239'
abstract:
- lang: eng
text: "Given a resolution of rational singularities π:X~→X over a field of characteristic
zero, we use a Hodge-theoretic argument to prove that the image of the functor
\ Rπ∗:Db(X~)→Db(X)\r\n between bounded derived categories of coherent sheaves
generates Db(X)\r\n as a triangulated category. This gives a weak version of
the Bondal–Orlov localization conjecture [BO02], answering a question from [PS21].
The same result is established more generally for proper (not necessarily birational)
morphisms π:X~→X , with X~\r\n smooth, satisfying Rπ∗(OX~)=OX ."
acknowledgement: "We thank Agnieszka Bodzenta-Skibińska, Paolo Cascini, Wahei Hara,
Sándor Kovács, Alexander Kuznetsov, Mircea Musta ă, Nebojsa Pavic, Pavel Sechin,
and Michael Wemyss for discussions and e-mail correspondence. We also thank the
anonymous referee for the helpful comments. M.M. was supported by the Institute
of Science and Technology Austria. This project has received funding from the European
Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie
grant agreement no. 101034413. E.S. was partially supported by the EPSRC grant EP/T019379/1
“Derived categories and algebraic K-theory of singularities”, and by the ERC Synergy
grant “Modern Aspects of Geometry: Categories, Cycles and Cohomology of Hyperkähler
Varieties.”\r\n\r\n"
article_number: e66
article_processing_charge: Yes
article_type: original
arxiv: 1
author:
- first_name: Mirko
full_name: Mauri, Mirko
id: 2cf70c34-09c1-11ed-bd8d-c34fac206130
last_name: Mauri
- first_name: Evgeny
full_name: Shinder, Evgeny
last_name: Shinder
citation:
ama: Mauri M, Shinder E. Homological Bondal-Orlov localization conjecture for rational
singularities. Forum of Mathematics, Sigma. 2023;11. doi:10.1017/fms.2023.65
apa: Mauri, M., & Shinder, E. (2023). Homological Bondal-Orlov localization
conjecture for rational singularities. Forum of Mathematics, Sigma. Cambridge
University Press. https://doi.org/10.1017/fms.2023.65
chicago: Mauri, Mirko, and Evgeny Shinder. “Homological Bondal-Orlov Localization
Conjecture for Rational Singularities.” Forum of Mathematics, Sigma. Cambridge
University Press, 2023. https://doi.org/10.1017/fms.2023.65.
ieee: M. Mauri and E. Shinder, “Homological Bondal-Orlov localization conjecture
for rational singularities,” Forum of Mathematics, Sigma, vol. 11. Cambridge
University Press, 2023.
ista: Mauri M, Shinder E. 2023. Homological Bondal-Orlov localization conjecture
for rational singularities. Forum of Mathematics, Sigma. 11, e66.
mla: Mauri, Mirko, and Evgeny Shinder. “Homological Bondal-Orlov Localization Conjecture
for Rational Singularities.” Forum of Mathematics, Sigma, vol. 11, e66,
Cambridge University Press, 2023, doi:10.1017/fms.2023.65.
short: M. Mauri, E. Shinder, Forum of Mathematics, Sigma 11 (2023).
date_created: 2023-08-27T22:01:16Z
date_published: 2023-08-03T00:00:00Z
date_updated: 2023-12-13T12:18:18Z
day: '03'
ddc:
- '510'
department:
- _id: TaHa
doi: 10.1017/fms.2023.65
ec_funded: 1
external_id:
arxiv:
- '2212.06786'
isi:
- '001041926700001'
file:
- access_level: open_access
checksum: c36241750cc5cb06890aec0ecdfee626
content_type: application/pdf
creator: dernst
date_created: 2023-09-05T06:43:11Z
date_updated: 2023-09-05T06:43:11Z
file_id: '14266'
file_name: 2023_ForumMathematics_Mauri.pdf
file_size: 280865
relation: main_file
success: 1
file_date_updated: 2023-09-05T06:43:11Z
has_accepted_license: '1'
intvolume: ' 11'
isi: 1
language:
- iso: eng
license: https://creativecommons.org/licenses/by/4.0/
month: '08'
oa: 1
oa_version: Published Version
project:
- _id: fc2ed2f7-9c52-11eb-aca3-c01059dda49c
call_identifier: H2020
grant_number: '101034413'
name: 'IST-BRIDGE: International postdoctoral program'
publication: Forum of Mathematics, Sigma
publication_identifier:
eissn:
- 2050-5094
publication_status: published
publisher: Cambridge University Press
quality_controlled: '1'
scopus_import: '1'
status: public
title: Homological Bondal-Orlov localization conjecture for rational singularities
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
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type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
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...