---
_id: '14694'
abstract:
- lang: eng
text: We study the unique solution m of the Dyson equation \( -m(z)^{-1} = z\1 -
a + S[m(z)] \) on a von Neumann algebra A with the constraint Imm≥0. Here, z lies
in the complex upper half-plane, a is a self-adjoint element of A and S is a positivity-preserving
linear operator on A. We show that m is the Stieltjes transform of a compactly
supported A-valued measure on R. Under suitable assumptions, we establish that
this measure has a uniformly 1/3-Hölder continuous density with respect to the
Lebesgue measure, which is supported on finitely many intervals, called bands.
In fact, the density is analytic inside the bands with a square-root growth at
the edges and internal cubic root cusps whenever the gap between two bands vanishes.
The shape of these singularities is universal and no other singularity may occur.
We give a precise asymptotic description of m near the singular points. These
asymptotics generalize the analysis at the regular edges given in the companion
paper on the Tracy-Widom universality for the edge eigenvalue statistics for correlated
random matrices [the first author et al., Ann. Probab. 48, No. 2, 963--1001 (2020;
Zbl 1434.60017)] and they play a key role in the proof of the Pearcey universality
at the cusp for Wigner-type matrices [G. Cipolloni et al., Pure Appl. Anal. 1,
No. 4, 615--707 (2019; Zbl 07142203); the second author et al., Commun. Math.
Phys. 378, No. 2, 1203--1278 (2020; Zbl 07236118)]. We also extend the finite
dimensional band mass formula from [the first author et al., loc. cit.] to the
von Neumann algebra setting by showing that the spectral mass of the bands is
topologically rigid under deformations and we conclude that these masses are quantized
in some important cases.
article_processing_charge: Yes
article_type: original
arxiv: 1
author:
- first_name: Johannes
full_name: Alt, Johannes
id: 36D3D8B6-F248-11E8-B48F-1D18A9856A87
last_name: Alt
- first_name: László
full_name: Erdös, László
id: 4DBD5372-F248-11E8-B48F-1D18A9856A87
last_name: Erdös
orcid: 0000-0001-5366-9603
- first_name: Torben H
full_name: Krüger, Torben H
id: 3020C786-F248-11E8-B48F-1D18A9856A87
last_name: Krüger
orcid: 0000-0002-4821-3297
citation:
ama: 'Alt J, Erdös L, Krüger TH. The Dyson equation with linear self-energy: Spectral
bands, edges and cusps. Documenta Mathematica. 2020;25:1421-1539. doi:10.4171/dm/780'
apa: 'Alt, J., Erdös, L., & Krüger, T. H. (2020). The Dyson equation with linear
self-energy: Spectral bands, edges and cusps. Documenta Mathematica. EMS
Press. https://doi.org/10.4171/dm/780'
chicago: 'Alt, Johannes, László Erdös, and Torben H Krüger. “The Dyson Equation
with Linear Self-Energy: Spectral Bands, Edges and Cusps.” Documenta Mathematica.
EMS Press, 2020. https://doi.org/10.4171/dm/780.'
ieee: 'J. Alt, L. Erdös, and T. H. Krüger, “The Dyson equation with linear self-energy:
Spectral bands, edges and cusps,” Documenta Mathematica, vol. 25. EMS Press,
pp. 1421–1539, 2020.'
ista: 'Alt J, Erdös L, Krüger TH. 2020. The Dyson equation with linear self-energy:
Spectral bands, edges and cusps. Documenta Mathematica. 25, 1421–1539.'
mla: 'Alt, Johannes, et al. “The Dyson Equation with Linear Self-Energy: Spectral
Bands, Edges and Cusps.” Documenta Mathematica, vol. 25, EMS Press, 2020,
pp. 1421–539, doi:10.4171/dm/780.'
short: J. Alt, L. Erdös, T.H. Krüger, Documenta Mathematica 25 (2020) 1421–1539.
date_created: 2023-12-18T10:37:43Z
date_published: 2020-09-01T00:00:00Z
date_updated: 2023-12-18T10:46:09Z
day: '01'
ddc:
- '510'
department:
- _id: LaEr
doi: 10.4171/dm/780
external_id:
arxiv:
- '1804.07752'
file:
- access_level: open_access
checksum: 12aacc1d63b852ff9a51c1f6b218d4a6
content_type: application/pdf
creator: dernst
date_created: 2023-12-18T10:42:32Z
date_updated: 2023-12-18T10:42:32Z
file_id: '14695'
file_name: 2020_DocumentaMathematica_Alt.pdf
file_size: 1374708
relation: main_file
success: 1
file_date_updated: 2023-12-18T10:42:32Z
has_accepted_license: '1'
intvolume: ' 25'
keyword:
- General Mathematics
language:
- iso: eng
license: https://creativecommons.org/licenses/by/4.0/
month: '09'
oa: 1
oa_version: Published Version
page: 1421-1539
publication: Documenta Mathematica
publication_identifier:
eissn:
- 1431-0643
issn:
- 1431-0635
publication_status: published
publisher: EMS Press
quality_controlled: '1'
related_material:
record:
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relation: earlier_version
status: public
status: public
title: 'The Dyson equation with linear self-energy: Spectral bands, edges and cusps'
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)
short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 25
year: '2020'
...
---
_id: '7436'
abstract:
- lang: eng
text: 'For an ordinary K3 surface over an algebraically closed field of positive
characteristic we show that every automorphism lifts to characteristic zero. Moreover,
we show that the Fourier-Mukai partners of an ordinary K3 surface are in one-to-one
correspondence with the Fourier-Mukai partners of the geometric generic fiber
of its canonical lift. We also prove that the explicit counting formula for Fourier-Mukai
partners of the K3 surfaces with Picard rank two and with discriminant equal to
minus of a prime number, in terms of the class number of the prime, holds over
a field of positive characteristic as well. We show that the image of the derived
autoequivalence group of a K3 surface of finite height in the group of isometries
of its crystalline cohomology has index at least two. Moreover, we provide a conditional
upper bound on the kernel of this natural cohomological descent map. Further,
we give an extended remark in the appendix on the possibility of an F-crystal
structure on the crystalline cohomology of a K3 surface over an algebraically
closed field of positive characteristic and show that the naive F-crystal structure
fails in being compatible with inner product. '
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Tanya K
full_name: Srivastava, Tanya K
id: 4D046628-F248-11E8-B48F-1D18A9856A87
last_name: Srivastava
citation:
ama: Srivastava TK. On derived equivalences of k3 surfaces in positive characteristic.
Documenta Mathematica. 2019;24:1135-1177. doi:10.25537/dm.2019v24.1135-1177
apa: Srivastava, T. K. (2019). On derived equivalences of k3 surfaces in positive
characteristic. Documenta Mathematica. EMS Press. https://doi.org/10.25537/dm.2019v24.1135-1177
chicago: Srivastava, Tanya K. “On Derived Equivalences of K3 Surfaces in Positive
Characteristic.” Documenta Mathematica. EMS Press, 2019. https://doi.org/10.25537/dm.2019v24.1135-1177.
ieee: T. K. Srivastava, “On derived equivalences of k3 surfaces in positive characteristic,”
Documenta Mathematica, vol. 24. EMS Press, pp. 1135–1177, 2019.
ista: Srivastava TK. 2019. On derived equivalences of k3 surfaces in positive characteristic.
Documenta Mathematica. 24, 1135–1177.
mla: Srivastava, Tanya K. “On Derived Equivalences of K3 Surfaces in Positive Characteristic.”
Documenta Mathematica, vol. 24, EMS Press, 2019, pp. 1135–77, doi:10.25537/dm.2019v24.1135-1177.
short: T.K. Srivastava, Documenta Mathematica 24 (2019) 1135–1177.
date_created: 2020-02-02T23:01:06Z
date_published: 2019-05-20T00:00:00Z
date_updated: 2023-10-17T07:42:21Z
day: '20'
ddc:
- '510'
department:
- _id: TaHa
doi: 10.25537/dm.2019v24.1135-1177
external_id:
arxiv:
- '1809.08970'
isi:
- '000517806400019'
file:
- access_level: open_access
checksum: 9a1a64bd49ab03fa4f738fb250fc4f90
content_type: application/pdf
creator: dernst
date_created: 2020-02-03T06:26:12Z
date_updated: 2020-07-14T12:47:58Z
file_id: '7438'
file_name: 2019_DocumMath_Srivastava.pdf
file_size: 469730
relation: main_file
file_date_updated: 2020-07-14T12:47:58Z
has_accepted_license: '1'
intvolume: ' 24'
isi: 1
language:
- iso: eng
month: '05'
oa: 1
oa_version: Published Version
page: 1135-1177
publication: Documenta Mathematica
publication_identifier:
eissn:
- 1431-0643
issn:
- 1431-0635
publication_status: published
publisher: EMS Press
quality_controlled: '1'
scopus_import: '1'
status: public
title: On derived equivalences of k3 surfaces in positive characteristic
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)
short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 24
year: '2019'
...
---
_id: '1451'
abstract:
- lang: eng
text: Extending work of Bielawski-Dancer 3 and Konno 14, we develop a theory of
toric hyperkähler varieties, which involves toric geometry, matroid theory and
convex polyhedra. The framework is a detailed study of semi-projective toric varieties,
meaning GIT quotients of affine spaces by torus actions, and specifically, of
Lawrence toric varieties, meaning GIT quotients of even-dimensional affine spaces
by symplectic torus actions. A toric hyperkähler variety is a complete intersection
in a Lawrence toric variety. Both varieties are non-compact, and they share the
same cohomology ring, namely, the Stanley-Reisner ring of a matroid modulo a linear
system of parameters. Familiar applications of toric geometry to combinatorics,
including the Hard Lefschetz Theorem and the volume polynomials of Khovanskii-Pukhlikov
11, are extended to the hyperkähler setting. When the matroid is graphic, our
construction gives the toric quiver varieties, in the sense of Nakajima 17.
acknowledgement: "Both authors were supported by the Miller Institute for Basic Research
in Science, in the form of a Miller Research Fellowship (1999-2002) for the first
author and a Miller Professorship (2000-2001) for the second author. The second
author was also supported by the National Science\r\nFoundation (DMS-9970254)."
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Tamas
full_name: Hausel, Tamas
id: 4A0666D8-F248-11E8-B48F-1D18A9856A87
last_name: Hausel
- first_name: Bernd
full_name: Sturmfels, Bernd
last_name: Sturmfels
citation:
ama: Hausel T, Sturmfels B. Toric hyperkähler varieties. Documenta Mathematica.
2002;7(1):495-534. doi:10.4171/DM/130
apa: Hausel, T., & Sturmfels, B. (2002). Toric hyperkähler varieties. Documenta
Mathematica. Deutsche Mathematiker Vereinigung. https://doi.org/10.4171/DM/130
chicago: Hausel, Tamás, and Bernd Sturmfels. “Toric Hyperkähler Varieties.” Documenta
Mathematica. Deutsche Mathematiker Vereinigung, 2002. https://doi.org/10.4171/DM/130.
ieee: T. Hausel and B. Sturmfels, “Toric hyperkähler varieties,” Documenta Mathematica,
vol. 7, no. 1. Deutsche Mathematiker Vereinigung, pp. 495–534, 2002.
ista: Hausel T, Sturmfels B. 2002. Toric hyperkähler varieties. Documenta Mathematica.
7(1), 495–534.
mla: Hausel, Tamás, and Bernd Sturmfels. “Toric Hyperkähler Varieties.” Documenta
Mathematica, vol. 7, no. 1, Deutsche Mathematiker Vereinigung, 2002, pp. 495–534,
doi:10.4171/DM/130.
short: T. Hausel, B. Sturmfels, Documenta Mathematica 7 (2002) 495–534.
date_created: 2018-12-11T11:52:06Z
date_published: 2002-01-01T00:00:00Z
date_updated: 2023-07-26T09:16:33Z
day: '01'
doi: 10.4171/DM/130
extern: '1'
external_id:
arxiv:
- math/0203096
intvolume: ' 7'
issue: '1'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://ems.press/journals/dm/articles/8965058
month: '01'
oa: 1
oa_version: Published Version
page: 495 - 534
publication: Documenta Mathematica
publication_identifier:
issn:
- 1431-0635
publication_status: published
publisher: Deutsche Mathematiker Vereinigung
publist_id: '5741'
quality_controlled: '1'
scopus_import: '1'
status: public
title: Toric hyperkähler varieties
type: journal_article
user_id: ea97e931-d5af-11eb-85d4-e6957dddbf17
volume: 7
year: '2002'
...