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dc.contributor.author
Bryan, Jim
dc.contributor.author
Oberdieck, Georg
dc.contributor.author
Pandharipande, Rahul
dc.contributor.author
Yin, Qizheng
dc.date.accessioned
2018-10-04T15:07:13Z
dc.date.available
2018-08-11T06:43:09Z
dc.date.available
2018-08-13T11:28:18Z
dc.date.available
2018-10-04T15:07:13Z
dc.date.issued
2018-07
dc.identifier.issn
2313-1691
dc.identifier.issn
2214-2584
dc.identifier.other
10.14231/AG-2018-012
en_US
dc.identifier.uri
http://hdl.handle.net/20.500.11850/281979
dc.identifier.doi
10.3929/ethz-b-000281979
dc.description.abstract
We study the enumerative geometry of algebraic curves on abelian surfaces and three- folds. In the abelian surface case, the theory is parallel to the well-developed study of the reduced Gromov–Witten theory of K3 surfaces. We prove complete results in all genera for primitive classes. The generating series are quasi-modular forms of pure weight. Conjectures for imprimitive classes are presented. In genus 2, the counts in all classes are proven. Special counts match the Euler characteristic calculations of the moduli spaces of stable pairs on abelian surfaces by G ̈ottsche–Shende. A formula for hyperelliptic curve counting in terms of Jacobi forms is proven (modulo a transversality statement). For abelian threefolds, complete conjectures in terms of Jacobi forms for the gen- erating series of curve counts in primitive classes are presented. The base cases make connections to classical lattice counts of Debarre, G ̈ottsche, and Lange–Sernesi. Further evidence is provided by Donaldson–Thomas partition function computations for abelian threefolds. A multiple cover structure is presented. The abelian threefold conjectures open a new direction in the subject.
en_US
dc.format
application/pdf
en_US
dc.language.iso
en
en_US
dc.publisher
Foundation Compositio Mathematica
en_US
dc.rights.uri
http://creativecommons.org/licenses/by-nc/3.0/
dc.subject
Gromov–Witten theory
en_US
dc.subject
Abelian varieties
en_US
dc.subject
Donaldson–Thomas theory
en_US
dc.title
Curve counting on abelian surfaces and threefolds
en_US
dc.type
Journal Article
dc.rights.license
Creative Commons Attribution-NonCommercial 3.0 Unported
ethz.journal.title
Algebraic Geometry
ethz.journal.volume
5
en_US
ethz.journal.issue
4
en_US
ethz.pages.start
398
en_US
ethz.pages.end
463
en_US
ethz.version.deposit
publishedVersion
en_US
ethz.grant
Moduli spaces of curves, sheaves, and K3 surfaces
en_US
ethz.identifier.scopus
ethz.publication.place
Amsterdam
en_US
ethz.publication.status
published
en_US
ethz.grant.agreementno
143274
ethz.grant.fundername
SNF
ethz.grant.funderDoi
10.13039/501100001711
ethz.grant.program
Projektförderung in Mathematik, Natur- und Ingenieurwissenschaften (Abteilung II)
ethz.date.deposited
2018-08-11T06:43:09Z
ethz.source
SCOPUS
ethz.eth
yes
en_US
ethz.availability
Open access
en_US
ethz.rosetta.installDate
2018-08-13T11:28:21Z
ethz.rosetta.lastUpdated
2018-10-04T15:07:22Z
ethz.rosetta.exportRequired
false
ethz.rosetta.versionExported
true
ethz.COinS
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