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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Wall divisors and algebraically coisotropic subvarieties of irreducible holomorphic symplectic manifolds


Authors: Andreas Leopold Knutsen, Margherita Lelli-Chiesa and Giovanni Mongardi
Journal: Trans. Amer. Math. Soc. 371 (2019), 1403-1438
MSC (2010): Primary 14J40; Secondary 14C25, 14E25, 14M20
DOI: https://doi.org/10.1090/tran/7340
Published electronically: September 20, 2018
MathSciNet review: 3885184
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Abstract: Rational curves on Hilbert schemes of points on $ K3$ surfaces and generalised Kummer manifolds are constructed by using Brill-Noether theory on nodal curves on the underlying surface. It turns out that all wall divisors can be obtained, up to isometry, as dual divisors to such rational curves. The locus covered by the rational curves is then described, thus exhibiting algebraically coisotropic subvarieties. This provides strong evidence for a conjecture by Voisin concerning the Chow ring of irreducible holomorphic symplectic manifolds. Some general results concerning the birational geometry of irreducible holomorphic symplectic manifolds are also proved, such as a non-projective contractibility criterion for wall divisors.


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Additional Information

Andreas Leopold Knutsen
Affiliation: Department of Mathematics, University of Bergen, Postboks 7800, 5020 Bergen, Norway
Email: andreas.knutsen@math.uib.no

Margherita Lelli-Chiesa
Affiliation: Department of Mathematics, University of Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy
Email: margherita.lelli@unipi.it

Giovanni Mongardi
Affiliation: Department of Mathematics, University of Bologna, Piazza di Porta San Donato 5, 40126 Bologna, Italy
Email: giovanni.mongardi2@unibo.it

DOI: https://doi.org/10.1090/tran/7340
Received by editor(s): March 17, 2016
Received by editor(s) in revised form: May 16, 2017, and May 31, 2017
Published electronically: September 20, 2018
Article copyright: © Copyright 2018 American Mathematical Society