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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Cohomology algebra of plane curves, weak combinatorial type, and formality


Authors: J. I. Cogolludo Agustín and D. Matei
Journal: Trans. Amer. Math. Soc. 364 (2012), 5765-5790
MSC (2010): Primary 14F25, 14F40, 14H50, 58A10, 58A12, 58A14, 14B05, 14E15, 32A27, 32S22, 55P62
Published electronically: June 22, 2012
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Abstract: We determine an explicit presentation by generators and relations of the cohomology algebra $ H^*(\mathbb{P}^2\setminus \mathcal {C},\mathbb{C})$ of the complement to an algebraic curve $ \mathcal {C}$ in the complex projective plane $ \mathbb{P}^2$ via the study of log-resolution logarithmic forms on $ \mathbb{P}^2$. As a first consequence, we derive that $ H^*(\mathbb{P}^2\setminus \mathcal {C},\mathbb{C})$ depends only on the following finite pieces of data: the number of irreducible components of $ \mathcal {C}$ together with their degrees and genera, the number of local branches of each component at each singular point, and the intersection numbers of every two distinct local branches at each singular point of $ \mathcal {C}$. This finite set of data is referred to as the weak combinatorial type of $ \mathcal {C}$. A further corollary is that the twisted cohomology jumping loci of $ H^*(\mathbb{P}^2\setminus \mathcal {C},\mathbb{C})$ containing the trivial character also depend on the weak combinatorial type of $ \mathcal {C}$. Finally, the explicit construction of the generators and relations allows us to prove that complements of plane projective curves are formal spaces in the sense of Sullivan.


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Additional Information

J. I. Cogolludo Agustín
Affiliation: Departamento de Matemáticas, IUMA, Universidad de Zaragoza, C/Pedro Cerbuna 12, CP50009 Zaragoza, Spain
Email: jicogo@unizar.es

D. Matei
Affiliation: Departamento de Matemáticas, Universidad de Zaragoza, C/Pedro Cerbuna 12, CP50009 Zaragoza, Spain – and – Institute of Mathematics “Simion Stoilow” of the Romanian Academy, P.O. Box 1-764, RO-014700 Bucharest, Romania
Email: daniel.matei@imar.ro

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05489-4
PII: S 0002-9947(2012)05489-4
Keywords: Plane algebraic curves, singularities
Received by editor(s): July 10, 2009
Received by editor(s) in revised form: October 14, 2010
Published electronically: June 22, 2012
Additional Notes: The first author was partially supported by the Spanish Ministry of Education MTM2010-21740-C02-02. The second author has been partially supported by SB2004-0181 and grant CNCSIS PNII-IDEI 1189/2008.
Article copyright: © Copyright 2012 American Mathematical Society