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

HTML articles powered by AMS MathViewer

- by J. I. Cogolludo Agustín and D. Matei PDF
- Trans. Amer. Math. Soc.
**364**(2012), 5765-5790 Request permission

## 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.## References

- V. I. Arnol′d,
*The cohomology ring of the group of dyed braids*, Mat. Zametki**5**(1969), 227–231 (Russian). MR**242196** - Enrique Artal Bartolo, José Ignacio Cogolludo, and Hiro-o Tokunaga,
*A survey on Zariski pairs*, Algebraic geometry in East Asia—Hanoi 2005, Adv. Stud. Pure Math., vol. 50, Math. Soc. Japan, Tokyo, 2008, pp. 1–100. MR**2409555**, DOI 10.2969/aspm/05010001 - Egbert Brieskorn,
*Sur les groupes de tresses [d’après V. I. Arnol′d]*, Séminaire Bourbaki, 24ème année (1971/1972), Exp. No. 401, Lecture Notes in Math., Vol. 317, Springer, Berlin, 1973, pp. 21–44 (French). MR**0422674** - Egbert Brieskorn and Horst Knörrer,
*Plane algebraic curves*, Birkhäuser Verlag, Basel, 1986. Translated from the German by John Stillwell. MR**886476**, DOI 10.1007/978-3-0348-5097-1 - José Ignacio Cogolludo-Agustín,
*Topological invariants of the complement to arrangements of rational plane curves*, Mem. Amer. Math. Soc.**159**(2002), no. 756, xiv+75. MR**1921584**, DOI 10.1090/memo/0756 - J.I. Cogolludo-Agustín and M.Á. Marco Buzunáriz,
*The Max Noether Fundamental Theorem is combinatorial*, Preprint available at arXiv:1002.2325v1 [math.AG], 2010. - Pierre Deligne, Phillip Griffiths, John Morgan, and Dennis Sullivan,
*Real homotopy theory of Kähler manifolds*, Invent. Math.**29**(1975), no. 3, 245–274. MR**382702**, DOI 10.1007/BF01389853 - Alexandru Dimca, Ştefan Papadima, and Alexander I. Suciu,
*Topology and geometry of cohomology jump loci*, Duke Math. J.**148**(2009), no. 3, 405–457. MR**2527322**, DOI 10.1215/00127094-2009-030 - Alan H. Durfee and Richard M. Hain,
*Mixed Hodge structures on the homotopy of links*, Math. Ann.**280**(1988), no. 1, 69–83. MR**928298**, DOI 10.1007/BF01474182 - David Eisenbud and Walter Neumann,
*Three-dimensional link theory and invariants of plane curve singularities*, Annals of Mathematics Studies, vol. 110, Princeton University Press, Princeton, NJ, 1985. MR**817982** - Phillip Griffiths and Wilfried Schmid,
*Recent developments in Hodge theory: a discussion of techniques and results*, Discrete subgroups of Lie groups and applicatons to moduli (Internat. Colloq., Bombay, 1973) Oxford Univ. Press, Bombay, 1975, pp. 31–127. MR**0419850** - Phillip A. Griffiths,
*Periods of integrals on algebraic manifolds: Summary of main results and discussion of open problems*, Bull. Amer. Math. Soc.**76**(1970), 228–296. MR**258824**, DOI 10.1090/S0002-9904-1970-12444-2 - Toshitake Kohno,
*On the holonomy Lie algebra and the nilpotent completion of the fundamental group of the complement of hypersurfaces*, Nagoya Math. J.**92**(1983), 21–37. MR**726138**, DOI 10.1017/S0027763000020547 - A. Libgober,
*Alexander invariants of plane algebraic curves*, Singularities, Part 2 (Arcata, Calif., 1981) Proc. Sympos. Pure Math., vol. 40, Amer. Math. Soc., Providence, RI, 1983, pp. 135–143. MR**713242** - F. Loeser and M. Vaquié,
*Le polynôme d’Alexander d’une courbe plane projective*, Topology**29**(1990), no. 2, 163–173 (French). MR**1056267**, DOI 10.1016/0040-9383(90)90005-5 - David Lubicz,
*Une description de la cohomologie du complément à un diviseur non réductible de $\textbf {P}^2$*, Bull. Sci. Math.**124**(2000), no. 6, 447–458 (French). MR**1796280**, DOI 10.1016/S0007-4497(00)01059-9 - Anca Daniela Măcinic,
*Cohomology rings and formality properties of nilpotent groups*, J. Pure Appl. Algebra**214**(2010), no. 10, 1818–1826. MR**2608110**, DOI 10.1016/j.jpaa.2009.12.025 - John W. Morgan,
*The algebraic topology of smooth algebraic varieties*, Inst. Hautes Études Sci. Publ. Math.**48**(1978), 137–204. MR**516917**, DOI 10.1007/BF02684316 - Peter Orlik and Louis Solomon,
*Combinatorics and topology of complements of hyperplanes*, Invent. Math.**56**(1980), no. 2, 167–189. MR**558866**, DOI 10.1007/BF01392549 - Dennis Sullivan,
*Infinitesimal computations in topology*, Inst. Hautes Études Sci. Publ. Math.**47**(1977), 269–331 (1978). MR**646078**, DOI 10.1007/BF02684341 - Oscar Zariski,
*On the irregularity of cyclic multiple planes*, Ann. of Math. (2)**32**(1931), no. 3, 485–511. MR**1503012**, DOI 10.2307/1968247 - Oscar Zariski,
*The Topological Discriminant Group of a Riemann Surface of Genus $p$*, Amer. J. Math.**59**(1937), no. 2, 335–358. MR**1507244**, DOI 10.2307/2371416

## Similar Articles

## 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
- 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.
- © Copyright 2012 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**364**(2012), 5765-5790 - MSC (2010): Primary 14F25, 14F40, 14H50, 58A10, 58A12, 58A14, 14B05, 14E15, 32A27, 32S22, 55P62
- DOI: https://doi.org/10.1090/S0002-9947-2012-05489-4
- MathSciNet review: 2946931