Geometry & Topology

Rational cohomology tori

Olivier Debarre, Zhi Jiang, and Martí Lahoz

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/gt.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We study normal compact varieties in Fujiki’s class C whose rational cohomology ring is isomorphic to that of a complex torus. We call them rational cohomology tori. We classify, up to dimension three, those with rational singularities. We then give constraints on the degree of the Albanese morphism and the number of simple factors of the Albanese variety for rational cohomology tori of general type (hence projective) with rational singularities. Their properties are related to the birational geometry of smooth projective varieties of general type, maximal Albanese dimension, and with vanishing holomorphic Euler characteristic. We finish with the construction of series of examples.

In an appendix, we show that there are no smooth rational cohomology tori of general type. The key technical ingredient is a result of Popa and Schnell on 1–forms on smooth varieties of general type.

Article information

Source
Geom. Topol., Volume 21, Number 2 (2017), 1095-1130.

Dates
Received: 14 September 2015
Revised: 11 April 2016
Accepted: 13 May 2016
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1510859175

Digital Object Identifier
doi:10.2140/gt.2017.21.1095

Mathematical Reviews number (MathSciNet)
MR3626598

Zentralblatt MATH identifier
1378.32013

Subjects
Primary: 32J27: Compact Kähler manifolds: generalizations, classification 32Q15: Kähler manifolds 32Q55: Topological aspects of complex manifolds
Secondary: 14F45: Topological properties 14E99: None of the above, but in this section

Keywords
complex tori compact Kähler manifolds rational cohomology ring

Citation

Debarre, Olivier; Jiang, Zhi; Lahoz, Martí. Rational cohomology tori. Geom. Topol. 21 (2017), no. 2, 1095--1130. doi:10.2140/gt.2017.21.1095. https://projecteuclid.org/euclid.gt/1510859175


Export citation

References

  • D Abramovich, J Wang, Equivariant resolution of singularities in characteristic $0$, Math. Res. Lett. 4 (1997) 427–433
  • V Ancona, B Gaveau, Differential forms on singular varieties: de Rham and Hodge theory simplified, Pure and Applied Mathematics 273, Chapman & Hall/CRC, Boca Raton, FL (2006)
  • A Blanchard, Sur les variétés analytiques complexes, Ann. Sci. Ecole Norm. Sup. 73 (1956) 157–202
  • F A Bogomolov, C Böhning, H-C G von Bothmer, Birationally isotrivial fiber spaces, Eur. J. Math. 2 (2016) 45–54
  • R Bott, L W Tu, Differential forms in algebraic topology, Graduate Texts in Mathematics 82, Springer, New York (1982)
  • G E Bredon, Introduction to compact transformation groups, Pure and Applied Mathematics 46, Academic Press, New York (1972)
  • F Catanese, Moduli and classification of irregular Kaehler manifolds (and algebraic varieties) with Albanese general type fibrations, Invent. Math. 104 (1991) 263–289
  • F Catanese, Topological methods in moduli theory, Bull. Math. Sci. 5 (2015) 287–449
  • J A Chen, O Debarre, Z Jiang, Varieties with vanishing holomorphic Euler characteristic, J. Reine Angew. Math. 691 (2014) 203–227
  • J A Chen, C D Hacon, Pluricanonical maps of varieties of maximal Albanese dimension, Math. Ann. 320 (2001) 367–380
  • J A Chen, C D Hacon, On the irregularity of the image of the Iitaka fibration, Comm. Algebra 32 (2004) 203–215
  • J A Chen, Z Jiang, Positivity in varieties of maximal Albanese dimension, J. Reine Angew. Math. (online publication July 2015)
  • J Chen, Z Jiang, Z Tian, Irregular varieites with geometric genus one, theta divisors, and fake tori, preprint (2016)
  • O Debarre, Cohomological characterizations of the complex projective space, preprint (2015)
  • P Deligne, Théorie de Hodge, III, Inst. Hautes Études Sci. Publ. Math. 44 (1974) 5–77
  • L Ein, R Lazarsfeld, Singularities of theta divisors and the birational geometry of irregular varieties, J. Amer. Math. Soc. 10 (1997) 243–258
  • A Fujiki, Closedness of the Douady spaces of compact Kähler spaces, Publ. Res. Inst. Math. Sci. 14 (1978/79) 1–52
  • A Fujiki, Duality of mixed Hodge structures of algebraic varieties, Publ. Res. Inst. Math. Sci. 16 (1980) 635–667
  • T Fujita, On Kähler fiber spaces over curves, J. Math. Soc. Japan 30 (1978) 779–794
  • T Fujita, On topological characterizations of complex projective spaces and affine linear spaces, Proc. Japan Acad. Ser. A Math. Sci. 56 (1980) 231–234
  • M Green, R Lazarsfeld, Deformation theory, generic vanishing theorems, and some conjectures of Enriques, Catanese and Beauville, Invent. Math. 90 (1987) 389–407
  • M Green, R Lazarsfeld, Higher obstructions to deforming cohomology groups of line bundles, J. Amer. Math. Soc. 4 (1991) 87–103
  • C D Hacon, A derived category approach to generic vanishing, J. Reine Angew. Math. 575 (2004) 173–187
  • F Hirzebruch, K Kodaira, On the complex projective spaces, J. Math. Pures Appl. 36 (1957) 201–216 Reprinted in Kunihiko Kodaira: collected works, II, Princeton University Press (1975) 744–759
  • Z Jiang, M Lahoz, S Tirabassi, On the Iitaka fibration of varieties of maximal Albanese dimension, Int. Math. Res. Not. 2013 (2013) 2984–3005
  • Z Jiang, Q Yin, On the Chow ring of certain rational cohomology tori, preprint (2016)
  • Y Kawamata, Characterization of abelian varieties, Compositio Math. 43 (1981) 253–276
  • S J Kovács, Rational, log canonical, Du Bois singularities: on the conjectures of Kollár and Steenbrink, Compositio Math. 118 (1999) 123–133
  • S J Kovács, A characterization of rational singularities, Duke Math. J. 102 (2000) 187–191
  • S J Kovács, K E Schwede, Hodge theory meets the minimal model program: a survey of log canonical and Du Bois singularities, from “Topology of stratified spaces” (G Friedman, E Hunsicker, A Libgober, L Maxim, editors), Math. Sci. Res. Inst. Publ. 58, Cambridge Univ. Press (2011) 51–94
  • A S Libgober, J W Wood, Uniqueness of the complex structure on Kähler manifolds of certain homotopy types, J. Differential Geom. 32 (1990) 139–154
  • J A Morrow, A survey of some results on complex Kähler manifolds, from “Global analysis: papers in honor of K Kodaira” (D C Spencer, S Iyanaga, editors), Univ. Tokyo Press (1969) 315–324
  • R Pardini, Abelian covers of algebraic varieties, J. Reine Angew. Math. 417 (1991) 191–213
  • R Pardini, Infinitesimal Torelli and abelian covers of algebraic surfaces, from “Problems in the theory of surfaces and their classification” (F Catanese, C Ciliberto, M Cornalba, editors), Sympos. Math. XXXII, Academic Press, London (1991) 247–257
  • C A M Peters, J H M Steenbrink, Mixed Hodge structures, Ergeb. Math. Grenzgeb. 52, Springer (2008)
  • M Popa, C Schnell, Kodaira dimension and zeros of holomorphic one-forms, Ann. of Math. 179 (2014) 1109–1120
  • C Simpson, Subspaces of moduli spaces of rank one local systems, Ann. Sci. École Norm. Sup. 26 (1993) 361–401
  • K Ueno, Classification theory of algebraic varieties and compact complex spaces, Lecture Notes in Mathematics 439, Springer, Berlin (1975)
  • K Ueno, Bimeromorphic geometry of algebraic and analytic threefolds, from “Algebraic threefolds” (A Conte, editor), Lecture Notes in Math. 947, Springer, Berlin (1982) 1–34
  • E Viehweg, Weak positivity and the additivity of the Kodaira dimension for certain fibre spaces, from “Algebraic varieties and analytic varieties” (S Iitaka, editor), Adv. Stud. Pure Math. 1, North-Holland, Amsterdam (1983) 329–353