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Commensurability classes of fake quadrics

  • Benjamin Linowitz
  • Matthew StoverEmail author
  • John Voight


A fake quadric is a smooth projective surface that has the same rational cohomology as a smooth quadric surface but is not biholomorphic to one. We provide an explicit classification of all irreducible fake quadrics according to the commensurability class of their fundamental group. To accomplish this task, we develop a number of new techniques that explicitly bound the arithmetic invariants of a fake quadric and more generally of an arithmetic manifold of bounded volume arising from a form of \({{\,\mathrm{SL}\,}}_2\) over a number field.

Mathematics Subject Classification

11F23 14J29 22E40 11F06 



The authors would like to thank Carl Pomerance and Alan Reid for advice and the referees for their constructive feedback and comments on the code. The first author was partially supported by an NSF RTG Grant DMS-1045119 and an NSF Mathematical Sciences Postdoctoral Fellowship. This material is based upon work supported by Grant Number 523197 from the SimonsFoundation/SFARI. The second author was supported by NSF Grant DMS-1361000, and acknowledges support from U.S. National Science Foundation Grants DMS 1107452, 1107263, 1107367 “RNMS: GEometric structures And Representation varieties” (the GEAR Network). The third author was supported by an NSF CAREER Award (DMS-1151047).


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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Benjamin Linowitz
    • 1
  • Matthew Stover
    • 2
    Email author
  • John Voight
    • 3
  1. 1.Department of MathematicsOberlin CollegeOberlinUSA
  2. 2.Department of MathematicsTemple UniversityPhiladelphiaUSA
  3. 3.Department of MathematicsDartmouth CollegeHanoverUSA

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