Real algebraic tori

Last updated: April 10, 2005

The following data are equivalent:

  1. a complex algebraic torus T with involution θ
  2. a real algebraic torus (see here)
  3. a lattice X with involution θ

It will be convenient to use 3. As explained in Fokko du Cloux's 2004 notes, it is always possible to find a basis

e1, ... ,ep,  ep+1, ... ,ep+q,  ep+q+1, ... , ep+q+2r

of X such that the vectors e1, ... ,ep are eigenvectors of θ for the eigenvalue +1, ep+1, ... ,ep+q are eigenvectors of theta for the eigenvalue -1, and the vectors ej and ej+r are interchanged by θ for p+q < j ≤ p+q+r.

Here the numbers p, q and r are entirely determined:

It follows that the group of real points of T is isomorphic to the product of p copies of the compact torus U(1), q copies of the multiplicative group of the reals, and r copies of the multiplicative group of the complex numbers. In particular, the component group of T(R) is isomorphic to the product of q copies of Z/2.

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