Quotients by complex conjugation of nonsingular quadrics and cubics in P_C³ defined over R

Maurizio Letizia

Vol. 110 (1984), No. 2, 307–314

DOI: 10.2140/pjm.1984.110.307

Abstract

If X is a complex algebraic variety defined over R, complex conjugation in C induces an involution τ : X → X which we shall still call complex conjugation. If X is nonsingular and of complex dimension 2, τ is an orientation preserving diffeomorphism and the quotient X∕τ of X by τ is, as X, a naturally oriented smooth manifold without boundary. Our aim is to describe X∕τ, up to diffeomorphisms, in case X is a nonsingular quadric or cubic in P_C³.

Our results can be summarized in the following:

Proposition. If X is a nonsingular quadric or cubic in P_C³ defined over R then X∕τ is, up to diffeomorphisms, obtained from the 4-sphere S⁴ by a connected sum with copies of P_C².

Mathematical Subject Classification 2000

Primary: 14J99

Secondary: 14J50, 14N99

Milestones

Received: 29 March 1982

Revised: 20 May 1982

Published: 1 February 1984