Vol. 110, No. 2, 1984

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Quotients by complex conjugation of nonsingular quadrics and cubics in PC3 defined over R

Maurizio Letizia

Vol. 110 (1984), No. 2, 307–314

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 PC3.

Our results can be summarized in the following:

Proposition. If X is a nonsingular quadric or cubic in PC3 defined over R then X∕τ is, up to diffeomorphisms, obtained from the 4-sphere S4 by a connected sum with copies of PC2.

Mathematical Subject Classification 2000
Primary: 14J99
Secondary: 14J50, 14N99
Received: 29 March 1982
Revised: 20 May 1982
Published: 1 February 1984
Maurizio Letizia