|
Organizers |
The Rokhlin Conjecture and Quotients of Real Algebraic Varieties by the Complex Conjugation
by
Sergey Finashin
Middle East Technical University
The orbit space [`X]=\C X/\conj of the complex conjugation on the complex point set, \C X, of a real algebraic variety carries certain topological properties which may be viewed as "shadows" of the corresponding properties of \C X. Some of these shadow properties have almost the same formulation (like the homotopy description of a Milnor fiber, or the Lefschetz hyperplane section theorem), in some other cases the formulations of the shadow properties become more sophisticated and have only a few features in common with the original ones: as an example one can take the "shadow" of the Hodge decomposition. V. A. Rokhlin suggested to study the properties of [`X] and raised a few questions, one of which is discussed in the talk.
A Rokhlin Conjecture If \C X is a non-singular complex surface, then [`X] is a 4-manifold and the quotient map q \C X®[`X] is a double covering branched along the real locus, \R X, of \C X. One can endow [`X] with an orientation and a smooth structure making q smooth and orientation preserving, for example [`\romanP]2=\Cp2/\conj is well known to be diffeomorphic to S4.
A curve \C A Ì \Cp2, defined by a real form f(x, y, z) of degree d=2k, with \R A ¹ \oo, divides \Rp2 into two regions, \Rp2±={[x:y:z] Î \Rp2 | ±f ³ 0}. If \R A is non-singular, then one of these regions is orientable, the other is not, and for the sake of definiteness it is usually assumed that \Rp2+ is the orientable region.
The unions \A±=\Rp2±È[`A] are closed surfaces in S4=[`\roman P]2, called by Rokhlin the Arnold surfaces. They consist of the two smooth pieces, \Rp2± and [`A], which meet normally along \R A, and therefore \A± can be smoothed along \R A. Rokhlin suggested to prove that the embeddings \A± Ì S4 are standard, in the sense that they can be obtained via ambient connected sum from a few copies of a standard embedding of a torus, T2 Ì S4, if \A± is orientable, or from standard embeddings of \Rp2 if \A± is not orientable. The surfaces \A± was considered by Arnold in connection with studying the topology of the double planes p X®\Cp2 branched along \C A. There exist 2 liftings, \conj±, of the complex conjugation from \Cp2 to X. Being endowed with an involution \conj±, the double plane X is denoted by \CX±. Thus, if the Arnold surface \A-± is standard, then [`X]± is decomposable into a connected sum of a few copies of \Cp2, \barCP2, if \A-± is non-orientable, or copies of S2×S2 if orientable. Such a splitting is called a complete decomposition. A weaker form of Rokhlin's question: is it true that [`X]± admits such a decomposition for any double plane \C X± with \RX± ¹ \oo. I refer to this statement as to CDQ-Conjecture ("Complete Decomposability of the Quotients").
A weaker version of the CDQ-Conjecture was suggested by Akbulut to prove that Seiberg-Witten invariants of [`X]± vanish if b2+([`X]±) > 1. As an intermediate variation, I present one more conjecture. Let us call 4-manifolds X and Y blow-up stable equivalent (BUS-equivalent) if X#n\barCP2 is diffeomorphic to Y#m\barCP2 for some n, m ³ 0. If X is BUS-equivalent to #n\Cp2, n ³ 0, then we call X BUS-trivial. Complete decomposability implies BUS-triviality, but not vice versa. On the other hand, BUS-triviality guarantees vanishing of the Donaldson and SW-invariants (more generally, vanishing of these invariants is a property which is preserved under BUS-equivalence). So, the BUS-Conjecture suggesting that the quotients [`X]± are BUS-trivial for the double planes, \C X± with \RX± ¹ \oo, is weaker then CDQ-conjecture, but stronger then Akbulut's conjecture.
Theorem Assume that a real plane curve, \C A0, of degree 2k and with \R A0 ¹ \oo, splits into a union \C A0=\CB0È\C C0 of transverse non-singular curves and a non-singular curve, \C A, is obtained from \C A0 by a small perturbation. Let \C X± denote the double plane branched along \C A. Then the both quotients, [`X]+ and [`X]-, are BUS-trivial. In particular, the SW-invariants of [`X]± vanish, if k > 3.
The proof is based on the analysis of the bifurcations which can experience [`X]± as we deform \C B0 and \C C0. A generic such a deformation may involve simple tangency points, i.e., simple singularities of the type A3, and the corresponding bifurcations of [`X]± can be easily analyzed.
Real Simple Singularities More generally, using the well-known classification of deformations of the real simple singularities, one can prove the following result
Theorem If a real surface \C X has a simple singularity at s Î \R X which is not equivalent to the singularity defined in \C3 by the equation x2n+y2+z2=0, and \C X¢ is obtained by a real non-singular perturbation of \C X, then the BUS-equivalence class of [`X]¢ is independent of the perturbation. ( For the above exceptional singularity, the same claim is true for all the perturbations except the one for which the real locus, \R X¢, vanishes near s).
In fact, [`X] has at s a singularity topologically equivalent to an algebraic surface singularity and [`X]¢ is BUS-equivalent to [`X]\res, obtained from [`X] by resolution of this singularity. This can be proved both by elementary means and using a more sophisticated technique, namely, variation of the complex structure in a singularity neighborhood so that the involution of complex conjugation becomes holomorphic (a similar trick was used by Donaldson in the case of real K3 surfaces).
SF-singularities The Rokhlin Conjecture and its versions admit an extension to the case of real surfaces with a certain type of singularities. Note that one or the both of the Arnold surfaces, \A±, may turn out to be smoothable even if we admit certain types of real singularities on \C A. The condition of smoothability is that the links of \A± Ì S4 at the singular points, x Î \R A, are unknots. These unknots are piecewise smooth and their smoothings can be easily extended to smoothings of the embeddings \A± Ì S4. This gives a smoothing of the double covering [`X]±® S4, branched along \A±, for the double planes, \C X±®\Cp2, branched along \C A.
A singularity at a point x Î \R X of a real algebraic surface will be called smoothly-folding, or briefly, SF-singularity, if the link of [`X] at x is a 3-sphere (so, [`X] is a manifold around x). Examples: any simple surface singularity has a real form which is an SF-singularity. SF-singularities can be characterized in terms of a minimal resolution, \res \C X\res®\C X, and the exceptional curve, \C E Ì \C X\res, over x. The complex conjugation gives an involution on the resolution graph, \G, and we denote by [`\G] the quotient graph with respect to this involution.
Theorem Assume that x Î \R X is an isolated normal real surface singularity, and \C E is the exceptional curve of its minimal resolution. Then the following conditions are equivalent.
Date received: May 18, 1999
Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cacy-07.