Rokhlin Conjecture and Topology of Quotients of Complex Surfaces by Complex Conjugation

Abstract

Quotients $Y=X/conj$ of complex surfaces by anti-holomorphic involutions $conjX\to X$ tend to be completely decomposable when they are simply connected, i.e., split into connected sums, $n CP^2\#m\barCP2$, if $w_2(Y)\ne 0$, or into $n(S^2\times S^2)$ if $w_2(Y)=0$. If $X$ is a double branched covering over $C{P}^2$, this phenomenon is related to unknottedness of Arnold surfaces in $S^4=C{P}^2/conj$, which was conjectured by V.Rokhlin. The paper contains proof of Rokhlin Conjecture and of decomposability of quotients for plenty of double planes and in certain other cases. This results give, in particular, an elementary proof of Donaldson's result on decomposability of $Y$ for K3 surfaces.

arXiv commentSome minor improvements and correction of misprints were done and some references were added

References in corpus (1)

• Decomposability of quotients by complex conjugation for rational and Enriques surfaces

Cited by in corpus (1)

• Proof of the Rokhlin's Conjecture on Arnold's surfaces