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

TL;DR

This paper proves Rokhlin's conjecture and demonstrates that quotients of complex surfaces by anti-holomorphic involutions are often decomposable into standard 4-manifolds, providing new insights into their topology.

Contribution

It provides a proof of Rokhlin's conjecture and establishes decomposability results for quotients of complex surfaces, including elementary proofs for K3 surfaces.

Findings

Proof of Rokhlin Conjecture.

Decomposability of quotients for double planes.

Elementary proof of Donaldson's result for K3 surfaces.

Abstract

Quotients $Y=X/conj$ of complex surfaces by anti-holomorphic involutions $conj\: X\to X$ tend to be completely decomposable when they are simply connected, i.e., split into connected sums, $n CP^2\#m\barCP2$, if $w_2(Y)\ne0$, or into $n(S^2\times S^2)$ if $w_2(Y)=0$. If $X$ is a double branched covering over $CP^2$, this phenomenon is related to unknottedness of Arnold surfaces in $S^4=CP^2/conj$, 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.