Projective models of the twistor spaces of Joyce metrics

TL;DR

This paper constructs explicit algebraic models of twistor spaces for Joyce's self-dual metrics on certain 4-manifolds, extending to complex projective planes, and describes their geometric properties.

Contribution

It provides a new explicit algebraic construction of twistor spaces for Joyce metrics on nCP^2 within a CP^4-bundle over CP^1.

Findings

Realized projective models of twistor spaces in CP^4-bundles

Demonstrated that twistor spaces of H^2 x T^2 are dense in the models

Simplified equations describe non-compact twistor spaces

Abstract

We provide a simple algebraic construction of the twistor spaces of arbitrary Joyce's self-dual metrics on the 4-manifold H^2 x T^2 that extend smoothly to nCP^2, the connected sum of complex projective planes. Indeed, we explicitly realize projective models of the twistor spaces of arbitrary Joyce metrics on nCP^2 in a CP^4-bundle over CP^1, and show that they contain the twistor spaces of H^2 x T^2 as dense non-Zariski open subsets. In particular, we see that the last non-compact twistor spaces can be realized in rank-4 vector bundles over CP^1 by quite simple defining equations.