A new series of compact minitwistor spaces and Moishezon twistor spaces over them

Abstract

In recent papers [10], [11], we gave explicit description of some new Moishezon twistor spaces. In this paper, developing the method in the papers much further, we explicitly give projective models of a number of new Moishezon twistor spaces, as conic bundles over some rational surfaces (called minitwistor spaces). These include the twistor spaces studied in the papers as very special cases.Our source of the result is a series of self-dual metrics with torus action constructed by D. Joyce [15]. Actually, for arbitrary Joyce metrics and Uð1Þ-subgroups of the torus which fixes a torus-invariant 2-sphere, we first determine the associated minitwistor spaces in explicit forms. Next by analyzing the meromorphic maps from the twistor spaces to the minitwistor spaces, we realize projective models of the twistor spaces of all Joyce metrics, as conic bundles over the minitwistor spaces. Then we prove that for any one of these minitwistor spaces, there exist Moishezon twistor spaces with only C Ã -action whose quotient space is the given minitwistor space. This result generates numerous Moishezon twistor spaces which cannot be found in the literature (including the author's papers), in quite explicit form.

Minitwistor spaces, Severi varieties, and Einstein–Weyl structure

“…Recall that T can be regarded as a quotient space with respect to the G 1 -action: namely for general points of T , the inverse image under Φ G 1 m are the closure of the G 1 -orbits. (More invariantly, T is exactly the 'canonical quotient space' of the Moishezon twistor space Z under the G 1 -action, as proved in [6,Appendix]). Also recall that T has a natural real structure induced by that on Z.…”

“…First we briefly recall some of the results in [6]. We consider (arbitrary) effective U (1) 2 -action on nCP 2 and take any one of Joyce's self-dual metrics on nCP 2 which are invariant under the U (1) 2 -action [7].…”

“…2.11], the system |mF | G is composed with the pencil |F | G . As in [6,Diagram (13)] we have the following commutative diagram of meromorphic maps:…”

Moishezon twistor spaces on 4ℂℙ²

“…(Although presence of a C * × C * -or a C * -action on the twistor spaces is supposed in [11], for the purpose of finding members of the multiple system |mF |, we just need an action of these groups on the divisor S and we do not need that on the twistor spaces. We also note that the divisors (2.17) are identical with those in [8,Lemma 2.7], where they were found rather incidentally.)…”

On pluri-half-anticanonical systems of LeBrun twistor spaces

“…Recently a large number of new Moishezon twistor spaces was obtained [2,3,4]. An important common property of these twistor spaces is that they all admit a C * -action.…”