[Paper Review] A new series of compact minitwistor spaces and Moishezon twistor spaces over them

Nobuhiro Honda|ArXiv.org|May 1, 2008

Geometric and Algebraic Topology11 references8 citations

TL;DR

This paper constructs a new infinite family of compact minitwistor spaces and associated Moishezon twistor spaces with $$\mathbf{C}^{*}$-action by leveraging self-dual metrics with torus symmetry from D. Joyce's work. By analyzing meromorphic quotients of twistor spaces under $$\mathbf{C}^{*}$-actions, the authors explicitly realize projective models of these twistor spaces as conic bundles over rational surfaces (the minitwistor spaces), which are embedded in projective space and exhibit controlled singularities. The key contribution is a systematic, explicit construction of previously unknown Moishezon twistor spaces, generalizing LeBrun’s earlier examples.

ABSTRACT

In recent papers math.DG/0701278 and arXiv:0705.0060, 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. Actually, for arbitrary Joyce metrics and U(1)-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*-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.

Motivation & Objective

• To systematically construct new compact Moishezon twistor spaces with $$\mathbf{C}^{*}$-action from self-dual metrics with torus symmetry.
• To explicitly determine the structure of minitwistor spaces as rational surfaces embedded in projective space, arising as quotients of twistor spaces under $$\mathbf{C}^{*}$-actions.
• To realize projective models of twistor spaces of arbitrary Joyce metrics as conic bundles over these minitwistor spaces.
• To demonstrate that the resulting minitwistor spaces form a moduli space parameterized by continuous parameters from Joyce's metrics, yielding an infinite countable family of new examples.

Proposed method

• Starting from an arbitrary Joyce metric on $n\mathbf{CP}^{2}$ with 2-torus action, the authors select a $U(1)$-subgroup of the torus that fixes a 2-sphere, inducing a $$\mathbf{C}^{*}$-action on the twistor space.
• The meromorphic map $$\Phi_{m}^{G_{1}}$ from the twistor space to the minitwistor space is constructed via a linear system, with the image surfaces explicitly described as complex surfaces in projective space.
• The structure of the minitwistor spaces is determined using the geometry of the pencil of curves and the action of the $$\mathbf{C}^{*}$-subgroup, leading to explicit rational conic bundle maps to rational normal curves.
• Indeterminacy loci in the meromorphic maps are partially resolved via blow-ups to construct $$\mathbf{CP}^{2}$-bundles over the resolved minitwistor spaces.
• Projective models of the original twistor spaces are realized as conic bundles inside these $$\mathbf{CP}^{2}$-bundles, generalizing LeBrun’s construction.
• The construction is shown to yield a canonical quotient space of the twistor space by the $$\mathbf{C}^{*}$-action, identified as the unique component of the Douady space of $G_1$-invariant curves.

Experimental results

Research questions

• RQ1 Can a systematic construction of new Moishezon twistor spaces with $$\mathbf{C}^{*}$-action be achieved from Joyce’s self-dual metrics with torus symmetry?
• RQ2 What is the explicit geometric structure of the minitwistor spaces obtained as quotients of twistor spaces under $$\mathbf{C}^{*}$-actions?
• RQ3 How can the projective models of the twistor spaces be realized as conic bundles over these minitwistor spaces?
• RQ4 Do the resulting minitwistor spaces form a moduli space parameterized by the continuous parameters of Joyce’s metrics?
• RQ5 Can the indeterminacy of the meromorphic quotient maps be resolved sufficiently to construct explicit projective models?

Key findings

• The authors construct an infinite countable family of new compact minitwistor spaces as rational surfaces embedded in projective space, arising from $U(1)$-subgroups of the torus action on Joyce’s self-dual metrics.
• The minitwistor spaces are shown to be rational surfaces with mild singularities, and their structure depends explicitly on the continuous parameters of Joyce’s metrics, forming a moduli space.
• Projective models of the twistor spaces of all Joyce metrics are realized as conic bundles over these minitwistor spaces, generalizing LeBrun’s construction over $\mathbf{CP}^{1}\times\mathbf{CP}^{1}$.
• The meromorphic quotient map from the twistor space to the minitwistor space is shown to be a morphism after partial resolution of indeterminacy, and the image is a unique component of the Douady space of $G_1$-invariant curves.
• The canonical quotient space of the twistor space by the $$\mathbf{C}^{*}$-action is identified as the minitwistor space, and the construction yields a flat, $G_1$-equivariant morphism to this quotient.
• The fibers of the quotient map are shown to be curves, and the map does not contract any divisors to points, ensuring the geometric integrity of the quotient construction.