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In this paper, we study the analogue of the Shafarevich conjecture for polarized Calabi–;Yau varieties. We use variations of Hodge structures and Higgs bundles to establish a criterion for the rigidity of families. We then apply the criterion to obtain that some important and typical families of Calabi–Yau varieties are rigid, for examples., Lefschetz pencils of Calabi–Yau varieties, strongly degenerated families (not only for families of Calabi–Yau varieties), families of Calabi–Yau varieties admitting a degeneration with maximal unipotent monodromy.
There is a canonical identification, due independently to the author and to F. Labourie, of a convex real projective structure on an orientable surface of genus g and a pair consisting of a conformal structure together with a holomorphic cubic differential on the surface. The Deligne–Mumford compactification of the moduli space of curves then suggests a partial compactification of the moduli space of convex real projective structures: Allow the Riemann surface to degenerate to a stable nodal curve on which there is a regular cubic differential. We construct convex real projective structures on open surfaces corresponding to this singular data and relate their holonomy to earlier work of Goldman. Also we have results for families degenerating toward the boundary of the moduli space. The techniques involve affine differential geometry results of Cheng–Yau and C.P. Wang and a result of Dunkel on the asymptotics of systems of ODEs.
We present a classification of compact Kähler manifolds admitting a hamiltonian 2-form (which were classified locally in part I of this work). This involves two components of independent interest.
The first is the notion of a rigid hamiltonian torus action. This natural condition, for torus actions on a Kähler manifold, was introduced locally in part I, but such actions turn out to be remarkably well behaved globally, leading to a fairly explicit classification: up to a blow-up, compact Kähler manifolds with a rigid hamiltonian torus action are bundles of toric Kähler manifolds.
The second idea is a special case of toric geometry, which we call orthotoric. We prove that orthotoric Kähler manifolds are diffeomorphic to complex projective space, but we extend our analysis to orthotoric orbifolds, where the geometry is much richer. We thus obtain new examples of Kähler–Einstein 4-orbifolds.
Combining these two themes, we prove that compact Kähler manifolds with hamiltonian 2-forms are covered by blow-downs of projective bundles over Kähler products, and we describe explicitly how the Kähler metrics with a hamiltonian 2-form are parameterized. We explain how this provides a context for constructing new examples of extremal Kähler metrics—in particular a subclass of such metrics which we call weakly Bochner-flat.
We also provide a self-contained treatment of the theory of compact toric Kähler manifolds, since we need it and find the existing literature incomplete.
We prove the existence of a local smooth Levi decomposition for smooth Poisson structures and Lie algebroids near a singular point. This Levi decomposition is a kind of normal form or partial linearization, which was established in the formal case by Wade  and in the analytic case by the second author . In particular, in the case of smooth Poisson structures with a compact semisimple linear part, we recover Conn's smooth linearization theorem , and in the case of smooth Lie algebroids with a compact semisimple isotropy Lie algebra, our Levi decomposition result gives a positive answer to a conjecture of Weinstein  on the smooth linearization of such Lie algebroids. In the appendix of this paper, we show an abstract Nash-Moser normal form theorem, which generalizes our Levi decomposition result, and which may be helpful in the study of other smooth normal form problems.