On the extension theorem of Hwang and Mok

Jun-Muk Hwang and Ngaiming Mok developed a framework for studying complex Fano (more generally, uniruled) manifolds in terms of their intrinsic local differential-geometric structure: the varieties of minimal rational tangents (VMRT). In particular, their ‘Cartan–Fubini’ extension theorem shows that a Fano manifold of Picard number 1 (satisfying certain technical conditions) is determined, up to biholomorphism, by an analytic germ of its VMRT at a general point. We prove a characteristic-free analogue of this result, replacing the VMRT with families of formal arcs.

Singularity of the varieties of representations of lattices in solvable Lie groups

For a lattice [Formula: see text] of a simply connected solvable Lie group [Formula: see text], we describe the analytic germ in the variety of representations of [Formula: see text] at the trivial representation as an analytic germ which is linearly embedded in the analytic germ associated with the nilpotent Lie algebra determined by [Formula: see text]. By this description, under certain assumption, we study the singularity of the analytic germ in the variety of representations of [Formula: see text] at the trivial representation by using the Kuranishi space construction. By a similar technique, we also study deformations of holomorphic structures of trivial vector bundles over complex parallelizable solvmanifolds.

REAL ANALYTIC GERMS $f \bar{g}$ AND OPEN-BOOK DECOMPOSITIONS OF THE 3-SPHERE

Let f,g: (C2,0) → (C,0) be two holomorphic germs with isolated singularities and no common branches and let Lf, [Formula: see text] be their links. We prove that the real analytic germ [Formula: see text] has an isolated singularity at 0 if and only if the link Lf ∪ -Lg is fibred. This was conjectured by Rudolph in [14]. If this condition holds, then the underlying Milnor fibration is an open-book fibration of the link Lf ∪ -Lg which coincides with [Formula: see text] in a tubular neighbourhood of this link. This enables one to realize a large family of fibrations of plumbing links in S3 as the Milnor fibrations of some real analytic germs [Formula: see text].