Deformation limit and bimeromorphic embedding of Moishezon manifolds

Let [Formula: see text] be a holomorphic family of compact complex manifolds over an open disk in [Formula: see text]. If the fiber [Formula: see text] for each nonzero [Formula: see text] in an uncountable subset [Formula: see text] of [Formula: see text] is Moishezon and the reference fiber [Formula: see text] satisfies the local deformation invariance for Hodge number of type [Formula: see text] or admits a strongly Gauduchon metric introduced by D. Popovici, then [Formula: see text] is still Moishezon. We also obtain a bimeromorphic embedding [Formula: see text]. Our proof can be regarded as a new, algebraic proof of several results in this direction proposed and proved by Popovici in 2009, 2010 and 2013. However, our assumption with [Formula: see text] not necessarily being a limit point of [Formula: see text] and the bimeromorphic embedding are new. Our strategy of proof lies in constructing a global holomorphic line bundle over the total space of the holomorphic family and studying the bimeromorphic geometry of [Formula: see text]. S.-T. Yau’s solutions to certain degenerate Monge–Ampère equations are used.

Rigidity properties of holomorphic Legendrian singularities

We study the singularities of Legendrian subvarieties of contact manifolds in the complex-analytic category and prove two rigidity results. The first one is that Legendrian singularities with reduced tangent cones are contactomorphically biholomorphic to their tangent cones. This result is partly motivated by a problem on Fano contact manifolds. The second result is the deformation-rigidity of normal Legendrian singularities, meaning that any holomorphic family of normal Legendrian singularities is trivial, up to contactomorphic biholomorphisms of germs. Both results are proved by exploiting the relation between infinitesimal contactomorphisms and holomorphic sections of the natural line bundle on the contact manifold. Comment: 21 pages, minor revision

Boundary asymptotics of the relative Bergman kernel metric for hyperelliptic curves

We survey variations of the Bergman kernel and their asymptotic behaviors at degeneration. For a Legendre family of elliptic curves, the curvature form of the relative Bergman kernel metric is equal to the Poincaré metric on ℂ \ {0,1}. The cases of other elliptic curves are either the same or trivial. Two proofs depending on elliptic functions’ special properties and Abelian differentials’ Taylor expansions are discussed, respectively. For a holomorphic family of hyperelliptic nodal or cuspidal curves and their Jacobians, we announce our results on the Bergman kernel asymptotics near various singularities. For genus-two curves particularly, asymptotic formulas with precise coefficients involving the complex structure information are written down explicitly.

DYNAMICS OF NONHOLOMORPHIC SINGULAR CONTINUATIONS: A CASE IN RADIAL SYMMETRY

This paper is primarily a study of special families of rational maps of the real plane of the form: [Formula: see text] where the dynamic variable z ∈ ℂ, and ℂ is identified with ℝ2. The parameter β is complex; n is a positive integer. For β small, this family can be considered a nonholomorphic singular perturbation of the holomorphic family z ↦ zn, although we consider large values of β as well. Compared to the more general family [Formula: see text], the special case where n = d and c = 0 is easier to analyze because the radial component in polar coordinates decouples from the angular component. This reduces a significant part of the analysis to the study of a family of one-real-dimensional unimodal maps. For each fixed n, the β parameter plane separates into three major regions, corresponding to maps which have one of the following behaviors: (i) all orbits go off to infinity, (ii) only an annulus of points stays bounded, and (iii) only a Cantor set of circles stays bounded. In cases (ii) and (iii), there is a transitive invariant set; this set is an attractor in case (ii). Some comparisons are made between [Formula: see text] and the holomorphic singularly perturbed maps: z ↦ zn + λ/zn, studied by Devaney and coauthors over the last decade. Additional results and observations are made about the more general family where c ≠ 0 and n ≠ d.

Natural invariant measures, divergence points and dimension in one-dimensional holomorphic dynamics

In this paper we discuss the dimension-theoretical properties of rational maps on the Riemann sphere. In particular, we study the existence and uniqueness of generalized physical measures for several classes of maps including hyperbolic, parabolic, non-recurrent and topological Collet–Eckmann maps. These measures have the property that their typical points have maximal Hausdorff dimension. On the other hand, we prove that the set of divergence points (the set of points which are non-typical for any invariant measure) also has maximal Hausdorff dimension. Finally, we prove that if (fa)ais a holomorphic family of stable rational maps, then the dimensiond(fa) is a continuous and plurisubharmonic function of the parametera. In particular,d(f) varies continuously and plurisubharmonically on an open and dense subset ofRatd, the space of all rational maps with degreed≥2.