Analytic classification of germs of parabolic antiholomorphic diffeomorphisms of codimension k

We investigate the local dynamics of antiholomorphic diffeomorphisms around a parabolic fixed point. We first give a normal form. Then we give a complete classification including a modulus space for antiholomorphic germs with a parabolic fixed point under analytic conjugacy. We then study some geometric applications: existence of real analytic invariant curves, existence of holomorphic and antiholomorphic roots of holomorphic and antiholomorphic parabolic germs, and commuting holomorphic and antiholomorphic parabolic germs.

Convergence of the Birkhoff normal form sometimes implies convergence of a normalizing transformation

Consider an analytic Hamiltonian system near its analytic invariant torus $\mathcal T_0$ carrying zero frequency. We assume that the Birkhoff normal form of the Hamiltonian at $\mathcal T_0$ is convergent and has a particular form: it is an analytic function of its non-degenerate quadratic part. We prove that in this case there is an analytic canonical transformation—not just a formal power series—bringing the Hamiltonian into its Birkhoff normal form.

On the Dimension of a New Class of Derivation Lie Algebras Associated to Singularities

Let (V,0)={(z1,…,zn)∈Cn:f(z1,…,zn)=0} be an isolated hypersurface singularity with mult(f)=m. Let Jk(f) be the ideal generated by all k-th order partial derivatives of f. For 1≤k≤m−1, the new object Lk(V) is defined to be the Lie algebra of derivations of the new k-th local algebra Mk(V), where Mk(V):=On/((f)+J1(f)+…+Jk(f)). Its dimension is denoted as δk(V). This number δk(V) is a new numerical analytic invariant. In this article we compute L4(V) for fewnomial isolated singularities (binomial, trinomial) and obtain the formulas of δ4(V). We also verify a sharp upper estimate conjecture for the δ4(V) for large class of singularities. Furthermore, we verify another inequality conjecture: δ(k+1)(V)<δk(V),k=3 for low-dimensional fewnomial singularities.

Pointed harmonic volume and its relation to the extended Johnson homomorphism

The period for a compact Riemann surface, defined by the integral of differential 1-forms, is a classical complex analytic invariant, strongly related to the complex structure of the surface. In this paper, we treat another complex analytic invariant called the pointed harmonic volume. As a natural extension of the period defined using Chen’s iterated integrals, it captures more detailed information of the complex structure. It is also one of a few explicitly computable examples of complex analytic invariants. We obtain its new value for a certain pointed hyperelliptic curve. An application of the pointed harmonic volume is presented. We explain the relationship between the harmonic volume and first extended Johnson homomorphism on the mapping class group of a pointed oriented closed surface. Moreover, we explicitly compute a certain restriction of this homomorphism.