Unobstructedness of hyperkähler twistor spaces

A family of irreducible holomorphic symplectic (ihs) manifolds over the complex projective line has unobstructed deformations if its period map is an embedding. This applies in particular to twistor spaces of ihs manifolds. Moreover, a family of ihs manifolds over a subspace of the period domain extends to a universal family over an open neighborhood in the period domain.

D-modules of pure Gaussian type and enhanced ind-sheaves

Differential systems of pure Gaussian type are examples of D-modules on the complex projective line with an irregular singularity at infinity, and as such are subject to the Stokes phenomenon. We employ the theory of enhanced ind-sheaves and the Riemann–Hilbert correspondence for holonomic D-modules of D’Agnolo and Kashiwara to describe the Stokes phenomenon topologically. Using this description, we perform a topological computation of the Fourier–Laplace transform of a D-module of pure Gaussian type in this framework, recovering and generalizing a result of Sabbah.

Principal co-Higgs bundles on ℙ1

For complex connected, reductive, affine, algebraic groups G, we give a Lie-theoretic characterization of the semistability of principal G-co-Higgs bundles on the complex projective line ℙ1 in terms of the simple roots of a Borel subgroup of G. We describe a stratification of the moduli space in terms of the Harder–Narasimhan type of the underlying bundle.

Algebraic families of groups and commuting involutions

Let [Formula: see text] be a complex affine algebraic group, and let [Formula: see text] and [Formula: see text] be commuting anti-holomorphic involutions of [Formula: see text]. We construct an algebraic family of algebraic groups over the complex projective line and a real structure on the family that interpolates between the real forms [Formula: see text] and [Formula: see text].

Quantum spectral curve for the Gromov–Witten theory of the complex projective line

We construct the quantum curve for the Gromov–Witten theory of the complex projective line.

Numerical calculation of three-point branched covers of the projective line

We exhibit a numerical method to compute three-point branched covers of the complex projective line. We develop algorithms for working explicitly with Fuchsian triangle groups and their finite-index subgroups, and we use these algorithms to compute power series expansions of modular forms on these groups.

GENERATING SERIES FOR SYMMETRIC PRODUCT SPACES

We consider the symmetric product spaces of closed manifolds. We introduce some geometric invariants and the topological properties of symmetric product spaces via the symmetric invariant ones of product spaces and apply to Gromov–Witten invariants. We examine the symmetric product spaces of the complex projective line, their Gromov–Witten invariants and compute the generating series induced by their Gromov–Witten invariants.

QUANTUM COHOMOLOGIES OF SYMMETRIC PRODUCTS

In this paper we investigate the quantum cohomologies of symmetric products of Kähler manifolds. To do this we study the moduli space of product space and symmetric group action on it, Gromov–Witten invariant and relative Gromov–Witten invariant. Also we investigate the relations between symmetric invariant properties on the products space and the corresponding ones on the symmetric product. As an example we examine the symmetric product of k copies complex projective line ℙ1, which is the k-dimensional complex projective space ℙk.