On the Negativity of Moduli Spaces for Polarized Manifolds

Given a log base space (Y, S), parameterizing a smooth family of complex projective varieties with semi-ample canonical line bundle, we briefly recall the construction of the deformation Higgs sheaf and the comparison map on (Y, S) made in the work by Viehweg–Zuo. While almost all hyperbolicities in the sense of complex analysis such as Brody, Kobayashi, big Picard and Viehweg hyperbolicities of the base U = Y ∖ S (under some technical assumptions) follow from the negativity of the kernel of the deformation Higgs bundle we pose a conjecture on the topological hyperbolicity on U. In order to study the rigidity problem we then introduce the notions of the length and characteristic varieties of a family f : X → Y, which provide an infinitesimal characterization of products of sub log pairs in (Y, S) and an upper bound for the number of subvarieties appearing as factors in such a product. We formulate a conjecture on a characterization of non-rigid families of canonically polarized varieties.

An application of a C2-estimate for a complex Monge–Ampère equation

By studying a complex Monge–Ampère equation, we present an alternate proof to a recent result of Chu-Lee-Tam concerning the projectivity of a compact Kähler manifold [Formula: see text] with [Formula: see text] for some integer [Formula: see text] with [Formula: see text], and the ampleness of the canonical line bundle [Formula: see text].

Very ampleness of the bicanonical line bundle on compact complex 2-ball quotients

The purpose of this note is to show that{2K}of any smooth compact complex 2-ball quotient is very ample, except possibly for four pairs of fake projective planes of minimal type, whereKis the canonical line bundle. For the four pairs of fake projective planes, the sections of{2K_{M}}give an embedding ofMexcept possibly for at most two points onM.

AFFINE GEOMETRY OF STRATA OF DIFFERENTIALS

Affine varieties among all algebraic varieties have simple structures. For example, an affine variety does not contain any complete algebraic curve. In this paper, we study affine-related properties of strata of $k$ -differentials on smooth curves which parameterize sections of the $k$ th power of the canonical line bundle with prescribed orders of zeros and poles. We show that if there is a prescribed pole of order at least $k$ , then the corresponding stratum does not contain any complete curve. Moreover, we explore the amusing question whether affine invariant manifolds arising from Teichmüller dynamics are affine varieties, and confirm the answer for Teichmüller curves, Hurwitz spaces of torus coverings, hyperelliptic strata as well as some low genus strata.

Qubit and fermionic Fock spaces from type II superstring black holes

Using Hodge diagram combinatorial data, we study qubit and fermionic Fock spaces from the point of view of type II superstring black holes based on complex compactifications. Concretely, we establish a one-to-one correspondence between qubits, fermionic spaces and extremal black holes in maximally supersymmetric supergravity obtained from type II superstring on complex toroidal and Calabi–Yau compactifications. We interpret the differential forms of the [Formula: see text]-dimensional complex toroidal compactification as states of [Formula: see text]-qubits encoding information on extremal black hole charges. We show that there are [Formula: see text] copies of [Formula: see text] qubit systems which can be split as [Formula: see text]. More precisely, [Formula: see text] copies are associated with even [Formula: see text]-brane charges in type IIA superstring and the other [Formula: see text] ones correspond to odd [Formula: see text]-brane charges in IIB superstring. This correspondence is generalized to a class of Calabi–Yau manifolds. In connection with black hole charges in type IIA superstring, an [Formula: see text]-qubit system has been obtained from a canonical line bundle of [Formula: see text] factors of one-dimensional projective space [Formula: see text]

Factorization of Generalized Theta Functions Revisited

This survey is based on my lectures given in the last few years. As a reference, constructions of moduli spaces of parabolic sheaves and generalized parabolic sheaves are provided. By a refinement of the proof of vanishing theorems, we show, without using vanishing theorems, a new observation that [Formula: see text] is independent of all of the choices for any smooth curves. The estimate of various codimensions and computation of canonical line bundle of moduli space of generalized parabolic sheaves on a reducible curve are provided in Section 6, which is completely new.

Homogeneous variational problems and Lagrangian sections

We define a canonical line bundle over the slit tangent bundle of a manifold, and define a Lagrangian section to be a homogeneous section of this line bundle. When a regularity condition is satisfied the Lagrangian section gives rise to local Finsler functions. For each such section we demonstrate how to construct a canonically parametrized family of geodesics, such that the geodesics of the local Finsler functions are reparametrizations.

Equivariant Forms: Structure and Geometry

In this paper we study the notion of equivariant forms introduced in the authors' previous works. In particular, we completely classify all the equivariant forms for a subgroup of SL2(ℤ) by means of the cross-ratio, weight 2 modular forms, quasimodular forms, as well as differential forms of a Riemann surface and sections of a canonical line bundle.