Algebraic approximations of compact Kähler manifolds of algebraic codimension 1

Algebraic Approximations

Stability of Algebraic Manifolds

In this paper, we study the stability of non-singular projective varieties. We will prove a geometric criterion for a non-singular projective variety to be GIT stable in the Hilbert scheme, and then relate the Gieseker-Mumford stability of polarized manifolds to the behavior of heat kernels. We will also discuss the stability notion used by Viehweg, and state some new semi-positivity results of...

Algebraic Structures on Hyperkähler Manifolds Algebraic Structures on Hyperkähler Manifolds

Let M be a compact hyperkähler manifold. The hy-perkähler structure equips M with a set R of complex structures parametrized by CP 1 , called the set of induced complex structures. It was known previously that induced complex structures are non-algebraic, except may be a countable set. We prove that a countable set of induced complex structures is algebraic, and this set is dense in R. A more g...

All compact manifolds are homeomorphic to totally algebraic real algebraic sets

The real algebraic set we obtain will in general be singular. The homeomorphism will be piecewise differentiable. Total algebraicity is a very useful concept since it eiliminates many obstructions to making a topological situation algebraic, c.f. [AK1], [AK2], [AK5]. At first glance our result does not seem to be very useful since current algebraic approximation lemmas require nonsingularity. T...

Non-singular Points on Algebraic Curves and Codimension-1 primes

Let K be a perfect field and K be a fixed algebraic closure of K. We begin our study of algebraic geometry with affine n-space and its subsets defined by zeros of polynomials.