arXiv subjects

Gordon Heier

Publications and source records attributed to Gordon Heier.

23 records · Page 2Linked to original sources

arXiv2006

Curve selection for finite-type ideals

Let $\mathfrak a$ be an ideal of holomorphic functions vanishing only at the origin in $\mathbb{C}^n$. The \textit{type} of $\mathfrak a$ is an invariant that measures the order of vanishing of the functions in $\mathfrak a$ along holomorphic curves; this invariant is of importance in the study of subelliptic estimates and subelliptic multiplier ideal sheaves. Recently there has been some interest in the question of which curves actually compute the type. In this note we prove that it is computed by one of the analytic irreducible components of the intersection of $n-1$ general functions in $\mathfrak a$.

math.CV↗

arXiv2006

Finite type and the effective Nullstellensatz

Improved local and global versions of the effective Nullstellensatz for ideal sheaves on non-singular complex varieties are obtained, based on a new invariant motivated by the notion of finite type from the theory of several complex variables. Two closely related curve selection theorems for curves with maximal adjusted tangency orders to a given ideal sheaf are established along the way, using normalized blow-ups and integral closures.

math.AG↗

arXiv2003

Uniformly effective Shafarevich Conjecture on families of hyperbolic curves over a curve with prescribed degeneracy locus

The paper's main result is an effective uniform bound for the finiteness statement of the Shafarevich Conjecture over function fields. Several results on the projective geometry of curves are established in the course of the proof. These results should be of independent interest. As a corollary, a uniform effective bound for the Mordell Conjecture over function fields is derived via Parshin's trick.

math.AG↗

arXiv2003

Effective finiteness theorems for maps between canonically polarized compact complex manifolds

Effective bounds for the finite number of surjective holomorphic maps between canonically polarized compact complex manifolds of any dimension with fixed domain are proven. Both the case of a fixed target and the case of varying targets are treated. In the case of varying targets, bounds on the complexity of Chow varieties are used.

math.AG↗

arXiv2001

Effective freeness of adjoint line bundles

In this note we establish a new Fujita-type effective bound for the base point freeness of adjoint line bundles on a compact complex projective manifold of complex dimension $n$. The bound we obtain (approximately) differs from the linear bound conjectured by Fujita only by a factor of the cube root of $n$. As an application, a new effective statement for pluricanonical embeddings is derived.

math.AG↗