Holomorphic flexibility properties of complex manifolds

We obtain results on approximation of holomorphic maps by algebraic maps, jet transversality theorems for holomorphic and algebraic maps, and the homotopy principle for holomorphic submersions of Stein manifolds to certain algebraic manifolds.Comment: To appear in Amer. J. Mat

Dominating and unbounded free sets

We prove that every analytic set in ωω × ωω with σ-bounded sections has a not σ-bounded closed free set. We show that this result is sharp. There exists a closed set with bounded sections which has no dominating analytic free set. and there exists a closed set with non-dominating sections which does not have a not σ-bounded analytic free set. Under projective determinacy analytic can be replaced in the above results by projectiv

Disjoint Borel Functions

For each $a \in \mathbb{R}$, we define a Borel function $f_a : \mathbb{R} \to \mathbb{R}$ which encodes $a$ in a certain sense. We show that for each Borel $g : \mathbb{R} \to \mathbb{R}$, $f_a \cap g = \emptyset$ implies $a \in \Delta^1_1(c)$ where $c$ is any code for $g$. We generalize this theorem for $g$ in larger pointclasses $\Gamma$. Specifically, if $\Gamma = \mathbf{\Delta}^1_2$, then $a \in L[c]$. Also for all $n \in \omega$, if $\Gamma = \mathbf{\Delta}^1_{3 + n}$, then $a \in \mathcal{M}_{1 + n}(c)$.Comment: 15 page