Can a small forcing create Kurepa trees

In this paper we probe the possibilities of creating a Kurepa tree in a generic extension of a ground model of CH plus no Kurepa trees by an ω₁ -preserving forcing notion of size at most ω₁ In Section 1 we show that in the Lévy model obtained by collapsing all cardinals between ω₁ and a strongly inaccessible cardinal by forcing with a countable support Lévy collapsing order, many ω₁-preserving forcing notions of size at most ω₁ including all ω-proper forcing notions and some proper but not ω-proper forcing notions of size at most ω₁ do not create Kurepa trees. In Section 2 we construct a model of CH plus no Kurepa trees, in which there is an ω-distributive Aronszajn tree such that forcing with that Aronszajn tree does create a Kurepa tree in the generic extension. At the end of the paper we ask three questions.