The Oka principle for tame families of Stein manifolds
<p> Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a smooth open manifold of even dimension, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding="application/x-tex">T</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a topological space, and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper J equals left-brace upper J Subscript t Baseline right-brace Subscript t element-of upper T"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="script">J</mml:mi> </mml:mrow> <mml:mo>=</mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:msub> <mml:mi>J</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:msub> <mml:mo fence="false" stretchy="false">}</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>t</mml:mi> <mml:mo> ∈ </mml:mo> <mml:mi>T</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathscr {J}=\{J_t\}_{t\in T}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a continuous family of smooth integrable Stein structures on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Under suitable additional assumptions on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding="application/x-tex">T</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper J"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="script">J</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathscr {J}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , we prove an Oka principle for continuous families of maps from the family of Stein manifolds <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper X comma upper J Subscript t Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>J</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(X,J_t)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t element-of upper T"> <mml:semantics> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo> ∈ </mml:mo> <mml:mi>T</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">t\in T</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , to any Oka manifold, showing that every family of continuous maps is homotopic to a family of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper J Subscript t"> <mml:semantics> <mml:msub> <mml:mi>J</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">J_t</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -holomorphic maps depending continuously on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t"> <mml:semantics> <mml:mi>t</mml:mi> <mml:annotation encoding="application/x-tex">t</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We also prove the Oka–Weil theorem for sections of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper J"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="script">J</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathscr {J}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -holomorphic vector bundles on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Z equals upper T times upper X"> <mml:semantics> <mml:mrow> <mml:mi>Z</mml:mi> <mml:mo>=</mml:mo> <mml:mi>T</mml:mi> <mml:mo> × </mml:mo> <mml:mi>X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">Z=T\times X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the Oka principle for isomorphism classes of such bundles. The assumption on the family <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper J"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="script">J</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathscr {J}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is that the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper J Subscript t"> <mml:semantics> <mml:msub> <mml:mi>J</mml:mi> <mml:mi>t</mml:mi>