Vaught’s conjecture for varieties

Abstract

We prove that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper V"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">V</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {V}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a superstable variety or one with few countable models then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper V"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">V</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {V}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the varietal product of an affine variety and a combinatorial variety. Vaught’s conjecture for varieties is an immediate consequence.

Locations

• Transactions of the American Mathematical Society - View - PDF