Locally trivial deformations of toric varieties

We study locally trivial deformations of toric varieties from a combinatorial point of view. For any fan <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Sigma"> <mml:semantics> <mml:mi mathvariant="normal"> Σ </mml:mi> <mml:annotation encoding="application/x-tex">\Sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , we construct a deformation functor <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D e f Subscript normal upper Sigma"> <mml:semantics> <mml:msub> <mml:mi>Def</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal"> Σ </mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\operatorname {Def}_{\Sigma }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> by considering Čech zero-cochains on certain simplicial complexes. We show that under appropriate hypotheses, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D e f Subscript normal upper Sigma"> <mml:semantics> <mml:msub> <mml:mi>Def</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal"> Σ </mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\operatorname {Def}_{\Sigma }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is isomorphic to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D e f prime Subscript upper X Sub Subscript normal upper Sigma"> <mml:semantics> <mml:msubsup> <mml:mi>Def</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>X</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal"> Σ </mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:mo>′</mml:mo> </mml:msubsup> <mml:annotation encoding="application/x-tex">\operatorname {Def}’_{X_{\Sigma }}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , the functor of locally trivial deformations for the toric variety <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X Subscript normal upper Sigma"> <mml:semantics> <mml:msub> <mml:mi>X</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal"> Σ </mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">X_{\Sigma }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> associated to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Sigma"> <mml:semantics> <mml:mi mathvariant="normal"> Σ </mml:mi> <mml:annotation encoding="application/x-tex">\Sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . In particular, for any complete toric variety <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> that is smooth in codimension <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</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="double-struck upper Q"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Q</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {Q}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -factorial in codimension <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3"> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding="application/x-tex">3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , there exists a fan <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Sigma"> <mml:semantics> <mml:mi mathvariant="normal"> Σ </mml:mi> <mml:annotation encoding="application/x-tex">\Sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D e f Subscript normal upper Sigma"> <mml:semantics> <mml:msub> <mml:mi>Def</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal"> Σ </mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\operatorname {Def}_{\Sigma }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is isomorphic to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D e f Subscript upper X"> <mml:semantics> <mml:msub> <mml:mi>Def</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>X</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\operatorname {Def}_{X}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , the functor of deformations of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X">