Hyperflächensingularitäten/Zweidimensional/Quasihomogen/Zusätzliche Differentialform/Kohomologieklasse/Fakt/Beweis

Beweis

Es ist

${}D={\frac {-\delta _{2}ydx+\delta _{1}xdy}{\partial _{3}(F)}}={\frac {\delta _{3}zdx-\delta _{1}xdz}{\partial _{2}(F)}}={\frac {-\delta _{3}zdy+\delta _{2}ydz}{\partial _{1}(F)}}\,$

Die Differenz der lokalen Urbilder von ${}D$ ist

${}{\begin{aligned}{\frac {-\delta _{2}ydx+\delta _{1}xdy}{\partial _{3}(F)}}-{\frac {\delta _{3}zdx-\delta _{1}xdz}{\partial _{2}(F)}}&={\frac {{\left(-\delta _{2}ydx+\delta _{1}xdy\right)}\partial _{2}(F)-{\left(\delta _{3}zdx-\delta _{1}xdz\right)}\partial _{3}(F)}{\partial _{2}(F)\partial _{3}(F)}}\\&={\frac {{\left(-\delta _{2}y\partial _{2}(F)-\delta _{3}z\partial _{3}(F)\right)}dx+\delta _{1}x\partial _{2}(F)dy+\delta _{1}x\partial _{3}(F)dz}{\partial _{2}(F)\partial _{3}(F)}}\\&={\frac {\delta _{1}x\partial _{1}(F)dx+\delta _{1}x\partial _{2}(F)dy+\delta _{1}x\partial _{3}(F)dz}{\partial _{2}(F)\partial _{3}(F)}}.\end{aligned}}$

Dies ist das Bild von

${\frac {\delta _{1}x}{\partial _{2}(F)\partial _{3}(F)}}.$

Zur bewiesenen Aussage