On Calabi-Yau supermanifolds

  title={On Calabi-Yau supermanifolds},
  author={Martin Ro{\vc}ek},
We prove that a Kähler supermetric on a supermanifold with one complex fermionic dimension admits a super Ricci-flat supermetric if and only if the bosonic metric has vanishing scalar curvature. As a corollary, it follows that Yau’s theorem does not hold for supermanifolds. Calabi[1] proposed that if a Kähler manifold has vanishing first Chern class, that is, the Ricci-form obeys Rij̄(g) = ∂iv̄j − ∂̄jvi for a globally defined 1-form v, or, equivalently, a complex n-dimensional Kähler manifold… CONTINUE READING
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