Bogomolov–Sommese vanishing theorem

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Short description: Theorem in algebraic geometry


In algebraic geometry, the Bogomolov–Sommese vanishing theorem is a result related to the Kodaira–Itaka dimension. It is named after Fedor Bogomolov and Andrew Sommese. Its statement has differing versions:

Bogomolov–Sommese vanishing theorem for snc pair:[1][2][3][4] Let X be a projective manifold (smooth projective variety), D a simple normal crossing divisor (snc divisor) and [math]\displaystyle{ A \subseteq \Omega ^{p} _ {X} (\log D) }[/math] an invertible subsheaf. Then the Kodaira–Itaka dimension [math]\displaystyle{ \kappa(A) }[/math] is not greater than p.

This result is equivalent to the statement that:[5]

[math]\displaystyle{ H^{0}\left(X,A^{- 1} \otimes \Omega ^{p}_{X} (\log D) \right) = 0 }[/math]

for every complex projective snc pair [math]\displaystyle{ (X, D) }[/math] and every invertible sheaf [math]\displaystyle{ A \in \mathrm{Pic}(X) }[/math] with [math]\displaystyle{ \kappa(A) \gt p }[/math].

Therefore, this theorem is called the vanishing theorem.

Bogomolov–Sommese vanishing theorem for lc pair:[6][7] Let (X,D) be a log canonical pair, where X is projective. If [math]\displaystyle{ A \subseteq\Omega ^{[p]}_{X} (\log \lfloor D \rfloor) }[/math] is a [math]\displaystyle{ \mathbb{Q} }[/math]-Cartier reflexive subsheaf of rank one,[8] then [math]\displaystyle{ \kappa(A) \leq p }[/math].

See also

Notes

  1. ↑ (Michałek 2012)
  2. ↑ (Greb Kebekus)
  3. ↑ (Esnault Viehweg)
  4. ↑ (Kebekus 2013)
  5. ↑ (Graf 2015)
  6. ↑ (Greb Kebekus)
  7. ↑ (Kebekus 2013)
  8. ↑ (Greb Kebekus)

References

Further reading



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