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Bogomolov–Sommese vanishing theorem

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Theorem in algebraic geometry Not to be confused with Le Potier's vanishing theorem.

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: Let X be a projective manifold (smooth projective variety), D a simple normal crossing divisor (snc divisor) and A ⊆ Ω X p ( log ⁡ D ) {\displaystyle A\subseteq \Omega _{X}^{p}(\log D)} an invertible subsheaf. Then the Kodaira–Itaka dimension κ ( A ) {\displaystyle \kappa (A)} is not greater than p.

This result is equivalent to the statement that:

H 0 ( X , A − 1 ⊗ Ω X p ( log ⁡ D ) ) = 0 {\displaystyle H^{0}\left(X,A^{-1}\otimes \Omega _{X}^{p}(\log D)\right)=0}

for every complex projective snc pair ( X , D ) {\displaystyle (X,D)} and every invertible sheaf A ∈ P i c ( X ) {\displaystyle A\in \mathrm {Pic} (X)} with κ ( A ) > p {\displaystyle \kappa (A)>p} .

Therefore, this theorem is called the vanishing theorem.

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

See also

Notes

  1. (Michałek 2012)
  2. (Greb, Kebekus & Kovács 2010)
  3. (Esnault & Viehweg 1992, Corollary 6.9)
  4. (Kebekus 2013, Theorem 2.17)
  5. (Graf 2015)
  6. (Greb et al. 2011, Theorem 7.2)
  7. (Kebekus 2013, Corollary 4.14)
  8. (Greb et al. 2011, Definition 2.20.)

References

Further reading

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