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[FTheoryTools] Fix LaTeX super indices in documentation
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HereAround committed Nov 1, 2024
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8 changes: 4 additions & 4 deletions experimental/FTheoryTools/src/G4Fluxes/special_attributes.jl
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By virtue of Theorem 12.4.1 in [CLS11](@cite), one can compute a monomial
basis of $H^4(X, \mathbb{Q})$ for a simplicial, complete toric variety $X$
by truncating its cohomology ring to degree $2$. Inspired by this, this
method identifies a basis of $H^(2,2)(X, \mathbb{Q})$ by multiplying
method identifies a basis of $H^{(2,2)}(X, \mathbb{Q})$ by multiplying
pairs of cohomology classes associated with toric coordinates.
By definition, $H^(2,2)(X, \mathbb{Q})$ is a subset of $H^(4)(X, \mathbb{Q})$.
By definition, $H^{(2,2)}(X, \mathbb{Q})$ is a subset of $H^{4}(X, \mathbb{Q})$.
However, by Theorem 9.3.2 in [CLS11](@cite), for complete and simplicial
toric varieties and $p \neq q$ it holds $H^(p,q)(X, \mathbb{Q}) = 0$. It follows
that for such varieties $H^(2,2)(X, \mathbb{Q}) = H^(4)(X, \mathbb{Q})$ and the
toric varieties and $p \neq q$ it holds $H^{(p,q)}(X, \mathbb{Q}) = 0$. It follows
that for such varieties $H^{(2,2)}(X, \mathbb{Q}) = H^4(X, \mathbb{Q})$ and the
vector space dimension of those spaces agrees with the Betti number $b_4(X)$.
Note that it can be computationally very demanding to check if a toric variety
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