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In the paper "A new cohomology theory for orbifold" by Chen/Ruan, they define the orbifold cohomology group of an orbifold $X$ by

$H^d_{orb}(X)=\bigoplus_{(g) \in T} H^{d-2\iota_{(g)}}(X_{(g)})$

where $H^\ast(X_{(g)})$ is the singular cohomology of $X_{(g)}$ with real coefficients and $\iota(g)$ is a rational number.

Also, the orbifold Dolbeault cohomology group is defined by

$H^{p,q}_{orb}(X)=\bigoplus_{(g) \in T}H^{p-\iota_{(g)},q-\iota_{(g)}}(X_{(g)})$.

How are these cohomologies with rational coefficients defined?

EDIT: In Adem/Leida/Ruan's "Orbifolds and stringy topology" they similarly define the Chen-Ruan cohomology group for orbifold groupoid as

$H^d_{CR}=\bigoplus_{(g) \in T}H^d(X_{(g)})[-2\iota_{(g)}]=\bigoplus_{(g) \in T}H^{d-2\iota_{(g)}}(X_{(g)})$,

but I am also unfamiliar with the notation $H^d(X_{(g)})[-2\iota_{(g)}]$.

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It is my understanding that $H^d$ is taken to be trivial unless $d$ is an integer. Hence this is just notation for a degree shift by a rational number.

Similarly, in the other formula, the square bracket notation is also just a degree-shift.

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  • $\begingroup$ In the paper, p. 27 they calculate the numbers $h^{d}=1$ for the weighted projective space with $d$ rational. $\endgroup$ Mar 16, 2015 at 19:34
  • $\begingroup$ There are two different cohomology theories going on. This (p. 27) is a calculation of the orbifold cohomology with $d$ rational. Elsewhere, the orbifold cohomology is defined in terms of ordinary cohomology, and that is extended from integer gradings to rational gradings by taking it to be trivial for non-integer degrees. So the theory they define has interesting behaviour in rational degrees, even though the theory it's defined in terms of doesn't. $\endgroup$ Mar 16, 2015 at 22:06

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