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Philosophy. If $f \colon X \dashrightarrow Y$ is a birational map, consider the cycles $\bar \Gamma_f \subseteq X \times Y$ and $\bar \Gamma_{f^{-1}} \subseteq Y \times X$. Then $\bar\Gamma_{f^{-1}} \circ \bar \Gamma_f$ and $\bar\Gamma_f \circ \bar\Gamma_{f^{-1}}$ differ from the identity by a cycle supported on $D \times D$ and $E \times E$ respectively, where $D \subseteq X$ and $E \subseteq Y$ are divisors. (For a proof, see e.g. this answer.)

Chow group of zero-cycles. The group $\operatorname{CH}_0(X)$ is a birational invariant over any field. Indeed, it suffices to show that a cycle supported on $D \times D$ induces the zero map on $\operatorname{CH}_0(X)$. This is because we can move any zero-cycle to a cycle not meeting $D$. See this answer for more details.

Philosophy. If $f \colon X \dashrightarrow Y$ is a birational map, consider the cycles $\bar \Gamma_f \subseteq X \times Y$ and $\bar \Gamma_{f^{-1}} \subseteq Y \times X$. Then $\bar\Gamma_{f^{-1}} \circ \bar \Gamma_f$ and $\bar\Gamma_f \circ \bar\Gamma_{f^{-1}}$ differ from the identity by a cycle supported on $D \times D$ and $E \times E$ respectively, where $D \subseteq X$ and $E \subseteq Y$ are divisors. (For a proof, see e.g. this answer.)

Chow group of zero-cycles. The group $\operatorname{CH}_0(X)$ is a birational invariant over any field. Indeed, it suffices to show that a cycle supported on $D \times D$ induces the zero map on $\operatorname{CH}_0(X)$. This is because we can move any zero-cycle to a cycle not meeting $D$. See this answer for more details.

It follows from Berthelot's work that the coniveau $r$ part has slopes $\geq r$ (see e.g. Lemma 2.1 of this paper by Hélène Esnault). Thus, the slope $[0,1)$ part is a birational invariant (hence by slope symmetry, so is the slope $(k-1,k]$ part).

Explicitly, this says that the multiset of Frobenius-eigenvalues not divisible by $q = |k|$ is a birational invariant. This was actually proven in an elementary way by Ekedahl in 1983 (Sur le groupe fondamental d'une variété unirationelle).

It follows from Berthelot's work that the coniveau $r$ part has slopes $\geq r$ (see e.g. Lemma 2.1 of this paper by Hélène Esnault; the argument is identical to the one given above in the Hodge realisation: use purity of the Gysin maps). Thus, the slope $[0,1)$ part is a birational invariant (hence by slope symmetry, so is the slope $(k-1,k]$ part).

Explicitly, this says that the multiset of Frobenius-eigenvalues not divisible by $q = |k|$ is a birational invariant. This was actually proven in an elementary way by Ekedahl in 1983 (Sur le groupe fondamental d'une variété unirationelle).

Hodge realisation. In the Hodge realisation, the coniveau $r$ part of the cohomology will land inside $$H^{k-r,r}(X) \oplus \ldots \oplus H^{r,k-r}(X) \subseteq H^k(X,\mathbb C).$$ This shows that $H^0(X,\Omega^k_X)$ and $H^k(X,\mathcal O_X)$ are birational invariants: $(\bar\Gamma_{f^{-1}} \circ \bar\Gamma_f) _* = \operatorname{Id} + Z$, with $Z$ supported on $D \times D$. By the lemma above, $Z_*$ maps everything into the coniveau $1$ part, so it acts as $0$ on $H^k(X,\mathcal O_X) \oplus H^0(X,\Omega^k_X)$. Hence, the composition $$H^k(X,\mathcal O_X) \to H^k(Y,\mathcal O_Y) \to H^k(X,\mathcal O_X)$$ is the identity, and similarly for the opposite composition, as well as for $H^0(-,\Omega^k)$. $\square$

Frobenius realisation. Over a finite field $k$, we have the slope filtration on crystalline cohomology (after inverting $p$), which is the closest analogue to the Hodge decomposition on de Rham cohomology. This gives a filtration on $$H^k_{\operatorname{crys}}(X/K) = H^k_{\operatorname{crys}}(X/W(k))[\tfrac{1}{p}]$$ whose successive subquotients are the spaces where Frobenius acts with slope $[i,i+1)$. These spaces can alternatively be described (cf. Illusie) as the de Rham-Witt cohomology $$H^k_{\operatorname{crys}}(X/K)_{[i,i+1)} = H^i(X,W\Omega^{k-i}_X)[\tfrac{1}{p}],$$ which is why it is similar to the Hodge decomposition of de Rhram cohomology in characteristic $0$.

It follows from Berthelot's work that the coniveau $r$ part has slopes $\geq r$ (see e.g. Lemma 2.1 of this paper by Hélène Esnault). Thus, the slope $[0,1)$ part is a birational invariant (hence by slope symmetry, so is the slope $(k-1,k]$ part). This was actually proven in an elementary way by Ekedahl in 1983 (Sur le groupe fondamental d'une variété unirationelle).

Explicitly, this says that the multiset of Frobenius-eigenvalues not divisible by $q = |k|$ is a birational invariant.

Hodge realisation. In the Hodge realisation, the coniveau $r$ part of the cohomology will land inside $$H^{k-r,r}(X) \oplus \ldots \oplus H^{r,k-r}(X) \subseteq H^k(X,\mathbb C).$$ Indeed, an element of $H^k(X,\mathbb C)$ that vanishes on $X-Y$ with $\operatorname{codim}(Y,X) = r$ comes from the cohomology $H^{k-2r}(Y,\mathbb C)$ under the Gysin map, at least when $Y$ is smooth (in general, replace $Y$ by a resolution $\tilde Y \to Y$). This is a morphism of Hodge structures of bidegree $(r,r)$, which proves the claim.

This shows that $H^0(X,\Omega^k_X)$ and $H^k(X,\mathcal O_X)$ are birational invariants: $(\bar\Gamma_{f^{-1}} \circ \bar\Gamma_f) _* = \operatorname{Id} + Z$, with $Z$ supported on $D \times D$. By the lemma above, $Z_*$ maps everything into the coniveau $1$ part, so it acts as $0$ on $H^k(X,\mathcal O_X) \oplus H^0(X,\Omega^k_X)$. Hence, the composition $$H^k(X,\mathcal O_X) \to H^k(Y,\mathcal O_Y) \to H^k(X,\mathcal O_X)$$ is the identity, and similarly for the opposite composition, as well as for $H^0(-,\Omega^k)$. $\square$

Frobenius realisation. Over a finite field $k$, we have the slope filtration on crystalline cohomology (after inverting $p$), which is the closest analogue to the Hodge decomposition on de Rham cohomology. This gives a filtration on $$H^k_{\operatorname{crys}}(X/K) = H^k_{\operatorname{crys}}(X/W(k))[\tfrac{1}{p}]$$ whose successive subquotients are the spaces where Frobenius acts with slope $[i,i+1)$. These spaces can alternatively be described (cf. Illusie) as the de Rham-Witt cohomology $$H^k_{\operatorname{crys}}(X/K)_{[i,i+1)} = H^i(X,W\Omega^{k-i}_X)[\tfrac{1}{p}],$$ which is why it is similar to the Hodge decomposition of de Rham cohomology in characteristic $0$.

It follows from Berthelot's work that the coniveau $r$ part has slopes $\geq r$ (see e.g. Lemma 2.1 of this paper by Hélène Esnault). Thus, the slope $[0,1)$ part is a birational invariant (hence by slope symmetry, so is the slope $(k-1,k]$ part).

Explicitly, this says that the multiset of Frobenius-eigenvalues not divisible by $q = |k|$ is a birational invariant. This was actually proven in an elementary way by Ekedahl in 1983 (Sur le groupe fondamental d'une variété unirationelle).