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Let $k$ be a perfect field of positive characteristic. Let $X$ be a smooth projective geometrically connected $k$-scheme with a $k$-point.

Can we reconstruct $X$ from its small crystalline topos $((X/W(k))_{\mathrm{cris}}, \mathcal{O}_{X/W(k)})$ considered with the structure morphism to $((\mathrm{Spec}\:k/W(k))_{\mathrm{cris}}, \mathcal{O}_{\mathrm{Spec}\:k/W(k)})$?

If not areCan we at least find the Hodge numbers uniquely determined?

Let $k$ be a perfect field of positive characteristic. Let $X$ be a smooth projective geometrically connected $k$-scheme with a $k$-point.

Can we reconstruct $X$ from its small crystalline topos $((X/W(k))_{\mathrm{cris}}, \mathcal{O}_{X/W(k)})$ considered with the structure morphism to $((\mathrm{Spec}\:k/W(k))_{\mathrm{cris}}, \mathcal{O}_{\mathrm{Spec}\:k/W(k)})$?

If not are the Hodge numbers uniquely determined?

Let $k$ be a perfect field of positive characteristic. Let $X$ be a smooth projective geometrically connected $k$-scheme with a $k$-point.

Can we reconstruct $X$ from its small crystalline topos $((X/W(k))_{\mathrm{cris}}, \mathcal{O}_{X/W(k)})$ considered with the structure morphism to $((\mathrm{Spec}\:k/W(k))_{\mathrm{cris}}, \mathcal{O}_{\mathrm{Spec}\:k/W(k)})$?

Can we at least find the Hodge numbers?

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Reconstruct a variety from its crystalline topos

Let $k$ be a perfect field of positive characteristic. Let $X$ be a smooth projective geometrically connected $k$-scheme with a $k$-point.

Can we reconstruct $X$ from its small crystalline topos $((X/W(k))_{\mathrm{cris}}, \mathcal{O}_{X/W(k)})$ considered with the structure morphism to $((\mathrm{Spec}\:k/W(k))_{\mathrm{cris}}, \mathcal{O}_{\mathrm{Spec}\:k/W(k)})$?

If not are the Hodge numbers uniquely determined?