Timeline for When does the rational Hodge structure determine the integral Hodge structure?

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May 9 at 15:32	comment	added	Will Sawin		@NickAddington One further observation: One can study the failure of injectivity in general using results about the Hodge locus, since an isomorphism of rational Hodge structures between $H^k(X)$ and $H^k(Y)$ gives a Hodge class on $X \times Y$ (or more precisely in $H^k(X)^\vee \otimes H^k(Y)$). There is some subtlety since the converse is not always true but some results in this direction should be helpful.
May 9 at 15:24	comment	added	Nick Addington		Ah ha, so I did.
May 9 at 6:58	comment	added	Will Sawin		@NickAddington Well, you said smooth projective complex variety, and the K3s in the moduli space of unpolarized K3s are neither projective nor varieties.
May 9 at 1:44	comment	added	Nick Addington		By the way, you can make the moduli space of unpolarized K3s: take the moduli space of marked K3s, discussed in Huybrechts' book section 7.2, and then divide by automorphisms of the lattice. But neither the marked moduli space nor the quotient is Hausdorff.
May 9 at 1:44	comment	added	Nick Addington		This is great, thanks! So it's important to focus on the map from a moduli space of varieties to either period domain, and not about the map between the two period domains.
May 9 at 1:40	vote	accept	Nick Addington
May 9 at 0:24	history	answered	Will Sawin	CC BY-SA 4.0