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Classical algebraic geometry is full of ingenious constructions and miraculous coincidences: 27 lines on a cubic surface are related to Weyl lattice of type $E_6,$ lines on an intersection of four-dimensional quadrics are parametrized by a Jacobian of genus two curve and so on.

Existence of some constructions of this type may be predicted by simple cohomological computations. For instance, consider a smooth intersection $X$ of two quadrics $Q_1$ and $Q_2$ in $\mathbb{P}^5.$ This example is explained in detail in the beautiful paper The complete intersection of two or more quadrics of Miles Reid. It is easy to see that $h^{p,q}(X)$ equals to $1$ for $$(p,q)\in \{(0,0),\ (1,1),\ (2,2),\ (3,3)\},$$ equals to 2 for $$(p,q)\in \{(1,2),\ (2,1)\}$$ and equals to $0$ in other cases. Generalized Hodge Conjecture predicts that there exists a certain curve $C$ and a correspondence $D\subset C\times X$ which induces a surjective morphism $$H^1(C,\mathbb{Z})\longrightarrow H^3(X,\mathbb{Z})(-3).$$ Explicit construction of this correspondence is based on an identification of the Hilbert scheme of lines on $X$ with Jacobian $J(C).$

Question 1: What other classical constructions in algebraic geometry can be a posteriori motivated by simple computations with (mixed) Hodge structures?

Question 2: It will be very interesting to see some concrete examples, when such a construction is expected to exist, but has not been worked out yet.

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  • $\begingroup$ I don't know about the durability of Semantic Scholar links, but this paper is also on Reid's home page: Reid - The complete intersection of two or more quadrics. $\endgroup$ – LSpice Aug 15 '19 at 11:39
  • $\begingroup$ There are many examples involving cubic fourfolds - see arxiv.org/abs/1601.05501. $\endgroup$ – dhy Aug 15 '19 at 13:26
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    $\begingroup$ As far as "Mathematics" (with a big M) is concerned, I believe things go the other way round. People usually first notice a more or less simple phenomenon occuring in some examples (geometric for instance). Then, they try to simplify and generalize it using more abstract mathematical structures (say Hodge structures for instance). This is at least the way Grothendieck thought about Algebraic Geometry. $\endgroup$ – Libli Aug 15 '19 at 14:37
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    $\begingroup$ Although it doesn't involve Hodge structures, maybe the following sort of situation interests you? There is a simply defined, natural group action on cohomology, and one suspects that it arises from a group action on the variety, but the latter is unknown. If so, then a concrete example is Tymoczko's "dot action" on the cohomology of certain subvarieties of the flag variety (Hessenberg varieties), which in general is not known to come from a group action on the varieties themselves. See arxiv.org/abs/0706.0460 especially Section 5.2. $\endgroup$ – Timothy Chow Aug 16 '19 at 2:25
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    $\begingroup$ @Libli I guess, agree with you in historical perspective. On the other hand, now for many people cohomology, MHS and motives are far more familiar objects than «quadratic line complexes», polar correspondence etc. The goal of this question is to collect a few examples of that. $\endgroup$ – Daniil Rudenko Aug 17 '19 at 22:27
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Cohomology, with the intersection form, can give interesting lattices. Geometry tells us that they contain many elements of square length $2$: they are given by vanishing cycles in a Lefschetz pencil. This also applies to Milnor fibers of deformations of isolated singularities. This is how one recognizes the $E_n$ in the orthogonal of the canonical class in the $\text{H}^2$ of $\mathbb{P}^2$ with $n$ points blown up ($n\le8$). Here, $n=6$ is the cubic surfaces case; the affine case $n=9$ when one blows up the intersection of two cubic curves is interesting too.

When a $\text{H}^{2n+1}$ is of type $\{(2n+1,0),(0,2n+1)\}$, one expects a principally polarized abelian variety is lurking around. A Jacobian? A Prym variety? For complete intersections in $\mathbb{P}^N$ there is a table in SGA 7 telling us when this Hodge level one case occurs. I do not know whether all have been unraveled.

Similarly, cubic fourfolds look very much like K3 surfaces (with cohomology one bigger). It follows that there are related Kuga-Satake abelian varieties attached to them (whose $\text{H}^2(-1)$ contains their $\text{H}^4$). This allowed to prove the Weil conjecture for them (as for the K3) early on, but remains quite unexplicit. The Milnor fibers story relates to which quadratic singularities one can have, and when many are imposed, the related abelian variety reduces to a lower dimensional one, possibly easier to see (case of Kummer surfaces among K3's).

A different game is guessing periods of differential forms ($\zeta(3)$ related to an extension of $\mathbb{Z}$ by $\mathbb{Z}(3)$, $\ldots$).

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    $\begingroup$ Should "$2n=1$" be "$2n+1$"? $\endgroup$ – Noah Schweber Aug 31 '19 at 0:22

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