When simple cohomological computations predict ingenious algebro-geometric constructions?

Question

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.

Answer 1

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 $\mathbb{Z}(3)$, $\ldots$).