a. Select target YYY: a singular K3 or a K3 with well-studied moduli of sheaves.
b. Gather FM data: identify moduli spaces MMM and universal kernels P\mathcal PP (or derived autoequivalences). Extract Mukai vectors.
c. Compute KT\Phi_{\mathcal K}|_TKT for kernels K\mathcal KK obtained by composing P\mathcal PP with its adjoint, or by using known autoequivalences.
d. Form TTT\otimes TTT vectors and test span; if span = 4, flag success and seek references/proofs that the kernels/classes are algebraic (Mukai/Huybrechts/others).
e. If unsuccessful, iterate: enlarge candidate set (other moduli, different stability conditions), consider twisted kernels, or add arithmetic specializations (CM/Kuga--Satake) to increase the chance of algebraicity.
Summary
Fourier--Mukai transforms and derived equivalences are central, principled sources of non-trivial algebraic correspondences that can act on the transcendental Hodge structure.
From the CAS-6 viewpoint they are precisely the interaction motifs needed to fill orphan topological nodes: they supply algebraic weights (via cohomological transforms), are often deformation-stable, and produce explicit geometric outputs (kernel classes in H4(YY)H^4(Y\times Y)H4(YY)).
Practically, integrating FM into CAS-6 reduces the transcendental realization problem to a finite cohomological computation (compute KT\Phi_{\mathcal K}|_TKT, test rank). When such computations succeed, they produce strong heuristic evidence --- and in many cases rigorous proof --- that the missing Hodge classes are algebraic.
C. Computational experiments for identifying candidate cycles (practical program)
This subsection gives a concrete, reproducible experiment plan (algorithms, checks, software recommendations, expected outputs) to search for algebraic correspondences that fill the transcendental block T(Y)T(Y)T(Y)\otimes T(Y)T(Y)T(Y) on a chosen K3 YYY. The plan turns the CAS-6 diagnostics into explicit computations: build explicit NS/T bases, model the H2,2H^{2,2}H2,2 space, represent candidate correspondences as cohomological operators, project their classes to TTT\otimes TTT, and test linear independence (rank = 4). Everything below is actionable and written so you (or I, on your go-ahead) can run it in Sage/Magma/Python.
1. Goals (succinct)
a. Produce an explicit rational model of
H2,2(YY)H4(YY,Q)H^{2,2}(Y\times Y)\cap H^4(Y\times Y,\mathbb Q)H2,2(YY)H4(YY,Q)
with decomposition into NSNS, NST TNS, and TT blocks.
b. For a library of candidate correspondences ZZZ (diagonal variants, graphs of automorphisms, FM kernels, Shioda--Inose-induced cycles), compute the cohomology class [Z][Z][Z] and its projection to TTT\otimes TTT.
c. Test whether the projected vectors span the entire 4-dimensional TTT\otimes TTT. If yes strong heuristic evidence that those correspondences restore CAS-6 closure; if no iterate with further candidates.
2. Required ingredients / data
A concrete K3 surface YYY with explicit NS generators and Gram matrix (e.g., Fermat quartic or a Shioda--Inose singular K3). (We already identified sources.)
An explicit integral basis for H2(Y,Q)H^2(Y,\mathbb Q)H2(Y,Q) adapted to decomposition NS(Y)T(Y)\operatorname{NS}(Y)\oplus T(Y)NS(Y)T(Y). For singular K3s dimT=2\dim T=2dimT=2.
Candidate correspondences and their geometric description:
diagonal / small diagonal / corrected diagonal;
graphs \Gamma_\sigma of automorphisms \sigma (when available);
FM kernels / composed kernels K\mathcal KK coming from universal families (Mukai vectors);
Shioda--Inose pushforwards from a Kummer/abelian surface.
Software: SageMath (ideal), Magma (if available), or Python with sympy/numpy for rational/numeric linear algebra. For heavy lattice work Sage/Magma recommended.
3. High-level algorithm (step-by-step)
Step 0 --- Choose YYY and load NS data
Choose YYY (recommend: Fermat quartic or specific Shioda--Inose example).
Load published NS generator list and Gram matrix GNSG_{\mathrm{NS}}GNS (or compute from geometry).
Using lattice routines, compute a Z\mathbb ZZ-basis {n1,...,n20}\{n_1,\dots,n_{20}\}{n1,...,n20} for NS(Y)\operatorname{NS}(Y)NS(Y) and its Gram matrix.
Step 1 --- Compute T(Y)T(Y)T(Y) (orthogonal complement)
Let K3\Lambda_{\mathrm{K3}}K3 be the standard K3 lattice. Embed GNSG_{\mathrm{NS}}GNS primitively and compute the orthogonal complement TTT (rank 2 for =20\rho=20=20). Output basis {t1,t2}\{t_1,t_2\}{t1,t2} and Gram matrix GTG_TGT.
Verify: signature, determinant (compare literature).
Step 2 --- Build basis of H2,2(YY)H^{2,2}(Y\times Y)H2,2(YY)
Use Knneth: a convenient basis for H4(YY,Q)H^4(Y\times Y,\mathbb Q)H4(YY,Q) is the tensor products eieje_i\otimes e_jeiej where {e}\{e_\alpha\}{e} runs over basis of H2(Y,Q)H^2(Y,\mathbb Q)H2(Y,Q) plus two extreme factors H0H4H^0\otimes H^4H0H4 and H4H0H^4\otimes H^0H4H0.
Identify the subspace of Hodge type (2,2)(2,2)(2,2) (for K3K3 this is the full H^4 of interest minus degree shifts). Practically, restrict to H2H2H^2\otimes H^2H2H2 block plus the two extremes.
Reorder basis so that the coordinates split as: NSNS (400), NST (40), TNS (40), TT (4), plus extremes (2).
Step 3 --- Represent cohomology pairing and push/pull operations
