Outputs: matrix representations of KT\Phi_{\mathcal K}|_{T}KT for candidate kernels KDb(YY)\mathcal K\in D^b(Y\times Y)KDb(YY), and the induced vectors in TTT\otimes TTT; a rank test for whether a collection of such candidates spans the full TTT\otimes TTT.
Steps:
a. Fix bases. Choose an explicit integral basis of H2(Y,Q)H^2(Y,\mathbb Q)H2(Y,Q) adapted to the decomposition H2=NSTH^2=\operatorname{NS}\oplus TH2=NST; for singular K3s the latter is rank 222. Denote basis vectors {n1,...,n,t1,t2}\{n_1,\dots,n_\rho, t_1,t_2\}{n1,...,n,t1,t2}.
b. Compute Chern characters. For the kernel P\mathcal PP (or kernel approximations), compute ch(P)\operatorname{ch}(\mathcal P)ch(P) in H(YM)H^\ast(Y\times M)H(YM). In practice for moduli of sheaves this is often described by a universal sheaf (or twisted universal family) whose Chern character is known symbolically (Mukai vector data). For YYY a K3 many terms in the Todd class simplify.
c. Form cohomological transform. Implement
PH()=pM(pY()ch(P)td(YM)).\Phi_{\mathcal P}^H(\alpha) \;=\; p_{M*}\big( p_Y^*(\alpha)\cup \operatorname{ch}(\mathcal P)\cup \sqrt{\mathrm{td}(Y\times M)} \big).PH()=pM(pY()ch(P)td(YM)).
Practically, this reduces to computing cup products and pushforwards (integrations along fibers) --- operations expressible linearly once bases are fixed.
d. Compose to return to YYY. If Q\mathcal QQ is the quasi-inverse kernel or the transpose, compute K=QP\mathcal K=\mathcal Q\star \mathcal PK=QP and derive KH=QHPH\Phi_{\mathcal K}^H = \Phi_{\mathcal Q}^H \circ \Phi_{\mathcal P}^HKH=QHPH. For autoequivalences one may work with K\mathcal KK directly.
e. Extract H2H^2H2 action. Compute the restriction of KH\Phi_{\mathcal K}^HKH to H2(Y)H^2(Y)H2(Y) and represent it as a rational matrix relative to the NST basis. Extract the lower-right 222\times222 block representing KT\Phi_{\mathcal K}|_TKT.
f. Form induced vectors in TTT\otimes TTT. The cohomology class of K\mathcal KK projected to TTT\otimes TTT corresponds to symmetric tensors coming from the entries of KT\Phi_{\mathcal K}|_TKT. Convert the 222\times222 matrices into 4-dimensional column vectors (e.g. flatten or take symmetric combinations) that live in the modeled TTT\otimes TTT vector space.
g. Test span. For a collection of candidate kernels K1,...,Km\mathcal K_1,\dots,\mathcal K_mK1,...,Km compute the matrix whose columns are these 4-vectors and check its rank. Rank =4=4=4 the FM candidates (heuristically) span TTT\otimes TTT.
h. Sanity checks. Verify that the transforms respect Mukai pairings / Hodge structures (for derived equivalences they should give Hodge isometries), and check compatibility with known invariants (determinants, traces).
Implementation notes:
For explicit K3 examples (Fermat quartic, Shioda--Inose models) much of the Mukai vector / universal family data is tabulated in the literature; these cases are the best starting points.
Symbolic algebra can be used for exact rational matrices; for more complicated kernels numeric approximations (period computations) can be used, though losing proof.
Libraries: use SageMath for lattice and cohomology manipulation; use sage + sympy or magma for exact linear algebra and Smith normal form.
4. Example archetypes where FM is effective (briefly)
K3 moduli of sheaves on K3: Mukai showed that certain moduli spaces MMM of stable sheaves on a K3 are themselves hyperkhler (often K3-type) and the universal family gives a derived equivalence. The induced cohomological action can be computed via Mukai vectors; these are classical settings where FM correspondences can and do act nontrivially on TTT.
Derived autoequivalences (spherical twists, P-twists): such autoequivalences have kernels whose classes are algebraic and whose action on H2H^2H2 can be nontrivial; they therefore generate candidate correspondences.
Compositions of FM kernels: composing transforms between YYY and moduli spaces can give highly nontrivial endomorphisms of H2(Y)H^2(Y)H2(Y) that are subtle enough to reach transcendental tensors.
5. Principal challenges & how CAS-6 modeling helps address them
(i) Existence of universal families. For some moduli spaces a universal family does not exist globally (only twisted families exist). When only a twisted universal sheaf P\mathcal PP exists, the cohomological transform still exists (using Brauer classes), but one must work with twisted Chern characters; the computational recipe adapts but is more delicate.
(ii) Algebraicity vs. Hodge-theoretic transform. The FM cohomological transform is Hodge-theoretic by construction, but showing that the corresponding class in H4(YY)H^4(Y\times Y)H4(YY) arises from an algebraic cycle sometimes needs extra input (though often the kernel is algebraic, so the class is algebraic). CAS-6 keeps this distinction explicit: we only accept an FM candidate as restoring closure if the kernel is algebraic (or known to be algebraic/motivated).
(iii) Rationality issues. Coefficients coming from ch(P)\operatorname{ch}(\mathcal P)ch(P) may involve denominators; one must check the rationality of the induced matrix entries. CAS-6 modeling uses exact rational linear algebra to detect whether the induced vectors truly live in the rational TTT\otimes TTT subspace.
(iv) Computational complexity. Computing push--pull integrals of Chern characters can be heavy; however, for K3s and small moduli spaces the relevant integrals reduce to manageable intersection pairings in the Mukai lattice.
CAS-6 helps by turning the problem into a finite, checkable linear-algebra task once the cohomological transforms are computed: it tells us exactly what numerical test to run (rank of T-block images) and what success looks like (rank 4).
6. Concrete CAS-6 experimental program using FM
