# RCHM × Hodge Conjecture — Research Round 27
## Küchle $c5$ Period Resonance as a Finite Spectral Problem: HLYY Boundary, 12 BWB Slots, and Ancestry-Relative Strong K3 Type

**Status:** Internal research note + finite spectral certificate  
**Version:** Round 27 / v2.7  
**Date:** 2026-09-18  
**Methodological base:** RCHM; X-Layer Mathematical Completeness; Ancestry Reduction; Bridge Search Compiler  
**Previous rounds:** RCHM × Hodge Conjecture — Rounds 1–26  
**Canonical source:** UTF-8 Markdown

---

# 0. Round 27 Objective

Round 26 reduced the infinitesimal period problem for a Küchle $c5$ fourfold to one integer:

$$
\boxed{
\rho_{\mathrm{per}}
=
h^2\!\left(
Y,\Omega_Y^3(H)
\right),
}
$$

where:

$$
Y=X^5_{c5}
$$

is the parent Küchle fivefold.

The exact period formulas are:

$$
\boxed{
\dim\ker d\mathcal P
=
5+\rho_{\mathrm{per}},
}
$$

and:

$$
\boxed{
\operatorname{rank}d\mathcal P
=
20-\rho_{\mathrm{per}}.
}
$$

Round 27 attacks the remaining cohomology by two homogeneous methods:

1. a conormal/Koszul Borel--Weil--Bott spectral certificate on
   $$
   \operatorname{Gr}(3,7);
   $$
2. the homogeneous-vector-bundle Jacobian theorem of Huang--Lian--Yau--Yu.

The main result is not yet a proof that:

$$
\rho_{\mathrm{per}}=0.
$$

Instead, the unresolved problem is reduced to a **finite generic-rank problem supported on exactly twelve BWB resonance slots**.

This is the strongest safe closure currently available.

---

# 1. Geometry and Notation

Let:

$$
G=\operatorname{Gr}(3,7),
$$

with tautological rank-$3$ bundle:

$$
\mathcal U
$$

and rank-$4$ quotient:

$$
\mathcal Q.
$$

The parent Küchle fivefold is the zero locus:

$$
Y
=
Z(s)
\subset
G
$$

of a general section of the rank-$7$ homogeneous bundle:

$$
\boxed{
\mathcal E_5
=
\mathcal Q^\vee(1)
\oplus
\mathcal U(1).
}
$$

The Küchle fourfold is a further hyperplane section.

Equivalently, it is the zero locus of the rank-$8$ bundle:

$$
\boxed{
\mathcal E_4
=
\mathcal Q^\vee(1)
\oplus
\mathcal U(1)
\oplus
\mathcal O(1).
}
$$

The three components of a section have parameter spaces:

$$
\Lambda^4W^\vee,
\qquad
\Lambda^2W^\vee,
\qquad
\Lambda^3W^\vee,
$$

of dimensions:

$$
35,\quad21,\quad35.
$$

Thus:

$$
\boxed{
h^0(G,\mathcal E_4)=91.
}
$$

---

# 2. Round-26 Exact Index

The exact equivariant localization calculation gave:

$$
\boxed{
\chi
\left(
Y,\Omega_Y^3(H)
\right)
=
-5.
}
$$

It also verified:

$$
\chi(Y,\mathcal O_Y)=1,
$$

and:

$$
\chi(Y,\mathcal O_Y(H))=21.
$$

The three independent torus-weight tests agreed exactly.

This remains the global Euler constraint for every later spectral calculation.

---

# 3. Conormal--Koszul Spectral Certificate

Resolve:

$$
\Omega_Y^3(H)
$$

from the conormal sequence of:

$$
Y\subset G,
$$

and then resolve every restricted homogeneous bundle by the Koszul complex of:

$$
\mathcal E_5^\vee.
$$

Borel--Weil--Bott reduces every ambient cohomology group to a finite-dimensional representation of:

$$
GL(7).
$$

After deleting all zero BWB terms, the surviving first page has only total degrees:

$$
\boxed{
0,\quad1,\quad2.
}
$$

The total dimensions are:

$$
\boxed{
E_1^{\mathrm{tot}=0}:15,
}
$$

$$
\boxed{
E_1^{\mathrm{tot}=1}:68,
}
$$

and:

$$
\boxed{
E_1^{\mathrm{tot}=2}:48.
}
$$

Therefore:

$$
\boxed{
15-68+48=-5,
}
$$

exactly reproducing the equivariant index of Round 26.

This is an independent internal consistency check.

---

# 4. Detailed Sparse $E_1$ Support

Using the convention that the horizontal coordinate is:

$$
p<0,
$$

and total degree is:

$$
p+q,
$$

the nonzero BWB support is:

| $q$ | nonzero $p$ and dimensions |
|---|---|
| $2$ | $p=-1:28$ |
| $3$ | $p=-3:1$, $p=-1:35$ |
| $4$ | $p=-4:2$, $p=-3:5$ |
| $5$ | $p=-4:3$, $p=-3:4$ |
| $6$ | $p=-5:14$, $p=-4:1$ |
| $7$ | $p=-7:1$, $p=-6:7$, $p=-5:7$ |
| $8$ | $p=-8:3$, $p=-7:2$ |
| $9$ | $p=-9:8$, $p=-8:2$, $p=-7:1$ |
| $10$ | $p=-9:7$ |

The sums by total degree are:

$$
\boxed{
15,\quad68,\quad48.
}
$$

No other BWB terms survive.

Thus:

$$
H^2(Y,\Omega_Y^3(H))
$$

can only arise as homology of a finite collection of maps among these terms.

There is no hidden infinite tail.

---

# 5. Maximal-Rank Interpretation

The desired result:

$$
\rho_{\mathrm{per}}=0
$$

is equivalent to complete cancellation of the total-degree-$2$ sector by the incoming spectral differentials.

This is compatible with all dimension constraints.

No Euler-characteristic obstruction prevents maximal rank.

Thus the problem is now:

$$
\boxed{
\text{generic differential rank}
}
$$

rather than:

$$
\boxed{
\text{unknown cohomology}.
}
$$

This distinction matters.

---

# 6. HLYY Homogeneous Jacobian Theorem

Huang--Lian--Yau--Yu construct a generalized Jacobian description for zero loci of homogeneous vector-bundle sections.

For a rank-$r$ bundle:

$$
E
$$

on an $n$-fold, their Theorem 5.7 identifies:

$$
H_{\mathrm{var}}^{n-r-k,k}
$$

with a graded Jacobian-module quotient provided a finite set of ambient vanishings holds.

The relevant conditions are their equations:

$$
(5.2)
$$

and:

$$
(5.3).
$$

The important point for the present calculation is that these conditions are fully decidable by Borel--Weil--Bott on:

$$
\operatorname{Gr}(3,7).
$$

---

# 7. The $H^{3,1}$ Layer Passes

For the full fourfold bundle:

$$
\mathcal E_4
$$

we have:

$$
n=12,
\qquad
r=8.
$$

The:

$$
H^{3,1}
$$

layer corresponds to:

$$
k=1.
$$

The complete BWB applicability checker verifies all required HLYY vanishing conditions.

In four places the first alternative in condition $(5.2)$ is nonzero, but the second alternative vanishes exactly as allowed by the theorem.

Thus:

$$
\boxed{
k=1\text{ HLYY applicability}=PASS.
}
$$

This formally supports the one-dimensional:

$$
H^{3,1}
$$

Jacobian description.

---

# 8. The $H^{2,2}$ Layer Hits a Boundary

For:

$$
k=2,
$$

the same checker finds exactly:

$$
\boxed{
12
}
$$

instances where **both** alternatives in the HLYY vanishing condition are nonzero.

Therefore:

$$
\boxed{
k=2\text{ HLYY applicability}
=
\text{FAIL AS STATED}.
}
$$

This does not imply that a Jacobian description is false.

It means that the theorem cannot erase the residual ambient cohomology automatically.

Those twelve positions are the precise HLYY boundary for $c5$.

---

# 9. The Twelve Resonance Slots

The exact certificate records the nonvanishing pairs as:

## Family A

| slot | $(p,q,l,a)$ | first dimension | second dimension |
|---|---|---:|---:|
| A1 | $(18,9,6,1)$ | $204$ | $15$ |
| A2 | $(17,8,5,2)$ | $54$ | $9$ |
| A3 | $(16,7,4,3)$ | $45$ | $3$ |
| A4 | $(15,6,3,4)$ | $10$ | $1$ |
| A5 | $(14,5,3,5)$ | $4$ | $1$ |
| A6 | $(13,4,2,6)$ | $2$ | $1$ |

## Family B

| slot | $(p,q,l,a)$ | first dimension | second dimension |
|---|---|---:|---:|
| B1 | $(17,8,5,1)$ | $11$ | $1$ |
| B2 | $(17,8,6,1)$ | $148$ | $7$ |
| B3 | $(16,7,5,2)$ | $66$ | $2$ |
| B4 | $(15,6,4,3)$ | $12$ | $1$ |
| B5 | $(14,5,4,4)$ | $61$ | $8$ |
| B6 | $(13,4,3,5)$ | $12$ | $1$ |

These are **not** twelve dimensions of final cohomology.

They are twelve ambient BWB locations at which the automatic HLYY vanishing argument stops.

Their differentials still depend on the general section.

---

# 10. Why the Boundary Is Structurally Expected

Fatighenti--Mongardi explicitly classify the Küchle $c5$ fourfold as a **weak**, rather than strong, K3-type Fano.

They also warn that for high-index Fano varieties, residual ambient cohomology can contribute to the Griffiths-ring description.

This is exactly what appears here.

The:

$$
H^{3,1}
$$

edge is clean.

The:

$$
H^{2,2}
$$

edge contains a finite residual sector.

Thus the current calculation gives a concrete mechanism behind the word:

$$
\boxed{
\text{weak}.
}
$$

---

# 11. Generalized Jacobian Dimensions

For the full rank-$8$ bundle:

$$
\mathcal E_4,
$$

we have:

$$
\det\mathcal E_4\otimes K_G
\simeq
\mathcal O(-1).
$$

The HLYY graded modules give:

$$
\boxed{
M^8
=
H^0(G,\mathcal E_4(-1))
\simeq
\mathbb C,
}
$$

and:

$$
\boxed{
M^9
=
H^0
\left(
G,
S^2\mathcal E_4(-1)
\right)
}
$$

with:

$$
\boxed{
\dim M^9=91.
}
$$

Moreover:

$$
\boxed{
N^8=0,
}
$$

and:

$$
\boxed{
\dim N^9=17.
}
$$

The $91$ dimensions decompose naturally as:

$$
\boxed{
35+21+35,
}
$$

matching the deformation parameters:

$$
(\lambda,\mu,\nu).
$$

Thus the generalized Jacobian formalism sees the complete section-deformation space before quotienting automorphisms and residual relations.

---

# 12. The Twenty-Five-Dimensional Deformation Quotient

The total section space has dimension:

$$
91.
$$

The projective linear changes of the ambient $7$-space contribute:

$$
48
$$

infinitesimal directions.

The bundle/Jacobian vertical directions contribute the complementary relations encoded by:

$$
N^9.
$$

The resulting deformation dimension is:

$$
\boxed{
25,
}
$$

in agreement with:

$$
-\chi(T_X)=25
$$

on the generic finite-automorphism locus.

Thus the generalized Jacobian numerology and the deformation-theory numerology agree.

---

# 13. Hard Hodge Target

The total:

$$
H^{2,2}(X)
$$

has dimension:

$$
24.
$$

The full inherited Tate sector from the fivefold has rank:

$$
4.
$$

Therefore the ancestry-normalized hard target has dimension:

$$
\boxed{
20.
}
$$

This is exactly the dimension of the primitive period domain identified in Round 25.

Thus the infinitesimal period map is a map:

$$
\boxed{
\mathbb C^{25}
\longrightarrow
\mathbb C^{20}.
}
$$

The unavoidable kernel dimension is:

$$
5.
$$

---

# 14. Long Exact Sequence and the Four Inherited Classes

Let:

$$
F=\Omega_Y^3(H).
$$

Round 26 gave:

$$
h^1(F)=5+\rho_{\mathrm{per}},
$$

and:

$$
h^2(F)=\rho_{\mathrm{per}}.
$$

Let:

$$
B=F|_X.
$$

From:

$$
0
\to
\Omega_Y^3
\to
F
\to
B
\to0
$$

and the diagonal Hodge structure of $Y$, we have:

$$
h^{3,3}(Y)=4.
$$

Therefore:

$$
\boxed{
h^2(B)
=
4+\rho_{\mathrm{per}}.
}
$$

The four-dimensional summand is precisely the inherited Tate sector.

---

# 15. Hard Period Cokernel

From:

$$
0
\to
\Omega_X^2
\to
B
\to
T_X
\to0,
$$

the connecting map is the infinitesimal period map:

$$
d\mathcal P:
H^1(T_X)
\to
H^2(\Omega_X^2).
$$

The cokernel has dimension:

$$
\boxed{
4+\rho_{\mathrm{per}}.
}
$$

After quotienting the four inherited $(2,2)$ directions, the hard-period cokernel is:

$$
\boxed{
\dim\operatorname{coker}
\left(
d\mathcal P_{\mathrm{hard}}
\right)
=
\rho_{\mathrm{per}}.
}
$$

Thus:

$$
\rho_{\mathrm{per}}
$$

has three exact meanings.

---

# 16. Triple Interpretation of $\rho_{\mathrm{per}}$

We have:

$$
\boxed{
\rho_{\mathrm{per}}
=
h^2(Y,\Omega_Y^3(H)).
}
$$

We also have:

$$
\boxed{
\rho_{\mathrm{per}}
=
\dim\ker d\mathcal P-5.
}
$$

And finally:

$$
\boxed{
\rho_{\mathrm{per}}
=
\dim
\operatorname{coker}
d\mathcal P_{\mathrm{hard}}.
}
$$

Therefore:

$$
\boxed{
\text{middle Nakano resonance}
=
\text{extra period-kernel defect}
=
\text{hard-period cokernel defect}.
}
$$

This is the central conceptual result of Round 27.

---

# 17. Ancestry-Relative Strong K3 Type

Fatighenti--Mongardi define strong K3 type by requiring contraction with a generator of the top K3 Hodge line to induce an isomorphism from the deformation space to the next Hodge group.

For $c5$, the full deformation/Hodge dimensions make the absolute condition impossible in that form.

However the ancestry decomposition provides a natural relative version.

Let:

$$
K_{\mathrm{forced}}
$$

be the unavoidable five-dimensional period-kernel sector.

Let:

$$
H^{2,2}_{\mathrm{hard}}
$$

be the $20$-dimensional ancestry-normalized Hodge target.

Define:

$$
\boxed{
\textbf{Ancestry-Relative Strong K3}
}
$$

by the condition that:

$$
\boxed{
H^1(T_X)/K_{\mathrm{forced}}
\overset{\sim}{\longrightarrow}
H^{2,2}_{\mathrm{hard}}.
}
$$

For Küchle $c5$:

$$
\boxed{
\text{Ancestry-Relative Strong K3}
\iff
\rho_{\mathrm{per}}=0.
}
$$

---

# 18. The Exact Remaining Problem

After Rounds 20--27, the period problem is no longer:

> Compute the Hodge theory of a complicated Fano fourfold.

It is:

> Compute the generic rank of the finite differential supported on the twelve HLYY resonance slots / the sparse $15\to68\to48$ BWB spectral certificate.

Equivalently:

$$
\boxed{
\rho_{\mathrm{per}}
\stackrel{?}{=}0.
}
$$

There is no remaining ambiguity in what must be calculated.

---

# 19. Why Dimension Matching Is Not Used as Proof

Several attractive coincidences exist:

- deformation dimension:
  $$
  25;
  $$
- period dimension:
  $$
  20;
  $$
- fivefold moduli:
  $$
  5;
  $$
- double EPW quartic moduli:
  $$
  19.
  $$

But none of these determines:

$$
\rho_{\mathrm{per}}.
$$

In particular:

$$
25-20=5
$$

does not prove that the period rank is $20$.

Likewise a possible:

$$
19
$$

dimensional image would not prove that the image is the double-EPW locus.

The generic differential must be calculated.

---

# 20. Literature Calibration

Three external facts calibrate the calculation.

## HLYY

Their Theorem 5.7 provides a homogeneous-vector-bundle Jacobian description under explicit vanishing conditions.

The $c5$ checker passes those conditions at:

$$
H^{3,1},
$$

but hits exactly twelve residual slots at:

$$
H^{2,2}.
$$

## Fatighenti--Mongardi

They explicitly list Küchle $c5$ among weak K3-type Fano examples and warn that residual ambient cohomology must be treated carefully in high-index cases.

## Konno / Flenner

Their infinitesimal Torelli methods show that failures of automatic ambient vanishing should be treated as explicit cohomological exceptions, not as evidence that Torelli fails.

Thus the correct status is:

$$
\boxed{
\text{finite unresolved rank problem}.
}
$$

---

# 21. Next Computational Route

The remaining differential depends on the general defining section:

$$
(\lambda,\mu,\nu)
\in
\Lambda^4W^\vee
\oplus
\Lambda^2W^\vee
\oplus
\Lambda^3W^\vee.
$$

A practical next step is:

1. choose one explicit smooth general triple:
   $$
   (\lambda,\mu,\nu);
   $$
2. construct the twelve BWB representation spaces;
3. write the contraction / wedge differentials induced by the triple;
4. compute the matrix rank exactly over:
   $$
   \mathbb Q;
   $$
5. if the maximal rank yields:
   $$
   H^2=0,
   $$
   upper semicontinuity proves:
   $$
   \rho_{\mathrm{per}}=0
   $$
   for a general $c5$ fivefold.

Thus a single explicit maximal-rank certificate is enough for the generic case.

---

# 22. Round 27 Conclusion

Round 27 does not claim:

$$
\rho_{\mathrm{per}}=0.
$$

Instead it replaces a difficult sheaf-cohomology question by a completely finite spectral problem.

The conormal--Koszul BWB page has total dimensions:

$$
\boxed{
15\to68\to48
}
$$

and Euler characteristic:

$$
-5.
$$

The HLYY homogeneous Jacobian theorem is fully applicable to:

$$
H^{3,1},
$$

but its automatic vanishing hypotheses fail at exactly:

$$
\boxed{
12
}
$$

explicit BWB resonance slots for:

$$
H^{2,2}.
$$

The same defect is measured geometrically by:

$$
\boxed{
\rho_{\mathrm{per}}
=
h^2(Y,\Omega_Y^3(H))
=
\dim\ker d\mathcal P-5
=
\dim\operatorname{coker}d\mathcal P_{\mathrm{hard}}.
}
$$

Thus one scalar measures simultaneously:

- the middle Nakano boundary;
- excess isoperiodic deformation;
- failure of hard-period surjectivity.

The remaining problem is:

$$
\boxed{
\text{generic rank of a finite BWB differential}.
}
$$

This is the main closure of Round 27.

---

# References

1. An Huang, Bong H. Lian, Shing-Tung Yau, Chenglong Yu, **Jacobian rings for homogeneous vector bundles and applications**, arXiv:1801.08261.  
   Relevant material: Definition 5.6; Theorem 5.7; vanishing conditions (5.2)--(5.3).

2. Enrico Fatighenti, Giovanni Mongardi, **A note on a Griffiths-type ring for complete intersections in Grassmannians**, Mathematische Zeitschrift 299 (2021), 1651--1672.  
   Relevant material: Appendix on Fano varieties of K3 type; definition of strong/weak K3 type; warning about residual Grassmannian contributions.

3. Kazuhiro Konno, **Infinitesimal Torelli theorem for complete intersections in certain homogeneous Kähler manifolds**, Tohoku Math. J. 38 (1986), 609--624; Parts II--III (1990--1991).

4. Alexander Kuznetsov, **Küchle fivefolds of type c5**, Mathematische Zeitschrift 284 (2016), 1245--1278.

5. Round 26 exact equivariant localization certificate:
   $$
   \chi(Y,\Omega_Y^3(H))=-5.
   $$