# RCHM × Hodge Conjecture — Research Round 26
## Küchle $c5$ Period Gate: Exact Euler Localization, Five-Dimensional Period Excess, and the Single Resonant Cohomology Obstruction

**Status:** Internal research note + exact equivariant-index verifier  
**Version:** Round 26 / v2.6  
**Date:** 2026-09-18  
**Methodological base:** RCHM; X-Layer Mathematical Completeness; Bridge Search Compiler; Ancestry Reduction  
**Previous rounds:** RCHM × Hodge Conjecture — Rounds 1–25  
**Canonical source:** UTF-8 Markdown

---

# 0. Round 26 Objective

Round 25 established that the ancestry-normalized hard integral lattice of a Küchle $c5$ fourfold is isometric, after sign change, to the primitive lattice of a degree-$4$, divisibility-$1$ polarized $K3^{[2]}$ fourfold.

The remaining problem has therefore moved from lattice theory to period geometry.

Round 26 asks:

> What is the rank of the local period map of the Küchle $c5$ fourfold family?

The answer is reduced to one cohomology group.

An exact equivariant $K$-theory localization calculation gives:

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

where:

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

is the Küchle fivefold.

After combining this with tangent stability, Kodaira--Akizuki--Nakano vanishing, the hyperplane conormal sequence, and the fivefold's diagonal Hodge structure, the infinitesimal period rank becomes:

$$
\boxed{
\operatorname{rank}d\mathcal P
=
20-
h^2\!\left(
Y,\Omega_Y^3(H)
\right).
}
$$

Thus one integer controls the next bridge branch.

---

# 1. The Fivefold and Fourfold

Let:

$$
Y=X^5_{\lambda,\mu}
$$

be a general Küchle fivefold of type $c5$.

Kuznetsov proves:

$$
\boxed{
\dim Y=5,
\qquad
-K_Y=2H.
}
$$

A general Küchle fourfold is a smooth hyperplane section:

$$
\boxed{
X\in|H|.
}
$$

By adjunction:

$$
\boxed{
-K_X=H.
}
$$

---

# 2. Fivefold Moduli

Kuznetsov's standard form for the pair:

$$
(\lambda,\mu)
$$

shows that the moduli space of Küchle $c5$ fivefolds is:

$$
\boxed{
5\text{-dimensional}.
}
$$

This number will reappear independently in the period-kernel calculation.

---

# 3. Hyperplane Parameters

For the fourfold database entry one has:

$$
\boxed{
h^0(X,-K_X)=20.
}
$$

Since:

$$
-K_X=H|_X,
$$

use:

$$
0
\to
\mathcal O_Y
\to
\mathcal O_Y(H)
\to
\mathcal O_X(H)
\to0.
$$

Because:

$$
H^1(Y,\mathcal O_Y)=0,
$$

we obtain:

$$
\boxed{
h^0(Y,H)=21.
}
$$

Thus hyperplanes form:

$$
\boxed{
\mathbb P^{20}.
}
$$

The natural family count is therefore:

$$
\boxed{
5+20=25.
}
$$

---

# 4. Tangent Euler Characteristic

The Fano-fourfold database gives:

$$
\boxed{
\chi(X,T_X)=-25.
}
$$

Since:

$$
T_X
\simeq
\Omega_X^3(H),
$$

Kodaira--Akizuki--Nakano vanishing gives:

$$
H^q(X,T_X)=0
$$

for:

$$
q\ge2.
$$

On the generic finite-automorphism locus:

$$
H^0(X,T_X)=0,
$$

hence:

$$
\boxed{
h^1(X,T_X)=25.
}
$$

This agrees exactly with the natural family count.

---

# 5. Period Domain Dimension

The ancestry-normalized hard Hodge structure has:

$$
\boxed{
h=(1,20,1),
\qquad
\operatorname{rank}=22.
}
$$

Its polarized period domain is type IV of complex dimension:

$$
\boxed{
20.
}
$$

Therefore any local period map from a $25$-dimensional deformation space has kernel dimension at least:

$$
\boxed{
5.
}
$$

Call:

$$
\boxed{
\epsilon_{\mathrm{per}}
=
25-20
=
5
}
$$

the **period excess**.

---

# 6. The Hyperplane Conormal Sequence

The conormal sequence is:

$$
0
\to
\mathcal O_X(-H)
\to
\Omega_Y^1|_X
\to
\Omega_X^1
\to0.
$$

Taking the third exterior power and twisting by $H$ gives:

$$
\boxed{
0
\to
\Omega_X^2
\to
\Omega_Y^3(H)|_X
\to
\Omega_X^3(H)
\to0.
}
$$

Using:

$$
\Omega_X^3(H)\simeq T_X,
$$

we obtain:

$$
\boxed{
0
\to
\Omega_X^2
\to
B
\to
T_X
\to0,
}
$$

where:

$$
B
=
\Omega_Y^3(H)|_X.
$$

---

# 7. Infinitesimal Period Map

Because:

$$
h^{3,1}(X)=1,
$$

the infinitesimal period map is the Kodaira--Spencer contraction:

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

against a generator of:

$$
H^1(X,\Omega_X^3).
$$

For a hypersurface section, the boundary morphism of the sequence above is the same standard infinitesimal-period morphism.

This is exactly the mechanism used by Debarre--Iliev--Manivel for Fano fourfolds of degree $10$.

Since:

$$
H^1(X,\Omega_X^2)
=
H^{2,1}(X)
=
0,
$$

we get:

$$
\boxed{
\ker d\mathcal P
\simeq
H^1(X,B).
}
$$

---

# 8. Move the Kernel to the Fivefold

Use the restriction sequence:

$$
0
\to
\Omega_Y^3
\to
\Omega_Y^3(H)
\to
B
\to0.
$$

Kuznetsov's integral Chow motive of the fivefold is of Lefschetz type.

Hence its Hodge diamond is diagonal.

In particular:

$$
H^1(Y,\Omega_Y^3)
=
H^{3,1}(Y)
=
0,
$$

and:

$$
H^2(Y,\Omega_Y^3)
=
H^{3,2}(Y)
=
0.
$$

Therefore:

$$
\boxed{
H^1(X,B)
\simeq
H^1\!\left(
Y,\Omega_Y^3(H)
\right).
}
$$

Thus:

$$
\boxed{
\ker d\mathcal P
\simeq
H^1\!\left(
Y,\Omega_Y^3(H)
\right).
}
$$

---

# 9. Exact Equivariant Index Calculation

The fivefold is the zero locus in:

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

of the rank-$7$ bundle:

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

At a torus fixed point indexed by:

$$
I\subset\{1,\ldots,7\},
\qquad
|I|=3,
$$

let:

- $W_I$ be the tangent weights of $G$;
- $R_I$ be the weights of $\mathcal E$;
- $h_I$ be the Plücker weight.

In equivariant $K$-theory:

$$
[T_Y]
=
[T_G]-[\mathcal E].
$$

The fixed-point contribution to:

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

is:

$$
\boxed{
e^{h_I}
\,
\operatorname{ch}
\left(
\Lambda^3 T_Y^\vee
\right)
\,
\frac{
\prod_{r\in R_I}(1-e^{-r})
}{
\prod_{w\in W_I}(1-e^{-w})
}.
}
$$

The non-equivariant index is the constant term as the torus parameter tends to zero.

---

# 10. Exact Result

The accompanying verifier expands each of the $35$ fixed-point contributions as an exact rational Laurent series.

Three unrelated sets of integer torus weights are used.

All three give:

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

Regression checks give:

$$
\boxed{
\chi(Y,\mathcal O_Y)=1,
}
$$

and:

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

The second agrees with:

$$
h^0(Y,H)=21.
$$

No floating-point arithmetic is used.

---

# 11. Vanishing of Global Sections

Since:

$$
K_Y=-2H,
$$

we have:

$$
\boxed{
\Omega_Y^3(H)
\simeq
\Lambda^2T_Y(-H).
}
$$

A Fano fivefold of Picard number one has stable tangent bundle by Hwang's low-dimensional stability theorem.

In characteristic zero, exterior powers of a semistable bundle are semistable.

The slope is:

$$
\mu(T_Y)
=
\frac25H^5.
$$

Hence:

$$
\mu(\Lambda^2T_Y)
=
\frac45H^5.
$$

After twisting by:

$$
-H,
$$

we get:

$$
\boxed{
\mu\!\left(
\Omega_Y^3(H)
\right)
=
-\frac15H^5<0.
}
$$

Semistability therefore gives:

$$
\boxed{
H^0\!\left(
Y,\Omega_Y^3(H)
\right)=0.
}
$$

---

# 12. High-Degree Vanishing

Kodaira--Akizuki--Nakano gives:

$$
H^q\!\left(
Y,\Omega_Y^3(H)
\right)
=
0
$$

whenever:

$$
3+q>5.
$$

Therefore:

$$
\boxed{
H^q\!\left(
Y,\Omega_Y^3(H)
\right)=0
\qquad
(q\ge3).
}
$$

The only unresolved groups are:

$$
H^1
$$

and:

$$
H^2.
$$

---

# 13. The Resonance Equation

Using:

$$
\chi=-5,
$$

$$
h^0=0,
$$

and:

$$
h^q=0
\quad(q\ge3),
$$

we obtain:

$$
-h^1+h^2=-5.
$$

Therefore:

$$
\boxed{
h^1
=
5+h^2.
}
$$

Since:

$$
\ker d\mathcal P
\simeq
H^1(Y,\Omega_Y^3(H)),
$$

we get:

$$
\boxed{
\dim\ker d\mathcal P
=
5+
h^2\!\left(
Y,\Omega_Y^3(H)
\right).
}
$$

---

# 14. Period Resonance Index

Define:

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

Then:

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

On the generic finite-automorphism locus:

$$
h^1(T_X)=25.
$$

Hence:

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

This is the central formula of Round 26.

---

# 15. Why $H^2$ Is a Natural Resonance Group

The pair:

$$
(p,q)=(3,2)
$$

satisfies:

$$
p+q=5=\dim Y.
$$

It lies exactly on the middle line where ordinary Nakano vanishing does not apply.

Thus:

$$
\boxed{
H^2(Y,\Omega_Y^3(H))
}
$$

is not an arbitrary leftover.

It is the unique **middle-resonance cohomology group** controlling period-rank loss.

This explains structurally why the entire problem collapses to one group.

---

# 16. Period-Rank Trichotomy

The value of:

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

selects the next geometry.

## Case A — $\rho_{\mathrm{per}}=0$

Then:

$$
\boxed{
\operatorname{rank}d\mathcal P=20.
}
$$

The local period map is a submersion onto the full degree-$4$, divisibility-$1$ polarized $K3^{[2]}$ period domain.

The kernel dimension is exactly:

$$
\boxed{
5.
}
$$

---

## Case B — $\rho_{\mathrm{per}}=1$

Then:

$$
\boxed{
\operatorname{rank}d\mathcal P=19.
}
$$

The period image is locally constrained to a hypersurface in the $20$-dimensional polarized period domain.

This is exactly the dimension of the family of double EPW quartics.

Thus double EPW quartics become a natural generic-image candidate only in this branch.

Dimension equality is not a proof of identification.

---

## Case C — $\rho_{\mathrm{per}}\ge2$

Then:

$$
\boxed{
\operatorname{rank}d\mathcal P\le18.
}
$$

The $c5$ Hodge structures satisfy additional hidden constraints beyond the lattice polarization type.

A smaller special period locus must be found.

---

# 17. Double EPW Quartic Test

Double EPW quartics are:

- hyperkähler fourfolds of $K3^{[2]}$ type;
- polarized with Beauville--Bogomolov square:
  $$
  q(h)=4;
  $$
- a:
  $$
  \boxed{
  19\text{-dimensional}
  }
  $$
  family.

Thus they occupy a proper special locus inside the $20$-dimensional degree-$4$ polarized $K3^{[2]}$ period space.

Consequently:

$$
\boxed{
\rho_{\mathrm{per}}=0
\Rightarrow
\text{generic }c5\text{ partner is not generically constrained to the double-EPW-quartic locus}.
}
$$

While:

$$
\boxed{
\rho_{\mathrm{per}}=1
}
$$

makes that locus dimensionally compatible with the generic period image.

This makes:

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

a discriminator between two concrete geometric programs.

---

# 18. Calibration by Degree-$10$ Fano Fourfolds

Debarre--Iliev--Manivel study degree-$10$ Fano fourfolds.

Their local deformation space has dimension:

$$
24,
$$

while their K3-type period domain has dimension:

$$
20.
$$

They prove that the local period map is a submersion with:

$$
\boxed{
4\text{-dimensional fibers}.
}
$$

Their proof uses the same structural mechanism:

$$
0
\to
\Omega_X^2
\to
\Omega_{\mathrm{ambient}}^3(\text{twist})|_X
\to
T_X
\to0.
$$

Thus the $c5$ conjectural:

$$
25\to20
$$

period behavior has a close proven analogue.

---

# 19. Parent-Kernel Coincidence

Kuznetsov proves that $c5$ fivefolds have:

$$
\boxed{
5
}
$$

moduli.

If:

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

the fourfold period kernel also has dimension:

$$
\boxed{
5.
}
$$

This motivates a stronger hypothesis:

$$
\boxed{
\textbf{Parent-Kernel Conjecture}
}
$$

> The infinitesimal period-kernel directions of a general $c5$ fourfold are induced by varying its parent $c5$ fivefold.

At present this is only a dimensionally and structurally motivated conjecture.

The period-kernel subspace could mix parent and hyperplane directions.

An explicit tangent-space comparison is required.

---

# 20. Exact Next Gate

Round 26 reduces the period question to:

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

The immediate computational target is:

$$
\boxed{
H^2\!\left(
Y,\Omega_Y^3(H)
\right)
\stackrel{?}{=}0.
}
$$

Possible routes include:

1. Borel--Weil--Bott plus the Koszul resolution of $Y\subset\operatorname{Gr}(3,7)$;
2. the birational Lagrangian-Grassmannian model;
3. a direct homogeneous-bundle spectral sequence;
4. derived-category / exceptional-collection methods.

This is now a finite cohomology calculation, not an open-ended bridge search.

---

# 21. Updated Bridge Frontier

## Frontier A — Generic degree-$4$ $K3^{[2]}$

Activation condition:

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

Then the period map has maximal rank $20$.

---

## Frontier B — Double EPW Quartic

Activation signal:

$$
\boxed{
\rho_{\mathrm{per}}=1.
}
$$

The dimensions become:

$$
25
\to
19,
$$

with kernel dimension:

$$
6.
$$

This matches the $19$-dimensional double-EPW family at the level of period dimension.

---

## Frontier C — 2026 Degeneration / Even-Nodal Geometry

This remains valuable in either case.

If the degeneration produces a double EPW or related special hyperkähler model, the value of:

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

will determine whether that geometry should describe:

- the generic $c5$ period;
- only a divisor/special locus;
- only a degeneration boundary.

---

# 22. Methodological Principle — Period Excess Decomposition

Define:

$$
\boxed{
\epsilon_{\mathrm{per}}
=
\dim\operatorname{Def}
-
\dim\mathcal D.
}
$$

Then write:

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

where:

- $\epsilon_{\mathrm{per}}$ is forced by dimension;
- $\rho_{\mathrm{per}}$ measures extra cohomological resonance.

For $c5$:

$$
\boxed{
\epsilon_{\mathrm{per}}=5.
}
$$

This separates unavoidable period fibers from genuinely special Hodge constraints.

---

# 23. Round 26 Conclusion

Round 26 converts the $c5$ period-realization problem into one exact cohomological gate.

The new exact localization result is:

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

Tangent stability gives:

$$
\boxed{
H^0(Y,\Omega_Y^3(H))=0.
}
$$

Nakano vanishing gives:

$$
\boxed{
H^q(Y,\Omega_Y^3(H))=0
\quad(q\ge3).
}
$$

Therefore:

$$
\boxed{
h^1
=
5+h^2.
}
$$

The hyperplane conormal sequence identifies:

$$
H^1
$$

with the infinitesimal period kernel.

Hence:

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

and:

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

where:

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

The next research problem is now completely explicit:

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

If yes, the generic $c5$ period fills the full $20$-dimensional degree-$4$ polarized $K3^{[2]}$ domain.

If:

$$
\rho_{\mathrm{per}}=1,
$$

the $19$-dimensional double-EPW-quartic locus becomes a natural generic-image candidate.

This is the main closure of Round 26.

---

# References

1. Alexander Kuznetsov, **Küchle fivefolds of type c5**, Mathematische Zeitschrift 284 (2016), 1245--1278.  
   Relevant items: Remark 3.13; Lemma 4.1; the hyperplane-section fourfold construction; Theorem 5.3 and Corollary 5.4.

2. Jun-Muk Hwang, **Stability of tangent bundles of low dimensional Fano manifolds with Picard number 1**, Mathematische Annalen 312 (1998), 599--606.  
   Relevant result: Fano fivefolds of Picard number one have stable tangent bundles.

3. Olivier Debarre, Atanas Iliev, Laurent Manivel, **Special prime Fano fourfolds of degree 10 and index 2**, in particular Theorem 4.3 on the local period map.

4. Carl Mazzanti, **On the Chow ring of double EPW quartics**, arXiv:2603.02251 (2026).  
   Relevant facts: double EPW quartics form a $19$-dimensional family of $K3^{[2]}$-type fourfolds with Beauville polarization square $4$.

5. Fano Fourfold Database, entry #321 (Küchle $c5$):  
   $h^0(-K)=20$, $h^{3,1}=1$, $h^{2,2}=24$, $\chi(T)=-25$.

6. Standard Kodaira--Akizuki--Nakano vanishing and the infinitesimal variation of Hodge structure for hypersurface sections.