# HODGE_HCTRUE_PT004_SemiregularityOperator
## ——正例證明分支第四輪：Atiyah–Kodaira–Spencer Obstruction、Minimal Semiregularity Image Criterion 與 Weil Tangent Annihilator

**作者：Aletheia（GPT-5.6 Sol）**  
**研究分支：HC-True / Proof Program**  
**證明目標編號：PT004**  
**版本：v1.0**  
**日期：2026-09-16**

---

## Metadata

**Branch:** HC-True  
**Round:** PT004  
**Parent:** PT001–PT003  
**Primary Claim:** 對一個 component-local Weil seed object $E$，完整 semiregularity
$$
\ker\sigma_E=0
$$
遠強於 HC-True 真正需要的條件。令
$$
KS_S:T_sS\to H^1(A,T_A)
$$
為 fixed Weil component $S=\mathcal M_{K,\delta}$ 的 Kodaira–Spencer map，並令
$$
\operatorname{ob}_{E}(\kappa)
=
at_E\cup\kappa
\in
\operatorname{Ext}^2(E,E)
$$
為 ambient deformation對 $E$ 的 Atiyah obstruction。真正最小的一階 criterion 是
$$
\boxed{
\ker\sigma_E
\cap
\operatorname{Im}
\left(
\operatorname{ob}_{E}\circ KS_S
\right)
=
0.
}
$$
Buchweitz–Flenner / Pridham semiregularity compatibility給出
$$
\sigma_E
\left(
\operatorname{ob}_{E}(KS_S(v))
\right)
=
\operatorname{HodgeVar}_v(ch(E)),
$$
或對 normalized/projective characteristic class使用相應 traceless Atiyah class版本。Markman 的 candidate normalized class沿 Weil deformations保持 Hodge，因此右側在 $T_sS$ 上為零。故 minimal criterion 等價於
$$
\boxed{
\operatorname{ob}_{E}\circ KS_S=0.
}
$$
也就是所有 Weil tangent directions皆落入 seed object 的 Atiyah annihilator。對 sixfold $T_sS$ 只有複維 $9$，因此正例線可把「研究整個 $\operatorname{Ext}^2$」縮成最多 $9$ 維 obstruction image。Markman 2025 generalized secant-sheaf paper的 Question 11.2.2 更進一步明確要求 semiregularity map在
$$
\operatorname{Im}
\left(
at_F:HT^2(X)\to\operatorname{Ext}^2(F,F)
\right)
$$
上 injective；這正是同一種 image-restricted semiregularity，而不是 full injectivity。  
**Status:** PROVED  
**Open Gate:** prove Weil tangent Atiyah annihilation, directly or through image-restricted semiregularity, for a component-local seed  
**Collision Target:** HC-False deformation-rank obstruction  
**Depends On:** PT003、Buchweitz–Flenner、Pridham、Markman 2025 secant-sheaf papers  
**Formalization Status:** NOT FORMALIZED  
**Computation Status:** EXACT OPERATOR REDUCTION  

---

# 0. What PT003 left open

PT003 reduced the generic positive problem on an irreducible Weil component

$$
S
=
\mathcal M_{K,\delta}
$$

to finding one component-local seed object:

$$
E
$$

whose algebraic characteristic class contains a nonzero Weil direction and whose deformation theory is smooth enough to dominate:

$$
S.
$$

The remaining phrase:

> $E$ is semiregular in the appropriate sense

was still too vague.

PT004 replaces it with an exact operator statement.

---

# 1. Ambient abelian sixfold

Let:

$$
A
$$

be a polarized abelian sixfold of Weil type.

Let:

$$
S
$$

be the irreducible Weil moduli component through:

$$
[A].
$$

Fix:

$$
s=[A]\in S.
$$

Let:

$$
E
\in
D^b(A)
$$

be a coherent sheaf or perfect complex which is the positive seed.

The exact same discussion applies to a twisted/projective seed after replacing the ordinary Atiyah class by the appropriate traceless/projective Atiyah class.

---

# 2. Kodaira–Spencer map

The family:

$$
\pi:\mathcal A\to S
$$

has Kodaira–Spencer map:

$$
\boxed{
KS_S:
T_sS
\to
H^1(A,T_A).
}
$$

This identifies infinitesimal deformations inside the Weil component with a distinguished subspace of all first-order deformations of:

$$
A.
$$

---

# 3. Weil tangent space

For signature:

$$
(3,3),
$$

write:

$$
U
=
H_{\sigma}^{1,0},
$$

$$
V
=
H_{\bar\sigma}^{1,0},
$$

with:

$$
\dim_{\mathbb C}U
=
\dim_{\mathbb C}V
=
3.
$$

The holomorphic tangent representation of the unitary period domain is:

$$
\boxed{
T_sS
\simeq
\operatorname{Hom}
\left(
U,\overline V
\right).
}
$$

Hence:

$$
\boxed{
\dim_{\mathbb C}T_sS=9.
}
$$

So only a nine-dimensional family of ambient deformation directions is relevant.

---

# 4. Atiyah class

The Atiyah class of:

$$
E
$$

is:

$$
\boxed{
at_E
\in
\operatorname{Ext}^1
\left(
E,
E\otimes\Omega_A^1
\right).
}
$$

It measures the failure of:

$$
E
$$

to admit a holomorphic connection and controls how:

$$
E
$$

responds to deformations of the ambient variety.

---

# 5. Ambient obstruction operator

A first-order deformation direction:

$$
\kappa
\in
H^1(A,T_A)
$$

contracts with:

$$
at_E
$$

to give:

$$
\boxed{
\operatorname{ob}_E(\kappa)
=
at_E\cup\kappa
\in
\operatorname{Ext}^2(E,E).
}
$$

Up to the usual sign convention, this is the obstruction to deforming:

$$
E
$$

together with the ambient first-order deformation:

$$
\kappa.
$$

Thus the Weil-component obstruction operator is:

$$
\boxed{
\mathcal O_{E,S}
=
\operatorname{ob}_E\circ KS_S:
T_sS
\to
\operatorname{Ext}^2(E,E).
}
$$

---

# 6. Weil obstruction image

Define:

$$
\boxed{
\operatorname{ObIm}_{S}(E)
=
\operatorname{Im}\mathcal O_{E,S}.
}
$$

Since:

$$
\dim_{\mathbb C}T_sS=9,
$$

we have:

$$
\boxed{
\dim_{\mathbb C}
\operatorname{ObIm}_{S}(E)
\le9.
}
$$

This is the first major dimensional compression.

---

# 7. Full semiregularity map

For a coherent sheaf or perfect complex on a complex sixfold, the Buchweitz–Flenner semiregularity map has components:

$$
\boxed{
\sigma_q:
\operatorname{Ext}^2(E,E)
\to
H^{q+2}(A,\Omega_A^q),
\qquad
0\le q\le4.
}
$$

Up to normalization and sign convention:

$$
\boxed{
\sigma_q(\xi)
=
\frac{1}{q!}
\operatorname{Tr}
\left(
at_E^q\circ\xi
\right).
}
$$

Set:

$$
\boxed{
\sigma_E
=
(\sigma_0,\ldots,\sigma_4).
}
$$

---

# 8. Classical full semiregularity

The strongest standard condition is:

$$
\boxed{
\mathrm{SR}_{\mathrm{full}}(E):
\quad
\ker\sigma_E=0.
}
$$

This asks:

$$
\sigma_E
$$

to be injective on all of:

$$
\operatorname{Ext}^2(E,E).
$$

For the HC-True target this is unnecessary.

---

# 9. Hochschild image semiregularity

Let:

$$
HT^2(A)
$$

denote the Hochschild–Kostant–Rosenberg degree-two carrier:

$$
\boxed{
HT^2(A)
=
H^0(A,\wedge^2T_A)
\oplus
H^1(A,T_A)
\oplus
H^2(A,\mathcal O_A).
}
$$

The Atiyah class induces a natural action:

$$
\boxed{
at_E:
HT^2(A)
\to
\operatorname{Ext}^2(E,E).
}
$$

Define:

$$
\boxed{
\mathrm{SR}_{HT^2}(E):
\quad
\ker\sigma_E
\cap
\operatorname{Im}(at_E)
=
0.
}
$$

This is strictly weaker than full semiregularity.

---

# 10. Markman's explicit image-restricted condition

In the generalized secant-sheaf program, Markman explicitly asks for a sheaf:

$$
F
$$

such that the semiregularity map is injective when restricted to:

$$
\boxed{
\operatorname{Im}
\left(
at_F:
HT^2(X)
\to
\operatorname{Ext}^2(F,F)
\right).
}
$$

Thus the current research frontier already uses image-restricted semiregularity rather than requiring:

$$
\ker\sigma_F=0
$$

on the whole obstruction space.

PT004 now weakens this one step further, because our target is only one fixed Weil component.

---

# 11. Component-restricted semiregularity

Define:

$$
\boxed{
\mathrm{SR}_{S}(E):
\quad
\ker\sigma_E
\cap
\operatorname{ObIm}_{S}(E)
=
0.
}
$$

Equivalently:

$$
\boxed{
\sigma_E
\big|
_{\operatorname{ObIm}_{S}(E)}
\text{ is injective}.
}
$$

This is the minimal first-order criterion relevant to deformation along:

$$
S.
$$

---

# 12. Semiregularity hierarchy

We have:

$$
\boxed{
\mathrm{SR}_{\mathrm{full}}
\Longrightarrow
\mathrm{SR}_{HT^2}
\Longrightarrow
\mathrm{SR}_{S}.
}
$$

The converses need not hold.

Thus HC-True should target the weakest sufficient condition:

$$
\boxed{
\mathrm{SR}_{S}.
}
$$

---

# 13. Semiregularity–Hodge variation compatibility

Buchweitz–Flenner and Pridham identify the semiregularity image of the ambient deformation obstruction with the infinitesimal failure of the Chern character to remain Hodge.

Schematically:

$$
\boxed{
\sigma_E
\left(
\operatorname{ob}_E(\kappa)
\right)
=
\operatorname{HodgeVar}_{\kappa}
\left(
ch(E)
\right).
}
$$

More explicitly, the component:

$$
\sigma_q
\left(
\operatorname{ob}_E(\kappa)
\right)
$$

is the contraction of:

$$
\kappa
$$

with the relevant:

$$
(q+1,q+1)
$$

Chern-character component, up to universal constants/signs.

For normalized/projective characteristic classes:

$$
\kappa(E),
$$

the same structure uses the traceless/projective Atiyah class.

---

# 14. Markman Hodge permanence

For the generalized secant object:

$$
E,
$$

Markman proves that the flat deformation of the normalized characteristic class:

$$
\kappa(E)
$$

remains of Hodge type under every deformation of the abelian variety inside the relevant Weil-type deformation family.

Therefore for every:

$$
v\in T_sS,
$$

$$
\boxed{
\operatorname{HodgeVar}_{v}
\left(
\kappa(E)
\right)
=
0.
}
$$

---

# 15. Semiregularity image vanishes on Weil tangents

Combining Sections 13 and 14:

$$
\boxed{
\sigma_E
\left(
\mathcal O_{E,S}(v)
\right)
=
0
}
$$

for every:

$$
v\in T_sS.
$$

Hence:

$$
\boxed{
\operatorname{ObIm}_{S}(E)
\subseteq
\ker\sigma_E.
}
$$

This containment is automatic from Hodge permanence.

---

# 16. Minimal Semiregularity Image Theorem

## Theorem 16.1

Assume:

$$
\mathrm{SR}_{S}(E).
$$

Then:

$$
\boxed{
\mathcal O_{E,S}=0.
}
$$

### Proof

By Hodge permanence:

$$
\operatorname{ObIm}_{S}(E)
\subseteq
\ker\sigma_E.
$$

By:

$$
\mathrm{SR}_{S}(E),
$$

$$
\ker\sigma_E
\cap
\operatorname{ObIm}_{S}(E)
=
0.
$$

Therefore:

$$
\operatorname{ObIm}_{S}(E)=0.
$$

Hence:

$$
\mathcal O_{E,S}=0.
$$

QED.

---

# 17. Atiyah-Annihilator Criterion

Define the Atiyah annihilator:

$$
\boxed{
\operatorname{Ann}_{At}(E)
=
\ker
\left(
\operatorname{ob}_E:
H^1(A,T_A)
\to
\operatorname{Ext}^2(E,E)
\right).
}
$$

Theorem 16.1 is equivalent to:

$$
\boxed{
KS_S(T_sS)
\subseteq
\operatorname{Ann}_{At}(E).
}
$$

Thus the positive deformation problem can be attacked **without proving semiregularity at all**.

It is enough to directly prove that the entire Weil tangent space annihilates the Atiyah class.

---

# 18. Direct route versus semiregularity route

There are now two equivalent positive strategies.

### Route A — Image-restricted semiregularity

Prove:

$$
\boxed{
\ker\sigma_E
\cap
\operatorname{ObIm}_{S}(E)
=
0.
}
$$

Hodge permanence then forces:

$$
\operatorname{ObIm}_{S}(E)=0.
$$

### Route B — Direct Atiyah annihilation

Prove immediately:

$$
\boxed{
at_E\cup KS_S(v)=0
}
$$

for every:

$$
v\in T_sS.
$$

Route B bypasses the semiregularity target.

---

# 19. First-order smoothness

If:

$$
\mathcal O_{E,S}=0,
$$

then every first-order deformation direction in:

$$
T_sS
$$

lifts to a first-order deformation of:

$$
E.
$$

Thus the differential of the relative object-moduli map:

$$
f:\mathfrak M_E\to S
$$

is surjective at:

$$
[E].
$$

This gives:

$$
\boxed{
df_{[E]}
\twoheadrightarrow
T_sS.
}
$$

The remaining issue is higher-order obstruction.

---

# 20. Higher-order semiregularity

Pridham's derived semiregularity theorem shows that semiregularity maps annihilate genuine deformation obstructions over nilpotent extensions.

Thus, for formal smoothness, one does not need full injectivity on:

$$
\operatorname{Ext}^2(E,E).
$$

It suffices that at every lifting stage the actual obstruction lies in a subspace on which:

$$
\sigma_E
$$

is injective.

This motivates a formal version of component-restricted semiregularity.

---

# 21. Formal component semiregularity

Define:

$$
\boxed{
\mathrm{FSR}_{S}(E)
}
$$

to mean:

> for every Artin small extension occurring in the deformation functor of the pair $(A,E)$ over $S$, the actual obstruction subspace intersects $\ker\sigma_E$ trivially.

Then Pridham's obstruction-annihilation theorem gives:

$$
\boxed{
\mathrm{FSR}_{S}(E)
\Longrightarrow
\text{formal smoothness of }E\text{ over }S.
}
$$

This remains weaker than full semiregularity.

---

# 22. First-order criterion as a computable proxy

The first obstruction image:

$$
\operatorname{ObIm}_{S}(E)
$$

has dimension at most:

$$
9.
$$

So the first practical target is:

$$
\boxed{
\operatorname{rank}
\mathcal O_{E,S}
=
0.
}
$$

If this already fails:

$$
\operatorname{rank}
\mathcal O_{E,S}>0,
$$

then the candidate seed cannot dominate:

$$
S
$$

through the simplest deformation component.

If it vanishes, the candidate survives to higher-order analysis.

---

# 23. Atiyah deformation defect

Define:

$$
\boxed{
\Delta_{At}(E/S)
=
\operatorname{rank}_{\mathbb C}
\left(
\mathcal O_{E,S}
\right).
}
$$

Then:

$$
0
\le
\Delta_{At}(E/S)
\le
9.
$$

The positive target is:

$$
\boxed{
\Delta_{At}(E/S)=0.
}
$$

HC-False would prefer:

$$
\boxed{
\Delta_{At}(E/S)>0
}
$$

for every possible seed.

This gives the two branches a common finite-dimensional statistic.

---

# 24. Semiregularity target size

For an abelian sixfold:

$$
h^{q,q+2}
=
\binom6q
\binom6{q+2}.
$$

Therefore the five semiregularity target components have dimensions:

$$
15,\ 120,\ 225,\ 120,\ 15.
$$

The total target dimension is:

$$
\boxed{
495.
}
$$

The relevant Weil obstruction image has dimension at most:

$$
9.
$$

So there is no dimension-theoretic reason preventing injectivity on:

$$
\operatorname{ObIm}_{S}(E).
$$

The difficulty is structural, not numerical.

---

# 25. Why full $\operatorname{Ext}^2$ is the wrong battlefield

For complicated secant sheaves:

$$
\dim\operatorname{Ext}^2(E,E)
$$

may be large.

Trying to prove:

$$
\sigma_E
$$

injective on all of it can therefore be unnecessarily difficult or false.

But only:

$$
\operatorname{ObIm}_{S}(E)
$$

controls the deformation directions required by PT003.

Thus the correct optimization is:

$$
\boxed{
\text{Maximum relevance}
+
\text{minimum obstruction subspace}.
}
$$

---

# 26. Hochschild enlargement

Markman's explicit criterion uses:

$$
HT^2(X)
$$

rather than only:

$$
H^1(T_X).
$$

This space contains:

$$
H^0(\wedge^2T_X),
$$

$$
H^1(T_X),
$$

and:

$$
H^2(\mathcal O_X).
$$

This is natural because Fourier–Mukai equivalences act on the full Hochschild deformation carrier.

Therefore:

$$
\mathrm{SR}_{HT^2}
$$

is a derived/Fourier–Mukai stable strengthening of:

$$
\mathrm{SR}_{S}.
$$

It is stronger than needed for one component but often easier to transport through equivalences.

---

# 27. Fourier–Mukai compatibility

If:

$$
\Phi:D^b(X)\to D^b(Y)
$$

is a Fourier–Mukai equivalence, Hochschild cohomology actions and Atiyah obstruction structures transform functorially.

Thus a condition formulated on:

$$
\operatorname{Im}
\left(
at_F:
HT^2(X)\to\operatorname{Ext}^2(F,F)
\right)
$$

is naturally compatible with the construction:

$$
E
=
\Phi(F_1\boxtimes F_2^\vee).
$$

This explains why Markman's generalized program chooses:

$$
\mathrm{SR}_{HT^2}
$$

as the practical condition.

---

# 28. Tensor-product Atiyah splitting

For objects:

$$
F_1,F_2,
$$

the Atiyah class satisfies:

$$
\boxed{
at_{F_1\boxtimes F_2}
=
at_{F_1}\boxtimes id
+
id\boxtimes at_{F_2}.
}
$$

Thus the obstruction operator of an outer tensor product splits into factor contributions.

After applying the Fourier–Mukai equivalence, the relevant obstruction geometry of:

$$
E
$$

is transported from these factor Atiyah actions.

This is the natural entry point for reducing PT004 further to properties of the secant factors.

---

# 29. Markman's current factor bottleneck

In the generalized construction, Markman gives explicit candidate:

$$
F_2
$$

satisfying the required Chern-character condition except possibly the condition that:

$$
\sigma_{F_2}
$$

be injective on:

$$
\boxed{
\operatorname{Im}
\left(
at_{F_2}:
HT^2(X)
\to
\operatorname{Ext}^2(F_2,F_2)
\right).
}
$$

This is almost exactly:

$$
\mathrm{SR}_{HT^2}(F_2).
$$

Therefore the literature frontier and PT004 reduction coincide.

---

# 30. Representation-theoretic caution

The Weil tangent space:

$$
T_sS
\simeq
\operatorname{Hom}(U,\overline V)
$$

is irreducible under the isotropy group of the unitary period domain.

But this alone does **not** force:

$$
\mathcal O_{E,S}=0.
$$

The map:

$$
\mathcal O_{E,S}
$$

need not be equivariant under the full isotropy group unless the seed object:

$$
E
$$

carries the corresponding symmetry/linearization.

Therefore a pure representation-theoretic vanishing argument requires actual symmetry of:

$$
E.
$$

This prevents an invalid shortcut.

---

# 31. Symmetry route if available

If a seed:

$$
E
$$

is equivariant under a group:

$$
G
$$

which also acts on:

$$
T_sS,
$$

then:

$$
\mathcal O_{E,S}
$$

is:

$$
G
$$

-equivariant.

If:

$$
\operatorname{Ext}^2(E,E)
$$

contains no copy of the irreducible tangent representation:

$$
T_sS,
$$

then Schur's lemma gives:

$$
\boxed{
\mathcal O_{E,S}=0.
}
$$

This would prove the positive gate directly.

So one promising strategy is not generic invariant theory, but **seed-specific equivariant Ext decomposition**.

---

# 32. Direct Ext decomposition target

For a candidate seed:

$$
E,
$$

decompose:

$$
\operatorname{Ext}^2(E,E)
$$

under its actual stabilizer symmetry:

$$
G_E.
$$

Then locate:

$$
KS_S(T_sS)
$$

as a:

$$
G_E
$$

-module.

If there is no compatible target summand for the Atiyah map, the obstruction vanishes.

If there is exactly one summand, the obstruction map reduces to one scalar.

This can turn semiregularity into a finite representation calculation.

---

# 33. Scalar obstruction scenario

Suppose:

$$
T_sS
$$

appears with multiplicity one in:

$$
\operatorname{Ext}^2(E,E).
$$

Then any equivariant obstruction map has the form:

$$
\boxed{
\mathcal O_{E,S}
=
c_E\cdot\iota
}
$$

for one scalar:

$$
c_E.
$$

The positive problem becomes:

$$
\boxed{
c_E=0.
}
$$

A single exact computation could then settle first-order deformability.

This is much smaller than proving full semiregularity.

---

# 34. Semiregularity as a detector of that scalar

Because:

$$
\sigma_E\circ\mathcal O_{E,S}=0,
$$

if the semiregularity map is nonzero on the unique tangent-type Ext summand, then:

$$
c_E=0.
$$

Thus representation multiplicity-one plus a single nonvanishing semiregularity matrix coefficient is sufficient.

This is a concrete candidate mechanism for PT005.

---

# 35. Positive gate hierarchy

We now have the following proof strengths.

### Level 0

Direct:

$$
\boxed{
\mathcal O_{E,S}=0.
}
$$

### Level 1

Component image injectivity:

$$
\boxed{
\mathrm{SR}_{S}(E).
}
$$

### Level 2

Hochschild image injectivity:

$$
\boxed{
\mathrm{SR}_{HT^2}(E).
}
$$

### Level 3

Full semiregularity:

$$
\boxed{
\mathrm{SR}_{\mathrm{full}}(E).
}
$$

Implications:

$$
\boxed{
\mathrm{SR}_{\mathrm{full}}
\Rightarrow
\mathrm{SR}_{HT^2}
\Rightarrow
\mathrm{SR}_{S}
\Rightarrow
\mathcal O_{E,S}=0
}
$$

where the final implication uses Weil Hodge permanence.

HC-True should stop as soon as any earlier level succeeds.

---

# 36. From first order to PT003 dominance

If:

$$
\mathcal O_{E,S}=0,
$$

then:

$$
df_{[E]}
$$

is surjective onto:

$$
T_sS.
$$

If higher-order obstruction images are likewise killed by the formal component semiregularity condition:

$$
\mathrm{FSR}_{S}(E),
$$

then the relative deformation functor is formally smooth.

Under algebraization/effectivity, PT003 gives a dominating object family.

Thus PT004 supplies the missing local operator behind PT003.

---

# 37. Exact collision with HC-False

HC-False wants to prove every possible Weil seed has a deformation defect.

One sharp form is:

$$
\boxed{
\Delta_{At}(E/S)>0
}
$$

for all candidate seeds.

HC-True wants one seed with:

$$
\boxed{
\Delta_{At}(E/S)=0
}
$$

and higher-order unobstructedness.

So the True/False collision is now reduced to an actual finite-dimensional Atiyah map.

This is stronger than comparing vague cycle families.

---

# 38. What PT004 proves

PT004 proves:

$$
\boxed{
\text{full semiregularity is unnecessary}.
}
$$

It proves the minimal first-order criterion:

$$
\boxed{
\ker\sigma_E
\cap
\operatorname{ObIm}_{S}(E)
=
0.
}
$$

It proves that under Weil Hodge permanence this criterion forces:

$$
\boxed{
\mathcal O_{E,S}=0.
}
$$

It identifies the relevant obstruction image as at most:

$$
9
$$

-dimensional.

It identifies Markman's current image-restricted semiregularity question as a stronger, Fourier–Mukai-stable version of exactly the same gate.

---

# 39. What PT004 does not prove

PT004 does not prove:

$$
\boxed{
\mathrm{SR}_{S}(E)
}
$$

for the generalized secant object.

It does not compute:

$$
\operatorname{Ext}^2(E,E)
$$

for the target seed.

It does not establish higher-order formal smoothness.

It therefore does not yet close a non-split Weil component.

The positive gate is now precise, not solved.

---

# 40. Next Interface

Next HC-True round:

```text
HODGE_HCTRUE_PT005_SecantExtRepresentation.md
```

Primary target:

$$
\boxed{
\mathcal O_{E,S}
=
at_E\circ KS_S
}
$$

for:

$$
E
=
\Phi(F_1\boxtimes F_2^\vee).
$$

Planned attacks:

1. transport the Hochschild/Atiyah action through Orlov's Fourier–Mukai equivalence;
2. use:
   $$
   at_{F_1\boxtimes F_2^\vee}
   =
   at_{F_1}\boxtimes1
   +
   1\boxtimes at_{F_2^\vee};
   $$
3. compute the Künneth decomposition of:
   $$
   \operatorname{Ext}^2;
   $$
4. identify which summands receive the Weil tangent representation;
5. exploit known semiregularity of the first secant factor where available;
6. isolate the unresolved:
   $$
   F_2
   $$
   obstruction block;
7. test multiplicity-one of the tangent representation;
8. reduce:
   $$
   \Delta_{At}(E/S)
   $$
   to one or a few scalar matrix coefficients if possible.

---

# References

1. R.-O. Buchweitz, H. Flenner, *A Semiregularity Map for Modules and Applications to Deformations*, arXiv:math/9912245. Constructs
   $$
   \sigma_E:\operatorname{Ext}^2(E,E)\to\prod_q H^{q+2}(\Omega^q)
   $$
   from Atiyah classes and traces and relates it to deformation theory.

2. J. P. Pridham, *Semiregularity as a consequence of Goodwillie's theorem*, arXiv:1208.3111. Shows semiregularity maps annihilate all obstructions and measure failure of Chern characters to remain Hodge under deformation.

3. E. Markman, *Cycles on abelian $2n$-folds of Weil type from secant sheaves on abelian $n$-folds*, arXiv:2502.03415, 2025. Defines semiregularity for coherent/projective/twisted sheaves in terms of Atiyah classes and proves the split sixfold theorem.

4. E. Markman, *Secant sheaves on abelian $n$-folds with real multiplication and Weil classes on abelian $2n$-folds with complex multiplication*, arXiv:2509.23079, 2025/2026. Proves Hodge permanence of the normalized secant-object class and, in Question 11.2.2, explicitly asks for injectivity of the semiregularity map restricted to
   $$
   \operatorname{Im}
   \left(
   at_F:HT^2(X)\to\operatorname{Ext}^2(F,F)
   \right).
   $$

5. Aletheia, *HODGE_HCTRUE_PT003_OneRelativeWeilCycle*, 2026-09-16.

---

## Canonical Source Declaration

本檔案為 HC-True 分支第四篇正式 UTF-8 Markdown canonical source。

數學原始碼只使用 `$...$` 與 `$$...$$`。

本輪沒有證明 generalized secant seed 已 semiregular；本輪證明的是 HC-True 只需要 image-restricted semiregularity，並將其在 Weil component 上等價壓縮為最多九維的 Atiyah-annihilator problem。