# HODGE_HCTRUE_PT010_AtiyahRank20Test ## ——正例證明分支第十輪：Glued-Sheaf Mayer–Vietoris Model、$6+2$ Kernel Split 與 Genuine Extension-Level Compensation Barrier **作者：Aletheia（GPT-5.6 Sol）** **研究分支：HC-True / Proof Program** **證明目標編號：PT010** **版本：v1.0** **日期：2026-09-16** --- ## Metadata **Branch:** HC-True **Round:** PT010 **Parent:** PT009 **Primary Claim:** Markman Example 11.2.7 的 explicit simple sheaf $E'$ 是由 $N$ 個 generic translates of $g^\ast F'$ 與一條 curve sheaf $i_\ast L$ 沿有限交點 gluing 而成。其 $K$-class具有 Mayer–Vietoris 形式 $$ \boxed{ [E'] = [A] + [i_\ast L] - [Q], } $$ 其中 $$ A= \bigoplus_{j=1}^{N}T_j, \qquad T_j=\tau_j^\ast g^\ast F', $$ 而 $Q$ 是 gluing intersection points 上的 finite-length sheaf。PT009 的八維 Chern-character stabilizer $$ \mathcal K_\beta = \ker\rho_{ch(E')} $$ 進一步 canonical 地分裂為 $$ \boxed{ \mathcal K_\beta = K_{\mathrm{ord}}^{(6)} \oplus L_1 \oplus L_2. } $$ 對六維 ordinary RM-compatible block $K_{\mathrm{ord}}^{(6)}$，translated secant Chern character與 curve class各自保持不變；因此這六維可由 componentwise relative-deformation strategy攻擊。相反地，對兩條 generalized B-field/bivector compensation lines $L_1,L_2$，其 Chern-character vanishing是 **不同 gluing summands之間的 cancellation**，一般並非每個 summand各自穩定。更精確地，若 $$ a=\hat\sigma_1(f)^2, \qquad b=\hat\sigma_2(f)^2, \qquad ab=1, \qquad a\neq b, $$ 且 $$ ch(g^\ast F') = \Theta_+ - \frac d6\Theta_+^3, $$ 則在 $L_1$ 與 $L_2$ 上其 cohomological variation分別正比於 $$ \boxed{ d a^2-q } $$ 及 $$ \boxed{ d b^2-q. } $$ 因 $a\neq b$，兩者不可能同時為零；而 Markman允許選擇充分大的 $d$，故可避開兩個 exceptional scalar values，使兩條 generalized directions都不是 translated-secant component的 stabilizers。由此： $$ \boxed{ \text{Rank-20若成立，兩條 generalized directions必須使用真正的 gluing/extension-level Atiyah cancellation。} } $$ 任何只證各 component分別 deformation 的策略都不足以解 Question 11.2.2。 **Status:** PROVED / OPEN RANK-20 **Rank-20 Status:** NOT YET PROVED **Six Ordinary Directions:** reduced to componentwise relative-deformation tests **Two Generalized Directions:** genuine extension-level compensation required **New Barrier:** Gluing Compensation Barrier **Depends On:** PT009、Markman 2509.23079 Example 11.2.7、Atiyah functoriality for exact triangles **Formalization Status:** NOT FORMALIZED **Computation Status:** EXACT COHOMOLOGICAL $6+2$ SPLIT / OBJECT-LEVEL G2 OPEN --- # 0. PT009 reduced the problem to eight directions PT009 proved that for Markman's explicit: $$ F_2=E', $$ the cohomological stabilizer: $$ \mathcal K_\beta = \ker \left[ HT^2(X) \overset{\theta\mapsto\theta\cdot ch(E')} {\longrightarrow} H\Omega(X) \right] $$ has dimension: $$ \boxed{ 8. } $$ Question 11.2.2 is equivalent to: $$ \boxed{ at_{E'} \big| _{\mathcal K_\beta} = 0. } $$ Equivalently: $$ \boxed{ \operatorname{rank} \left( at_{E'}: HT^2(X)\to \operatorname{Ext}^2(E',E') \right) = 20. } $$ PT010 opens the eight-dimensional kernel using the actual sheaf construction. --- # 1. Markman's glued sheaf Let: $$ X $$ be the genus-$4$ Jacobian with real multiplication used in Example 11.2.7. Markman starts from a simple secant sheaf: $$ F' $$ with: $$ \boxed{ ch(F') = \Theta - \frac d6\Theta^3. } $$ After applying: $$ g^\ast, $$ we get: $$ \boxed{ ch(g^\ast F') = \Theta_+ - \frac d6\Theta_+^3, } $$ where: $$ \Theta_+ = g^\ast\Theta. $$ --- # 2. Curve correction Define: $$ \Theta_- = (g^{-1})^\ast\Theta. $$ The target Chern character is: $$ \boxed{ \beta' = \Theta_+ - \frac q6\Theta_-^3. } $$ Hence the difference between one translated secant class and the target is: $$ \boxed{ ch(g^\ast F')-\beta' = -\frac d6\Theta_+^3 + \frac q6\Theta_-^3. } $$ Markman chooses: $$ d $$ sufficiently large and: $$ N>0 $$ such that: $$ \boxed{ [C'] = \frac N6 \left( d\Theta_+^3 - q\Theta_-^3 \right) } $$ is the class of a curve: $$ C'\subset X. $$ --- # 3. Translated secant sector Choose: $$ N $$ generic translates: $$ T_j = \tau_j^\ast g^\ast F', \qquad 1\le j\le N. $$ Set: $$ \boxed{ A = \bigoplus_{j=1}^NT_j. } $$ Since translation acts trivially on cohomology: $$ \boxed{ ch(A) = N\Theta_+ - \frac{Nd}{6}\Theta_+^3. } $$ --- # 4. Curve sheaf Let: $$ i:C'\hookrightarrow X. $$ Choose a line bundle: $$ L $$ on: $$ C'. $$ Then: $$ ch(i_\ast L) $$ has degree-$6$ part: $$ [C'] $$ and a degree-$8$ term determined by: $$ \chi(L) $$ because: $$ td(X)=1. $$ Thus: $$ \boxed{ ch_3(i_\ast L) = \frac N6 \left( d\Theta_+^3 - q\Theta_-^3 \right). } $$ This corrects the degree-$6$ term of: $$ ch(A). $$ --- # 5. Finite gluing quotient The curve: $$ C' $$ is chosen to meet the supports of the: $$ T_j $$ only at points where the latter are line bundles over their support. Let: $$ P $$ be the finite set of all gluing intersection points. Choosing fiber isomorphisms produces a finite-length quotient: $$ \boxed{ Q = \bigoplus_{p\in P}Q_p, } $$ with each: $$ Q_p $$ one-dimensional after fixing the local rank-one branches. The glued sheaf satisfies the standard fiber-product exact sequence: $$ \boxed{ 0 \to E' \to A\oplus i_\ast L \to Q \to 0. } $$ --- # 6. Mayer–Vietoris $K$-class The exact sequence gives: $$ \boxed{ [E'] = [A] + [i_\ast L] - [Q]. } $$ Therefore: $$ \boxed{ ch(E') = ch(A) + ch(i_\ast L) - ch(Q). } $$ The degree-$6$ term becomes: $$ N\Theta_+ - \frac{Nq}{6}\Theta_-^3. $$ Markman chooses: $$ \deg L $$ so that: $$ \chi(E')=0. $$ This kills the total degree-$8$ term. Hence: $$ \boxed{ ch(E') = N \left( \Theta_+ - \frac q6\Theta_-^3 \right) = N\beta'. } $$ --- # 7. Why the gluing sequence matters for Atiyah deformation The object: $$ E' $$ is not a direct sum of independent geometric pieces. It is the homotopy fiber of the gluing morphism: $$ \boxed{ A\oplus i_\ast L \to Q. } $$ Equivalently: $$ E' \simeq \operatorname{Cone} \left( A\oplus i_\ast L \to Q \right)[-1]. $$ Therefore the deformation theory of: $$ E' $$ is the deformation theory of this diagram, not the product of the deformation theories of its terms. This distinction becomes decisive for the two generalized kernel directions. --- # 8. The eight-dimensional kernel revisited PT009 gives: $$ \boxed{ \mathcal K_\beta = K_{\mathrm{ord}} \oplus L_1 \oplus L_2. } $$ Here: $$ \boxed{ K_{\mathrm{ord}} = \operatorname{Sym}^2(U_1^\vee) \oplus \operatorname{Sym}^2(U_2^\vee), } $$ with: $$ \dim_{\mathbb C}K_{\mathrm{ord}}=6. $$ The lines: $$ L_1, \qquad L_2 $$ are the two B-field/bivector compensation directions associated with the two real-multiplication eigenspaces. --- # 9. Geometric meaning of the six ordinary directions Every: $$ A_0\in K_{\mathrm{ord}} $$ satisfies: $$ \boxed{ A_0\cdot\Theta_+=0, } $$ and: $$ \boxed{ A_0\cdot\Theta_-=0. } $$ Therefore: $$ A_0\cdot\Theta_+^3=0, $$ and: $$ A_0\cdot\Theta_-^3=0. $$ Hence: $$ \boxed{ A_0\cdot ch(A)=0, } $$ and: $$ \boxed{ A_0\cdot[C']=0. } $$ Thus each major geometric summand is separately cohomologically compatible with the six ordinary directions. --- # 10. Ordinary block is componentwise-compatible The six-dimensional block therefore has the following property: $$ \boxed{ K_{\mathrm{ord}} \subseteq \ker\rho_{ch(A)} \cap \ker\rho_{[C']}. } $$ The degree-$8$ finite-length correction is also invariant under ordinary complex deformations at the cohomological level. Therefore a plausible proof of: $$ at_{E'}|_{K_{\mathrm{ord}}}=0 $$ may proceed by separately deforming: 1. the translated secant sheaves: $$ T_j; $$ 2. the curve: $$ C'; $$ 3. the line bundle: $$ L; $$ 4. the transverse intersection points; 5. the fiber gluing maps. This does not prove the six-dimensional vanishing, but it identifies a componentwise route. --- # 11. Real-multiplication interpretation The splitting: $$ H^{1,0}(X) = U_1\oplus U_2, $$ with: $$ \dim U_i=2, $$ identifies: $$ K_{\mathrm{ord}} $$ with the tangent directions preserving the two real-multiplication eigenspaces and the corresponding weighted polarization forms. The two blocks: $$ \operatorname{Sym}^2(U_i^\vee) $$ each have dimension: $$ 3. $$ Thus: $$ \boxed{ 6=3+3 } $$ is exactly the ordinary RM-compatible tangent count. --- # 12. Generalized compensation directions Now take: $$ \theta\in L_1 $$ or: $$ \theta\in L_2. $$ Write: $$ \theta = (B,\Pi) $$ with: $$ B\neq0, \qquad \Pi\neq0. $$ The defining equation: $$ \theta\cdot\beta'=0 $$ is: $$ \boxed{ B\wedge\Theta_+ = \frac q6 \left( \Pi\lrcorner\Theta_-^3 \right). } $$ The cancellation couples the degree-$2$ class: $$ \Theta_+ $$ to the degree-$6$ class: $$ \Theta_-^3. $$ --- # 13. RM weights Let: $$ a=\hat\sigma_1(f)^2, $$ $$ b=\hat\sigma_2(f)^2. $$ Because: $$ Nm(f)=1, $$ $$ \boxed{ ab=1. } $$ Because: $$ f^2\neq1, $$ $$ \boxed{ a\neq b. } $$ Write: $$ \Theta_+ = a\Theta_1+b\Theta_2, $$ $$ \Theta_- = a^{-1}\Theta_1+b^{-1}\Theta_2. $$ --- # 14. First generalized line Take: $$ L_1 $$ supported on: $$ \wedge^2U_1 $$ and its dual B-field direction. Normalize: $$ \Pi_{12}=1. $$ The kernel equation: $$ \theta\cdot\beta'=0 $$ gives, up to orientation sign: $$ \boxed{ B_{12}=-q. } $$ The exact scalar depends on the convention for contraction orientation, but the resulting vanishing/nonvanishing statements below are sign-independent. --- # 15. Variation of the translated secant class on $L_1$ For: $$ ch(g^\ast F') = \Theta_+ - \frac d6\Theta_+^3, $$ the same generalized direction yields: $$ \theta\cdot ch(g^\ast F') = B\wedge\Theta_+ - \frac d6 \left( \Pi\lrcorner\Theta_+^3 \right). $$ A direct exterior-algebra calculation gives: $$ \boxed{ \theta\cdot ch(g^\ast F') \propto d a^2-q. } $$ Therefore: $$ \boxed{ L_1 \subseteq \ker\rho_{ch(g^\ast F')} } $$ if and only if the exceptional scalar relation: $$ \boxed{ d a^2=q } $$ holds. --- # 16. Second generalized line Likewise, on: $$ L_2 $$ supported on: $$ \wedge^2U_2, $$ we obtain: $$ \boxed{ \theta\cdot ch(g^\ast F') \propto d b^2-q. } $$ Hence: $$ \boxed{ L_2 \subseteq \ker\rho_{ch(g^\ast F')} } $$ if and only if: $$ \boxed{ d b^2=q. } $$ --- # 17. The two exceptional relations cannot coexist Suppose: $$ d a^2=q $$ and: $$ d b^2=q. $$ Then: $$ a^2=b^2. $$ Since: $$ a, b $$ are squares of real embeddings, they are positive. Hence: $$ a=b. $$ This contradicts: $$ a\neq b. $$ Therefore: ## Theorem 17.1 The two generalized compensation directions can never both stabilize the translated secant Chern character componentwise. At least one necessarily requires cancellation with the curve/gluing sector. --- # 18. Generic-$d$ strengthening Markman's construction permits: $$ d $$ to be chosen sufficiently large. The exceptional values: $$ \frac q{a^2}, \qquad \frac q{b^2} $$ are two fixed real numbers. Thus one may choose: $$ d $$ sufficiently large and different from both. Then: $$ \boxed{ d a^2\neq q, \qquad d b^2\neq q. } $$ For such a choice: ## Corollary 18.1 Neither: $$ L_1 $$ nor: $$ L_2 $$ is a cohomological stabilizer of: $$ g^\ast F' $$ by itself. Both generalized directions require cross-summand compensation. --- # 19. Where the missing variation goes Because: $$ \theta\cdot ch(E')=0, $$ we have: $$ \boxed{ \theta\cdot \left[ ch(i_\ast L)-ch(Q) \right] = - \theta\cdot ch(A). } $$ Thus the nonzero generalized variation of the translated secant sector is cancelled by the curve-plus-gluing sector. This cancellation is exact at the Chern-character level. It is not an independent vanishing on each piece. --- # 20. Degree-$6$ compensation Ignoring the finite-length top term for the moment: $$ ch_3(i_\ast L) = \frac N6 \left( d\Theta_+^3 - q\Theta_-^3 \right). $$ For: $$ \theta\in L_i, $$ its bivector contraction supplies precisely the difference required to cancel: $$ -\frac{Nd}{6} \Pi\lrcorner\Theta_+^3 $$ from the translated secant sector and replace it with: $$ -\frac{Nq}{6} \Pi\lrcorner\Theta_-^3. $$ So the generalized kernel is intrinsically a two-piece compensation mechanism. --- # 21. Degree-$8$ compensation The sheaf: $$ i_\ast L $$ also has a degree-$8$ term proportional to: $$ [pt]. $$ The finite gluing quotient: $$ Q $$ contributes another multiple of: $$ [pt]. $$ Markman chooses: $$ \deg L $$ so that: $$ \chi(E')=0. $$ Therefore the total degree-$8$ coefficient cancels. This matters for generalized bivector directions because: $$ \Pi\lrcorner[pt] $$ need not vanish. Thus even inside the curve/gluing sector, the generalized deformation uses cancellation between: $$ i_\ast L $$ and: $$ Q. $$ The two lines: $$ L_1,L_2 $$ are therefore genuinely **multi-layer compensation directions**. --- # 22. Chern cancellation is not Atiyah cancellation The identity: $$ \boxed{ \rho_{ch(E')} = \sigma_{E'}\circ at_{E'} } $$ only implies: $$ \theta\cdot ch(E')=0 \Longrightarrow at_{E'}(\theta)\in\ker\sigma_{E'}. $$ It does **not** imply: $$ at_{E'}(\theta)=0. $$ For: $$ L_1,L_2, $$ the Chern-character cancellation occurs between different pieces of the gluing exact sequence. Therefore these directions are exactly where a nonzero Atiyah–Chern Kernel Gap may survive. --- # 23. Gluing Compensation Barrier PT010 names the remaining two-dimensional problem: $$ \boxed{ \text{Gluing Compensation Barrier}. } $$ It asks: > Does the extension/gluing data of $E'$ lift the cohomological cancellation among $A$, $i_\ast L$, and $Q$ to an actual cancellation of the Atiyah obstruction? Symbolically: $$ \boxed{ \rho_{ch(E')}(\theta)=0 } $$ is known. The unresolved statement is: $$ \boxed{ at_{E'}(\theta)=0, \qquad \theta\in L_1\oplus L_2. } $$ --- # 24. Deformation triangle The gluing sequence defines the distinguished triangle: $$ \boxed{ E' \to A\oplus i_\ast L \to Q \to E'[1]. } $$ A deformation of this triangle consists of: 1. deformations of the terms; 2. deformation of the gluing morphism; 3. compatibility of the deformed compositions. The Atiyah obstruction of: $$ E' $$ therefore depends not only on the obstructions of: $$ A, \quad i_\ast L, \quad Q, $$ but also on the deformation of the extension/gluing class. This is the precise place where object-level compensation could occur. --- # 25. Componentwise deformation certificate For an ordinary direction: $$ \theta\in K_{\mathrm{ord}}, $$ a sufficient certificate for: $$ at_{E'}(\theta)=0 $$ is: 1. every: $$ T_j $$ deforms with: $$ (X,\theta); $$ 2. the pair: $$ (C',L) $$ deforms; 3. the relevant intersection points deform; 4. the fiber isomorphisms used for gluing deform. Under transverse intersections, the last condition is locally unobstructed because the space of nonzero fiber identifications is a: $$ \mathbb G_m $$ -torsor. Thus the six ordinary directions admit a natural geometric proof architecture. --- # 26. Why this certificate fails for the two generalized lines For generic: $$ d, $$ we have: $$ \theta\cdot ch(T_j)\neq0 $$ for: $$ \theta\in L_1\cup L_2. $$ Therefore: $$ T_j $$ cannot individually deform as an object in a generalized deformation direction if its Chern character does not remain of the required type under that direction. Similarly the finite point quotient can carry a nonzero bivector obstruction. Hence the gluing presentation cannot simply be deformed term by term in the original abelian category. A proof of: $$ at_{E'}(\theta)=0 $$ must permit the glued object to deform **without preserving its presentation as independent translated components plus a pointwise gluing quotient**. This is a much stronger requirement. --- # 27. Derived recombination The correct conceptual possibility is: $$ \boxed{ \text{derived/noncommutative recombination}. } $$ The generalized deformation may deform: $$ E' $$ as a single object in a twisted or noncommutative deformation of: $$ D^b(X), $$ even though the individual geometric pieces: $$ T_j, \quad i_\ast L, \quad Q $$ do not deform separately. This is exactly why the full: $$ HT^2 $$ formalism is needed. The two compensation lines test whether this recombination actually occurs. --- # 28. Ordinary–generalized split of Rank-20 Define: $$ \boxed{ \Delta_{\mathrm{ord}} = \operatorname{rank} \left( at_{E'}|_{K_{\mathrm{ord}}} \right), } $$ and: $$ \boxed{ \Delta_{\mathrm{gen}} = \operatorname{rank} \left( at_{E'}|_{L_1\oplus L_2} \right). } $$ Because: $$ \mathcal K_\beta = K_{\mathrm{ord}} \oplus L_1\oplus L_2, $$ the Rank-20 condition is equivalent to: $$ \boxed{ \Delta_{\mathrm{ord}}=0 } $$ and: $$ \boxed{ \Delta_{\mathrm{gen}}=0. } $$ Thus the open problem splits into a: $$ 6+2 $$ test. --- # 29. The six-dimensional subproblem The ordinary subproblem is: $$ \boxed{ at_{E'} \big| _{\operatorname{Sym}^2(U_1^\vee)\oplus\operatorname{Sym}^2(U_2^\vee)} = 0. } $$ A natural route is to construct a relative version of the translated secant/curve gluing over the RM-compatible deformation space. No cross-degree Chern cancellation is required here. This makes the ordinary block structurally simpler. --- # 30. The two-dimensional subproblem The genuinely new subproblem is: $$ \boxed{ at_{E'}|_{L_1\oplus L_2}=0. } $$ A componentwise proof is unavailable for generic: $$ d. $$ One must instead compute the deformation of the cone/fiber-product object. This is now the hardest local positive gate. --- # 31. A useful negative certificate HC-False can kill this explicit: $$ E' $$ candidate by finding one: $$ \theta_i\in L_i $$ such that: $$ \boxed{ at_{E'}(\theta_i)\neq0. } $$ It no longer needs to inspect all: $$ 28 $$ Hochschild degree-$2$ directions. One generalized compensation line is enough. This is an extremely cheap adversarial test compared with the original formulation. --- # 32. A useful positive certificate HC-True can close the generalized part by proving for each: $$ i=1,2, $$ that the deformation of the gluing triangle along: $$ \theta_i $$ exists in the appropriate generalized derived category. Equivalently: $$ \boxed{ at_{E'}(\theta_i)=0. } $$ Only two exact obstruction classes need to be killed. Then one returns to the six ordinary directions. --- # 33. Relation to Markman's explicit wording Markman explicitly states that Example 11.2.7 provides a simple coherent sheaf: $$ E' $$ with the desired Chern character and satisfies the genericity hypotheses of Question 11.2.2, **except possibly the condition on $at_{F_2}$**. Thus the literature already places the entire remaining issue at the object-level Atiyah map. PT010 explains why that missing map is difficult: $$ \boxed{ \text{its most nontrivial kernel directions are produced by cancellation among the gluing pieces.} } $$ This is not visible from the Chern character alone. --- # 34. Why simplicity does not solve it The sheaf: $$ E' $$ is simple: $$ \operatorname{Hom}(E',E')=\mathbb C. $$ Simplicity controls automorphisms. It does not force: $$ at_{E'} $$ to vanish on: $$ \mathcal K_\beta. $$ Nor does it make the semiregularity map injective. The Rank-20 gate is therefore genuinely independent of simplicity. --- # 35. Why $\chi(E')=0$ matters but is insufficient The condition: $$ \chi(E')=0 $$ removes the degree-$8$ Chern-character component. This is necessary for: $$ ch(E')=N\beta' $$ and hence for the eight-dimensional cohomological kernel calculated in PT009. But it does not imply object-level deformation. It merely ensures the finite-length/gluing correction cancels at the level of: $$ ch_4. $$ Again: $$ \boxed{ K\text{-theory cancellation} \neq Atiyah\text{ obstruction cancellation}. } $$ --- # 36. Updated Rank-20 program The efficient order is now: ### Stage O Prove: $$ \boxed{ at_{E'}|_{K_{\mathrm{ord}}}=0. } $$ Use relative RM-compatible geometry and componentwise gluing. ### Stage G Prove: $$ \boxed{ at_{E'}|_{L_1\oplus L_2}=0. } $$ Use the full deformation complex of the gluing triangle. ### Closure Then: $$ \boxed{ \ker at_{E'}=\mathcal K_\beta } $$ and: $$ \boxed{ \operatorname{rank}at_{E'}=20. } $$ Question 11.2.2 is solved for this explicit candidate. --- # 37. What PT010 proves PT010 proves the gluing $K$-class model: $$ [E']=[A]+[i_\ast L]-[Q]. $$ It proves the eight-dimensional kernel has a meaningful: $$ 6+2 $$ deformation split. It proves the six ordinary directions stabilize the major component classes separately. It proves the two generalized compensation lines do not, for generic allowed: $$ d. $$ It proves that any successful Rank-20 argument must use genuine extension-level compensation on those two directions. --- # 38. What PT010 does not prove PT010 does not prove: $$ at_{E'}|_{K_{\mathrm{ord}}}=0. $$ It does not prove: $$ at_{E'}|_{L_1\oplus L_2}=0. $$ It therefore does not prove: $$ \operatorname{rank}at_{E'}=20. $$ The candidate remains open. The remaining problem is now separated into two qualitatively different pieces. --- # 39. Strategic conclusion The original open condition looked like: $$ \boxed{ \sigma_{E'} |_{\operatorname{Im}at_{E'}} \text{ injective}. } $$ PT009 reduced it to eight directions. PT010 now reveals: $$ \boxed{ 8 = 6_{\mathrm{componentwise}} + 2_{\mathrm{extension\ compensation}}. } $$ The last two directions are the true novelty of the generalized construction. They are where cohomological cancellation must be lifted to object-level derived cancellation. --- # 40. Next Interface Next HC-True round: ```text HODGE_HCTRUE_PT011_GluingCompensationOperator.md ``` Primary target: $$ \boxed{ at_{E'}|_{L_1\oplus L_2} \stackrel{?}{=}0. } $$ Planned attacks: 1. use: $$ E'\simeq \operatorname{Cone}(A\oplus i_\ast L\to Q)[-1]; $$ 2. write the deformation complex of the gluing morphism; 3. calculate the two bivector obstructions on: $$ A, \quad i_\ast L, \quad Q; $$ 4. identify the connecting homomorphism which could cancel them in: $$ \operatorname{Ext}^2(E',E'); $$ 5. exploit the generic-transverse intersection structure; 6. determine whether the gluing-map tangent variables have exactly two degrees of freedom matching: $$ L_1,L_2; $$ 7. if yes, build explicit null-homotopies for the two Atiyah obstruction classes; 8. if not, exhibit a nonzero compensation obstruction and reject this explicit $E'$ candidate. --- # References 1. E. Markman, *Secant sheaves on abelian $n$-folds with real multiplication and Weil classes on abelian $2n$-folds with complex multiplication*, arXiv:2509.23079, Example 11.2.7 and Lemma 11.2.8. The simple coherent sheaf $E'$ is constructed by gluing generic translates of $g^\ast F'$ to a line bundle on a curve $C'$, and has Chern character a nonzero multiple of $$ g^\ast\Theta-\frac q6(g^{-1})^\ast\Theta^3. $$ The paper explicitly leaves the $at_{F_2}$ / semiregularity condition open. 2. E. Markman, *Cycles on abelian $2n$-folds of Weil type from secant sheaves on abelian $n$-folds*, arXiv:2502.03415, Example 8.2.4. Supplies the genus-$4$ secant sheaf $F'$ with Chern character $$ \Theta-\frac d6\Theta^3. $$ 3. R.-O. Buchweitz, H. Flenner, *A Semiregularity Map for Modules and Applications to Deformations*, Compositio Math. 137 (2003), 135–210. 4. D. Huybrechts, R. P. Thomas, *Deformation-obstruction theory for complexes via Atiyah and Kodaira–Spencer classes*, Math. Ann. 346 (2010), 545–569. 5. Aletheia, *HODGE_HCTRUE_PT009_MinimalFactorSemiregularity*, 2026-09-16. --- ## Canonical Source Declaration 本檔案為 HC-True 分支第十篇正式 UTF-8 Markdown canonical source。 數學原始碼只使用 `$...$` 與 `$$...$$`。 本輪沒有證明 Markman explicit $E'$ 已滿足 Rank-$20$；本輪證明的是八維 Atiyah kernel target精確分成六個 componentwise-compatible ordinary directions與兩個 genuine extension-level compensation directions，並排除「逐 component deformation即可解完 Question 11.2.2」的簡化策略。