Nodal Jacobians, Diagonal Nearby Cycles, and Local P = W for an Endoscopic Prym Sector

Manuscript author: Mohamed Ben Ammar Chraiti (Independent Researcher, Tunis, Tunisia). Abstract of this summary. This document lists, in an organized form, the mathematical results that the manuscript claims as its own contribution, together with the lemmas that the manuscript proves relative to cited theorems and the classical tools it uses. The manuscript has three parts: Part I (nodal Jacobians, orientation, saturation and limit weights, including explicit Hitchin slices), Part II (a categorical comparison of diagonal and iterated nearby cycles under exact product charts), and Part III (a conditional local, sectorwise and blockwise P=W statement for the determinant-fixed endoscopic Prym sector). Novelty and complete validity of the claims require independent bibliographic and mathematical verification; this summary reports what the manuscript asserts and under which hypotheses. 1. Conventions and status labels----------------------------------- 1.1 Status labels. Each result below carries one of three labels. - Status: original. A result claimed as new by the manuscript.- Status: relative. A lemma proved in the manuscript, but relative to cited theorems (for example the Beauville-Narasimhan-Ramanan correspondence, the deformation theory of compactified Jacobians, Artin approximation, Saito's formalism).- Status: classical input. A known tool (Picard-Lefschetz theory, Schmid-Steenbrink theorem, Saito's mixed Hodge modules, and so on), recalled for use. 1.2 Normalized cycles. For a function \(g\) and a complex \(F\) of mixed Hodge modules, Saito's normalized unipotent functors satisfy \[rat(ψg,1F)=Rψg,1(ratF)[-1], rat(φg,1F)=Rφg,1(ratF)[-1].\] They are \(t\)-exact and commute with proper direct image. The canonical and variation morphisms are \[can:ψg,1F→φg,1F, Var:φg,1F→ψg,1F(-1),\] with \(Var=Nψ\) and \(can(-1)=Nφ\). For unipotent \(T\) one sets \(L=log T\) (a finite sum), and in the Hodge-module normalization \(N:V→ V(-1)\), \(NC=(2π i)⁻¹LC\), \(TC=exp(2π i NC)\). The logarithm must not be replaced by \(T-I\). 1.3 Degree convention. If \(X\) is smooth of dimension \(d\) and \(F_X= Q_X^H[d]\), then ordinary degree \(q\) corresponds to hypercohomological degree \(r=q-d+1\): \[V^qₗᵢₘ= Hq-d+1(X_0,ψf,1F_X).\] For a family of relative dimension \(G\) over a disk, \(d=G+1\) and \(r=q-G\). On a bidisk, \(d=G+2\) and two normalized functors give again \(q-G\). 1.4 Monodromy filtration. For a nilpotent \(L\), the monodromy filtration \(M(L;q)\) centered at \(q\) is the unique finite increasing filtration with \[L\,M_j⊂ Mⱼ₋₂, L^r:Gr^Mq+rV{~}Gr^Mq-rV(r≥ 0).\] For a polarized pure variation of weight \(q\) with unipotent monodromy, the Schmid-Steenbrink theorem gives \(W'•V^qₗᵢₘ=M(N_q;q)•V^qₗᵢₘ\), where \(W'\) denotes the limit weight filtration. This is applied only to blocks known to come from such a variation. Status: classical input. 1.5 Notation. \( D\) is a disk, \( D×= D\{0\}\). Binomial coefficients \( ab\) vanish for \(b<0\) or \(b>a\). The symbol \(Γ=Pic^0(C)[2]\) is used only in Part III. 2. Part I: Nodal Jacobians and limit weights---------------------------------------------- 2.1 Elementary lemmas on filtrations and ranks. Lemma 2.1 (image-kernel description). Let \(N\) be nilpotent on a finite-dimensional rational space \(V\). - If \(N^2=0\), then \(M(N;q)_j\) equals \(0\) for \(j≤ q-2\), \(ImN\) for \(j=q-1\), \(KerN\) for \(j=q\), and \(V\) for \(j≥ q+1\).- If \(N^3=0\), then \(M(N;q)_j\) equals \(0\) for \(j≤ q-3\), \(ImN^2\) for \(j=q-2\), \(ImN\) for \(j=q-1\), \(ImN+KerN\) for \(j=q\), \(KerN^2\) for \(j=q+1\), and \(V\) for \(j≥ q+2\). Status: original (elementary, proved by Jordan decomposition). Lemma 2.2 (rank saturation). Let \(V{a}B{b}V'\) be linear maps. If \( B=rank(ba)\), then \(a\) is surjective, \(b\) is injective, and \[Imb=Im(ba), Kera=Ker(ba).\] Status: original (elementary). It is the mechanism that turns a dimension count into maximal rank in every later saturation statement. 2.2 The naive local filtration has a single jump. Definition 2.3. For \(K=Rf_{}F_X\) and \(r=q-d+1\), the naive local perverse-truncation filtration is \[Lˡᵒᶜ_aV^qₗᵢₘ=Im[H^r(ψt,1\,τ≤ aK)→ H^r(ψt,1K)].\] Proposition 2.4 (degeneration of the naive filtration). One has \[Lˡᵒᶜ_aV^qₗᵢₘ={cases}0,&a<r,\\ V^qₗᵢₘ,&a≥ r.{cases}\] The statement holds for every bounded rational constructible complex \( K\) on \( D\) and every integer \(r\), with \(Ψ=Rψt,1[-1]\) or the full functor \(Rψ_t[-1]\); no Hodge-module lift and no decomposition-theorem splitting is used. In particular, if \(N≠0\) on \(V^qₗᵢₘ\), then \(Lˡᵒᶜ•\) cannot recover the intermediate levels of the monodromy filtration. Status: original. The proof uses only \(t\)-exactness of normalized nearby cycles and the fact that perverse and ordinary \(t\)-structures agree on a point. Remark 2.5 (compatibility with the literature). The statement does not contradict the specialization results of de Cataldo and Maulik for perverse Leray filtrations in families, which concern a different variation parameter, nor the global P=W theorems, which compare a perverse filtration on global Dolbeault cohomology with weights on a character variety. Those weights are not the limit weights \(W'\). Corollary 2.6 (nodal elliptic counterexample). For an integral one-nodal elliptic degeneration, for instance \(y^2z=x^2(x+z)+tz^3\) in \( P^2× D\), no reindexing of \(Lˡᵒᶜ\) equals \(W'₂ₖH^1ₗᵢₘ\). Indeed \(0 W'_0 W'_2=H^1ₗᵢₘ\) with dimensions \(1\) and \(2\), whereas \(Lˡᵒᶜ\) jumps only from \(0\) to the whole space. Status: original. 2.3 The pinching filtration and nodal comparison in all degrees. Hypothesis 2.7 (integral nodal degeneration). Let \(p: S→ D\) be a flat projective family of curves, smooth of genus \(G\) over \( D×\), with integral central fiber having exactly \(δ\) ordinary nodes and no other singularity. Near the \(i\)-th node the local equation is \(xy=τm_iε_i(τ)\) with \(m_i≥ 1\) and \(ε_i(0)≠ 0\). Definition 2.8 (pinching flag). Let \(Σ_{}\) be a reference smooth fiber, \(J_{}=Jac(Σ_{})\), \(α_1,,α_δ\) disjoint vanishing circles, \(H=H^1(Σ_{}, Q)\), and \[Aₚᵢₙ=(α_1),,PD(α_δ), Bₚᵢₙ=Aₚᵢₙ⊥.\] Put \(Fᵖⁱⁿ_jH=0\) for \(j<0\), \(Aₚᵢₙ\) for \(j=0\), \(Bₚᵢₙ\) for \(j=1\), and \(H\) for \(j≥ 2\). On \(V^q_{}=H^q(J_{}, Q)=Λ^qH\) the filtration is extended by exterior convolution: \[Fᵖⁱⁿ_jV^q_{}=∑a_1+⋯+a_q≤ jIm(Fᵖⁱⁿa_1H⊗⋯⊗ Fᵖⁱⁿa_qH{}Λ^qH).\] It is defined from the topology of the degeneration, not from weights or logarithms. Theorem 2.9 (nodal comparison in every degree). Under Hypothesis 2.7, after transport to the reference fiber, for every \(q\) and \(j\), \[Fᵖⁱⁿ_jV^q_{}=M(N_q;q)_jV^q_{}=W'_jV^qₗᵢₘ.\] The filtration is multiplicative and, for an identification preserving the vanishing-cycle flag, independent of the positive multiplicities \(m_i\). This independence concerns the rational filtration, not the full limit mixed Hodge structure. Status: original. Main steps of the proof: - Because the central curve is integral, the normalization is connected, so there are auxiliary cycles \(β_i\) with \([α_i]·[β_j]=±δᵢⱼ\). Hence the \(α_i\) are independent, \(Aₚᵢₙ\) is isotropic of dimension \(δ\), and \(δ≤ G\).- An adapted splitting \(H=_i( Qu_i⊕ Qv_i)⊕ E\) with \( E=e=2G-2δ\), and Picard-Lefschetz logarithm \(L_1=∑_i m_iD_i\) with \(L_1^2=0\), \(ImL_1=Aₚᵢₙ\), \(KerL_1=Bₚᵢₙ\).- On \(Λ^qH\) the logarithm is the degree-zero derivation \(L_q\) extending \(L_1\) (not \(Λ^qL_1\)).- An \(sl_2\) triple \(D,L,R\) with \([D,L]=-2L\), \([D,R]=2R\), \([R,L]=D\), acting by derivations, gives the symmetry \(L_q^r:Grq+r{~}Grq-r\) for the pinching filtration, so it equals \(M(L_q;q)\) by uniqueness. Schmid-Steenbrink then gives the equality with \(W'\). Proposition 2.10 (bigraded dimension polynomial). With \(e=2G-2δ\) and \(dq,j=W'_jV^qₗᵢₘ\), \[∑q,jdq,jz^qs^j=(1+z)δ(1+zs)ᵉ(1+zs^2)δ, dq,j=∑_{{a+b+c=q\\ b+2c=j}}{δ}{a}{e}{b}{δ}{c}.\] Status: original. Proposition 2.11 (Jordan multiplicities and ranks). The number of Jordan blocks of length \(+1\) of \(L_q\) is \[bq,=dq,q+-dq,q++2, rankL_q^r=∑≥ r(+1-r)\,bq,.\] In particular \(L_qδ+1=0\) and \