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Math-2577 · mathematics · Algebraic geometry

P=W for fixed-determinant trace-free coprime moduli

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For a smooth connected projective complex curve of genus at least two, any composite rank $r\geq 4$ and degree d coprime to r, fix a degree-d determinant L. Under nonabelian Hodge correspondence between stable trace-free L-determinant Higgs moduli and its twisted $SL_r$ character variety, does the Hitchin perverse filtration $P_k$ equal $W_{2k}=W_{2k+1}$ on rational cohomology in every degree?

Short English statement

For a smooth connected projective complex curve of genus at least two, any composite rank $r\geq 4$ and degree d coprime to r, fix a degree-d determinant L. Under nonabelian Hodge correspondence between stable trace-free L-determinant Higgs moduli and its twisted $SL_r$ character variety, does the Hitchin perverse filtration $P_k$ equal $W_{2k}=W_{2k+1}$ on rational cohomology in every degree?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2577-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2577

Manuscript 1 (family 043, 2026-09-24): The problem selects the mathematical statement summarized here, including its stated hypotheses and parameter range. The cited theorem gives a claimed proof of that assertion in full.

P=W for fixed-determinant trace-free coprime moduli

OpenAI's claim

For fixed determinant and trace-free Higgs moduli in coprime composite rank, OpenAI claims equality of the perverse and weight filtrations on the full rational cohomology, completing all coprime ranks with known cases. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
Let $C$ be a complex curve, smooth and projective, with genus $g\ge2$, composite $n\ge4$, $\gcd(n,d)=1$, and $L\in\operatorname{Pic}^d(C)$. Let $X$ parametrize stable rank-$n$, determinant-$L$, trace-free Higgs bundles, with Hitchin map $h:X\to A=\bigoplus_{i=2}^nH^0(C,K_C^i)$ and $R=(n^2-1)(g-1)$. The corresponding Betti variety is \[ X^B=\{(A_j,B_j)\in\operatorname{SL}_n(\mathbb C)^{2g}:\prod_j[A_j,B_j]=e^{2\pi id/n}I\}\mathbin{/\mkern-6mu/}\operatorname{SL}_n. \] Transport the perverse filtration $P_kH^m(X)=\operatorname{im}[H^{m-R}(A,{}^p\tau_{\le k}Rh_*\mathbb Q_X[R])\to H^m(X)]$ by nonabelian Hodge theory. OpenAI claims \[ P_kH^m(X^B,\mathbb Q)=W_{2k}H^m(X^B,\mathbb Q)=W_{2k+1}H^m(X^B,\mathbb Q) \] for every $m,k\ge0$, where $W$ is Deligne's weight filtration. The determinant is fixed and the composite-rank coprime-degree restrictions are part of the theorem.

Outline of the argument

Manuscript 1 (family 043, 2026-09-24): Determinant-root covers transfer toric stratifications and establish curious hard Lefschetz beyond the invariant summand. Finite algebraic-fiber diagrams carry the required weight-zero moving-base cochains, furnishing lowering operators compatible with the full cohomology and the common Lie-algebra structure.

Formal verification

Family 043 has no Lean scope page in this snapshot; no formalization of these manuscript results is listed.

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Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

Sources

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