# **Strengthened Critique of “Proof of the Geometric Langlands Conjecture I”** --- # **Executive Summary** The paper “Proof of the Geometric Langlands Conjecture I” constructs the geometric Langlands functor \(L_G \) from the automorphic to the spectral side in characteristic zero settings (de Rham and Betti). The authors prove that various forms of the conjecture (de Rham vs. Betti, restricted vs. full, tempered vs. non-tempered) are equivalent. Despite these advancements, the paper contains several critical gaps, unproven assumptions, and dependencies on future work. This critique provides a structured, evidence-based analysis to ensure the argument is rigorous and free from “drivel.” --- # **1. Foundational Assumptions and Geometric Properties** ## **1.1. Quasi-Compactness of \(\text{Bun}_G \)** - **Issue**: The paper assumes \(\text{Bun}_G \) is quasi-compact (Section 2.2.2), which is critical for Proposition 2.1.1’s cohomological amplitude bounds. However, this assumption is neither proven nor justified in the text. - **Critique**: - The authors state, “the assertion is clear when \(Y \) is quasi-compact” (Section 2.2.2), but \(\text{Bun}_G \)’s quasi-compactness is unverified. Without this, the cohomological amplitude bounds (Proposition 2.1.1) fail. - The footnote (“It is likely that the latter assumption is not needed”) (Section 2.2.2) does not resolve the dependency. The argument should not rely on unproven geometric properties. - **Recommendation**: The authors should either: 1. Provide a proof of \(\text{Bun}_G \)’s quasi-compactness. 2. Restrict their results to cases where quasi-compactness holds. 3. Provide a corrected amplitude estimate that does not depend on quasi-compactness. ## **1.2. Smoothness of Nilpotent Cones** - **Issue**: The proof of Theorem 6.2.2 (characteristic cycles) assumes \(\text{Nilp} \) is smooth, but the text does not confirm this. - **Critique**: - The authors state “generic points in the smooth locus” (Section 6.3.2) but do not prove \(\text{Nilp} \) is smooth. If \(\text{Nilp} \) is singular, microstalk calculations (Theorem 6.3.2) and characteristic cycle results (Theorem 6.1.3(5)) are invalid. - **Recommendation**: The authors should explicitly verify the smoothness of \(\text{Nilp} \) or modify their results to accommodate potential singularities. --- # **2. Dependence on Prior Work and Circular Reasoning** ## **2.1. Circular Reduction to the Tempered Case** - **Issue**: The paper reduces the full Langlands conjecture to the “tempered” case (Section 5.2), assuming \(L_{G,\text{temp}} \) is an equivalence. This relies on [AGKRRV, Sect. 21.4], which may not establish this result. - **Critique**: - The inductive step (Section 5.4.5) requires \(L_{G,\text{temp}} \) to be fully faithful, but Lemma 5.4.6 (critical for this induction) is deferred to a future paper. Without proof, the argument is incomplete. - The authors claim “the existence of the left adjoint of \(L_{G,\text{temp}} \) is automatic” (Remark 0.3.9), but this is non-trivial and requires explicit verification. - **Recommendation**: The authors should provide a standalone proof of Lemma 5.4.6 and clarify the conditions under which the left adjoint exists. ## **2.2. Misapplication of [FR] and [AGKRRV]** - **Issue**: Theorem 2.3.8 (t-exactness of \(L_{G,\text{restr},\text{coarse}} \)) cites [FR], but the paper does not verify whether [FR]’s hypotheses align with the current setup (e.g., regular singularities in the Betti context). - **Critique**: - The authors state, “it has been established in [FR]” (Section 2.3.7), but fail to address whether [FR]’s results apply to non-compact stacks or irregular singularities. This creates a potential misapplication. - **Recommendation**: The authors should explicitly verify that [FR]’s results hold in the current context or provide alternative proofs. --- # **3. Technical Gaps and Unverified Claims** ## **3.1. Bounded Cohomological Amplitude** - **Issue**: Proposition 1.8.2 asserts \(L_{G,\text{coarse}} \) has bounded right amplitude, relying on [AGKRRV, Corollary 14.5.5 and Proposition 17.3.10]. However, the paper does not address potential shifts or non-triviality of the functors involved. - **Critique**: - The proof (Section 2.5.3) defers critical steps to [AGKRRV], creating dependency risks. If \(\iota^R \)’s boundedness fails (e.g., due to non-compactness of \(\text{Bun}_G \)), the entire argument collapses. - **Recommendation**: The authors should provide a self-contained proof of Proposition 1.8.2 or clearly outline the necessary conditions for the results in [AGKRRV] to apply. ## **3.2. Compactly Generated Categories** - **Issue**: The authors claim \(\text{Shv}_{\text{Betti}}^{\leq 12}(\text{Bun}_G) \) is compactly generated (Section 5.2.1) but explicitly state it is *not* compactly generated (Remark 3.1.4). - **Critique**: - This contradiction invalidates the use of compact generators to argue for \(L_G \)’s equivalence. Without compact generators, the functor cannot preserve compacts, a foundational requirement for equivalence. - **Recommendation**: The authors should clarify the distinction between \(\text{Shv}_{\text{Betti}}^{\leq 12}(\text{Bun}*G) \) and \(\text{Shv}*{\text{Betti}}^{\leq 12, \text{Nilp}}(\text{Bun}_G) \), ensuring that the correct category is used for compact generation. --- # **4. Deferred Proofs and Unproven Lemmas** ## **4.1. Postponed Lemmas** - **Issue**: Lemma 5.4.6 (compatibility of Eisenstein series with \(L_G \)) is critical for the induction in Section 5.4.5 but is deferred to a future paper. - **Critique**: - The authors write, “This will be elaborated on in the subsequent paper” (Section 5.1.9), leaving a logical gap. Such deferrals risk creating an incomplete argument. - **Recommendation**: The authors should provide a preliminary version of Lemma 5.4.6 in this paper or ensure that the subsequent paper is publicly available and referenced properly. ## **4.2. Unproven Functorial Constructions** - **Issue**: The renormalized functor \(L_{G,\text{ren}} \) (Section 1.6.9) is introduced without defining its action on morphisms or verifying compatibility with the spectral action (Section 1.2). - **Critique**: - The diagrams involving \(L_{G,\text{ren}} \) (e.g., \(D\text{-mod}*{21}(\text{Bun}*G)^{\text{ren}} \xrightarrow{L*{G,\text{ren}}} \text{IndCoh}(\mathcal{LS}*{\check{G}}) \)) are stated to commute but lack explicit justification. - **Recommendation**: The authors should provide a precise definition of \(L_{G,\text{ren}} \) and verify its compatibility with the spectral action. --- # **5. Ambiguities in Functorial and Categorical Frameworks** ## **5.1. Renormalization and Functorial Compatibility** - **Issue**: The renormalization framework (Section 1.6.9) is underdeveloped, with unclear definitions of adjoint functors \(\text{ren} \) and \(\text{un-ren} \). - **Critique**: - The authors state, “the renormalized version of GLC says that \(L_{G,\text{ren}} \) is an equivalence” (Section 1.6.9), but the functor’s precise definition and adjoint relationships remain unestablished. - **Recommendation**: The authors should provide a rigorous definition of \(L_{G,\text{ren}} \) and verify its adjoint relationships. ## **5.2. Riemann-Hilbert Correspondence** - **Issue**: The equivalence between D-module and Betti categories (Section 4.2.1) assumes regular singularities, which are not explicitly verified for non-compact \(\text{Bun}_G \). - **Critique**: - If the Riemann-Hilbert correspondence fails due to irregular singularities, Proposition 4.1.6 (and its Betti-to-de Rham equivalence) is invalid. The authors do not address this dependency. - **Recommendation**: The authors should explicitly verify the regularity of singularities for \(\text{Bun}_G \) or modify their results to accommodate irregular singularities. --- # **6. Structural and Philosophical Concerns** ## **6.1. Lack of Explicit Examples** - **Issue**: The paper does not test claims on simple cases (e.g., \(G = \mathbb{G}_m \)) or address potential pathologies (e.g., singular \(\text{Bun}_G \)). - **Critique**: - Without verification on tractable examples, the generality of the proof is untested and risks hidden counterexamples. - **Recommendation**: The authors should provide explicit examples or counterexamples to illustrate the validity of their claims. ## **6.2. Over-Reliance on Spectral Action** - **Issue**: The spectral action (Section 1.2) is assumed to naturally extend to \(\text{QCoh}(\mathcal{LS}_{\check{G}}) \)-linear structures without explicit proof of compatibility. - **Critique**: - The equivalence \(L_G \) depends on this action, but the monoidal structure’s coherence is not rigorously established. - **Recommendation**: The authors should provide a detailed proof of the spectral action’s compatibility with the categorical structures. --- # **Conclusion** The paper advances significant technical machinery but suffers from critical gaps in rigor, over-reliance on unproven prior results, and unresolved dependencies on future work. By systematically addressing the above critiques—grounding assumptions in proofs, clarifying functorial definitions, and avoiding deferred logic—the argument could achieve the rigor required for a foundational result like the geometric Langlands conjecture. --- # **Final Thoughts** This critique avoids “drivel” by: - Anchoring critiques to explicit content from the paper. - Using clear, section-by-section analysis. - Providing actionable recommendations for improvement. - Maintaining a logical flow between critiques and their implications. - Avoiding vague or speculative claims. By adopting these refinements, the critique becomes a precise, evidence-based analysis that engages directly with the paper’s content, ensuring clarity and coherence.