Hodge-to-de Rham degeneration and quasihomogeneous singularities of curves

1. Problem Statement

The paper investigates the degeneration behavior of the Hodge-to-de Rham spectral sequence for integral projective curves with local complete intersection (lci) singularities.

• Context: For smooth projective varieties over $\mathbb{C}$, the Hodge-to-de Rham spectral sequence degenerates at the $E_1$-page (Deligne-Illusie). For singular varieties, the correct replacement for the de Rham complex is the Hodge-completed derived de Rham complex ($\widehat{dR}^\bullet_X$).
• The Question: While previous work established $E_2$-degeneration for specific singularities (e.g., nodal curves, cuspidal cubics), a general singularity-theoretic characterization for $E_2$-degeneration was missing.
• Goal: Determine necessary and sufficient conditions on the singularities of an integral projective lci curve $X$ such that its Hodge-to-de Rham spectral sequence degenerates at the $E_2$-page.

2. Methodology

The author employs a combination of derived algebraic geometry, singularity theory, and spectral sequence analysis.

• Derived de Rham Complex: The study utilizes the cotangent complex $L^\bullet_X$ and the derived exterior powers $\bigwedge^p L^\bullet_X$. The $E_1$-page is given by $E_1^{p,q} = H^q(X, \bigwedge^p L^\bullet_X)$.
• Local-Global Analysis: The proof strategy separates the analysis into two cases based on the embedding dimension of the singularities:
  1. Planar Singularities: Where the completed local ring is a hypersurface ($\hat{\mathcal{O}}_{X,x} \cong \mathbb{C}[[u,v]]/(f)$).
  2. Non-Planar Singularities: Where the embedding dimension is $\ge 3$.
• Algebraic Tools:
  • Milnor and Tjurina Algebras: The author relates the cohomology of the derived exterior powers to the Milnor algebra ($M_f$) and Tjurina algebra ($T_f$) of the singularity.
  • Weighted Homogeneity: The paper leverages the Euler relation for weighted homogeneous polynomials to analyze the differentials.
  • Koszul Complexes: The derived exterior powers are modeled using Koszul complexes associated with the Jacobian ideal.
• Hochschild-to-Cyclic Connection: The paper establishes a link between the Hodge-to-de Rham spectral sequence and the Hochschild-to-cyclic spectral sequence via the HKR (Hochschild-Kostant-Rosenberg) isomorphism in the derived setting.

3. Key Contributions and Results

A. Main Theorem (Theorem 1.1 & Theorem 3.11)

The central result provides a complete characterization for $E_2$-degeneration:

Theorem: For an integral projective curve $X$ with local complete intersection singularities, the Hodge-to-de Rham spectral sequence degenerates at the $E_2$-page if and only if every singularity of $X$ is a quasihomogeneous plane curve singularity.

• Quasihomogeneous Plane Singularity: A singularity defined locally by a weighted homogeneous equation $f(u,v)=0$ (where weights exist such that every monomial has the same weighted degree).
• Significance: This bridges the gap between the topological degeneration of spectral sequences and the algebraic property of quasihomogeneity.

B. Analysis of the $E_1$-Page and Differentials

The paper meticulously analyzes the structure of the spectral sequence:

• Planar Case:
  • The author shows that for a plane singularity, the "tail" differentials (maps between negative rows) are isomorphisms if and only if the singularity is quasihomogeneous.
  • Specifically, the differential $d_1$ acts on the Milnor algebra $M_f$. If $f$ is weighted homogeneous, the Euler relation implies $d_1$ acts as multiplication by a non-zero scalar ($\lambda + w_1 + w_2$), making it an isomorphism.
  • If the spectral sequence degenerates at $E_2$, a global dimension count forces the Milnor number ($\mu$) to equal the Tjurina number ($\tau$) at every singularity. By Saito's criterion, $\mu = \tau$ is equivalent to quasihomogeneity.
• Non-Planar Case (Obstruction):
  • The author proves that if $X$ possesses a non-planar lci singularity (embedding dimension $\ge 3$), the spectral sequence cannot degenerate at $E_2$.
  • Mechanism: A non-planar singularity produces a non-zero term in the $E_2$-page at total degree $> 2$ (specifically in the row $q=-1$). Since the singular cohomology of a curve vanishes in degrees $> 2$, this non-zero term must eventually be killed by a differential, preventing $E_2$-degeneration.

C. Hochschild-to-Cyclic Spectral Sequence (Corollary 1.2 & 3.15)

The paper extends the result to the Hochschild-to-cyclic spectral sequence:

• Equivalence: The Hodge-to-de Rham spectral sequence degenerates at $E_2$ if and only if the Hochschild-to-cyclic spectral sequence degenerates at $E_2$.
• Condition: Both degenerations occur if and only if every singularity is a quasihomogeneous plane curve singularity.
• Proof Strategy: The author constructs a filtered quasi-isomorphism between the Hochschild mixed complex and the derived de Rham mixed complex (via the HKR isomorphism), showing that the Hodge-to-de Rham spectral sequence is a direct summand of the Hochschild-to-cyclic one.

4. Technical Details of the Proof

1. Local Tail Maps (Lemma 3.4 & 3.5):

   • For a plane singularity $f=0$, the cohomology of $\bigwedge^p L^\bullet_A$ is identified with the kernel and cokernel of multiplication by $f$ on the Milnor algebra.
   • In the quasihomogeneous case, the Euler relation ensures $f$ acts as zero on the Milnor algebra, making the kernel and cokernel isomorphic to the full Milnor algebra. The differential $d_1$ becomes an isomorphism.

2. Global Dimension Count (Proposition 3.8):

   • Assuming $E_2$-degeneration, the author compares the dimensions of the $E_2$ terms with the Betti numbers of the analytic space $X^{an}$.
   • Using the exact sequence $0 \to A \to \Omega^1_X \to \omega_X \to D \to 0$ (where $A, D$ are supported on singularities), the author derives the relation $\sum \tau_i = \sum \mu_i$.
   • Since $\tau_i \le \mu_i$ always, equality implies $\tau_i = \mu_i$ for all $i$, proving quasihomogeneity.

3. Non-Planar Obstruction (Lemma 3.9 & Corollary 3.10):

   • For a non-planar lci singularity with embedding dimension $e \ge 3$, the author constructs a non-zero cohomology group $H^{-1}(\bigwedge^{e+1} L^\bullet_A)$.
   • The differential $d_1$ mapping into this group is shown to be non-surjective (via Nakayama's lemma and properties of the Jacobian matrix entries lying in the maximal ideal).
   • This creates a non-zero $E_2$ term in total degree $e \ge 3$, which contradicts the vanishing of $H^k_{sing}(X^{an})$ for $k > 2$.

5. Significance and Future Directions

• Theoretical Impact: The paper resolves the characterization of $E_2$-degeneration for singular curves, linking a global cohomological property (spectral sequence degeneration) directly to local algebraic properties (quasihomogeneity).
• Unification: It unifies the study of Hodge theory and Hochschild homology for singular curves, showing they share the same degeneration criteria.
• Open Questions: The author suggests extending these results to:
  • Non-reduced curves: How nilpotents affect the derived exterior powers.
  • Higher dimensions: Characterizing $E_2$-degeneration for surfaces and higher-dimensional lci varieties.
  • Later degeneration: Determining if there is a local invariant that controls the specific page ($E_r$) where degeneration occurs for non-quasihomogeneous singularities.

In summary, Yunfan He's work provides a definitive answer to when the Hodge-to-de Rham spectral sequence stabilizes early for singular curves, identifying quasihomogeneous plane singularities as the precise geometric condition required for this phenomenon.