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Hodge to de Rham degeneration of nodal curves

Yunfan He

TL;DR

The paper analyzes singular projective curves, focusing on nodal curves and, in particular, the nodal cubic. It proves that the HdR spectral sequence degenerates at $E_2$ in this singular setting, and computes both Hochschild and negative cyclic homology, including a detailed liftability classification for Hochschild classes. Using HKR and local nodal models, the authors derive explicit formulas for $HH_*(Y)$ and $HN_*(Y)$ for general nodal curves with normalization, and they reveal how liftable Hochschild classes inform categorical enumerative invariants (CEI). The results pave the way for numerical CEI calculations for elliptic curves by reducing to the nodal cubic and extend to cuspidal curves, advancing noncommutative de Rham theory in singular settings.

Abstract

We compute the Hochschild and negative cyclic homology of the nodal curves, and we show that the (noncommutative) Hodge to de Rham spectral sequence degenerates at the second page. We also classify all the Hochschild classes that can be lifted to negative cyclic homology. In particular, the result in the nodal cubic curve case is important for computation of categorical enumerative invariants.

Hodge to de Rham degeneration of nodal curves

TL;DR

The paper analyzes singular projective curves, focusing on nodal curves and, in particular, the nodal cubic. It proves that the HdR spectral sequence degenerates at in this singular setting, and computes both Hochschild and negative cyclic homology, including a detailed liftability classification for Hochschild classes. Using HKR and local nodal models, the authors derive explicit formulas for and for general nodal curves with normalization, and they reveal how liftable Hochschild classes inform categorical enumerative invariants (CEI). The results pave the way for numerical CEI calculations for elliptic curves by reducing to the nodal cubic and extend to cuspidal curves, advancing noncommutative de Rham theory in singular settings.

Abstract

We compute the Hochschild and negative cyclic homology of the nodal curves, and we show that the (noncommutative) Hodge to de Rham spectral sequence degenerates at the second page. We also classify all the Hochschild classes that can be lifted to negative cyclic homology. In particular, the result in the nodal cubic curve case is important for computation of categorical enumerative invariants.
Paper Structure (7 sections, 20 theorems, 54 equations, 2 tables)

This paper contains 7 sections, 20 theorems, 54 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Hochschild homology
  3. Hodge to de Rham spectral sequence
  4. Negative cyclic homology
  5. General cases
  6. Liftable classes
  7. Appendix: cuspidal curve

Key Result

Theorem 1.1

The Hodge to de Rham spectral sequence of any projective curve with nodal singularities degenerates at $\prescript{2}{}E$.

Theorems & Definitions (39)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Corollary 2.3
  • proof
  • Lemma 2.4
  • proof
  • Corollary 2.5
  • proof
  • ...and 29 more