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Hodge conjecture for projective hypersurface

Johann Bouali

TL;DR

This work develops the theory of complex analytic logarithmic De Rham classes and exploits a motivic purity framework to connect analytic $(d,d)$-classes with algebraic cycles. For smooth hypersurfaces $X$ with odd ambient dimension, the authors show that any Hodge class $\lambda\in F^pH^{2p}(X^{an},\mathbb Q)$ arises from an algebraic cycle, by expressing $\lambda$ via a residue from the hypersurface complement and using log De Rham tools to descend to a cycle on $X$. The approach unifies analytic and algebraic perspectives through period maps, residue morphisms, and transfers in the motivic setting, culminating in a full Hodge conjecture result for hypersurfaces. Overall, the paper extends known cases of the Hodge conjecture to broad classes of hypersurfaces and introduces analytic logarithmic De Rham classes as a key bridge between analysis and algebraic geometry.

Abstract

We show that a Hodge class of a complex smooth projective hypersurface is an analytic logarithmic De Rham class. On the other hand we show that for a complex smooth projective variety an analytic logarithmic De Rham class of of type $(d,d)$ is the class of codimension $d$ algebraic cycle. We deduce the Hodge conjecture for smooth projective hypersurfaces.

Hodge conjecture for projective hypersurface

TL;DR

This work develops the theory of complex analytic logarithmic De Rham classes and exploits a motivic purity framework to connect analytic -classes with algebraic cycles. For smooth hypersurfaces with odd ambient dimension, the authors show that any Hodge class arises from an algebraic cycle, by expressing via a residue from the hypersurface complement and using log De Rham tools to descend to a cycle on . The approach unifies analytic and algebraic perspectives through period maps, residue morphisms, and transfers in the motivic setting, culminating in a full Hodge conjecture result for hypersurfaces. Overall, the paper extends known cases of the Hodge conjecture to broad classes of hypersurfaces and introduces analytic logarithmic De Rham classes as a key bridge between analysis and algebraic geometry.

Abstract

We show that a Hodge class of a complex smooth projective hypersurface is an analytic logarithmic De Rham class. On the other hand we show that for a complex smooth projective variety an analytic logarithmic De Rham class of of type is the class of codimension algebraic cycle. We deduce the Hodge conjecture for smooth projective hypersurfaces.
Paper Structure (11 sections, 10 theorems, 160 equations)

This paper contains 11 sections, 10 theorems, 160 equations.

Table of Contents

  1. Introduction
  2. Complex analytic De Rham logarithmic classes
  3. Hodge conjecture for smooth projective hypersurfaces
  4. Preliminaries and Notations
  5. Notations
  6. Complex integral periods
  7. Algebraic cycles and motives
  8. The logaritmic De Rham complexes
  9. Complex analytic logarithmic de Rham classes
  10. Hypersurface De Rham cohomology and its complex periods
  11. Hodge Conjecture for complex smooth projective hypersurfaces

Key Result

Proposition 1

Let $k$ be a field a characteristic zero. Let $X\in\mathop{\rm SmVar}\nolimits(k)$ and $i:Z\subset X$ a smooth subvariety of pure codimension d. Then $C_{Z/X}=N_{Z/X}\to Z$ is a vector bundle of rank $d$. The closed embeddings $i_1:(X,Z)\hookrightarrow (D_ZX,\mathbb A^1_Z)$ and $i_0:(C_{Z/X},Z)\hook We have

Theorems & Definitions (25)

  • Proposition 1
  • proof
  • Definition 1
  • Definition 2
  • Remark 1
  • Definition 3
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • ...and 15 more