[Paper Review] Nonabelian mixed Hodge structures

Ludmil Katzarkov, Tony Pantev+1 more|ArXiv.org|Jun 28, 2000

Algebraic Geometry and Number Theory38 references3 citations

TL;DR

This paper introduces a new framework for nonabelian mixed Hodge structures (namhs) by defining a weight filtration and constructing nonabelian cohomology $ H = \mathrm{Hom}(X_M, V) $ for a smooth projective variety $ X $ and a nonabelian mixed Hodge structure $ V $. The key contribution is proving that when $ V $ has the homotopy type of the complexified 2-sphere $ \mathbb{P}^1 $, the resulting cohomology $ H $ is indeed a namhs, establishing a foundational example of the theory.

ABSTRACT

We propose a definition of ``nonabelian mixed Hodge structure'' together with a construction associating to a smooth projective variety $X$ and to a nonabelian mixed Hodge structure $V$, the ``nonabelian cohomology of $X$ with coefficients in $V$'' which is a (pre-)nonabelian mixed Hodge structure denoted $H=Hom(X_M, V)$. We describe the basic definitions and then give some conjectures saying what is supposed to happen. At the end we compute an example: the case where $V$ has underlying homotopy type the complexified 2-sphere, and mixed Hodge structure coming from its identification with $\pp ^1$. For this example we show that $Hom (X_M,V)$ is a namhs for any smooth projective variety $X$.

Motivation & Objective

• To define a notion of nonabelian mixed Hodge structure (namhs) that extends classical mixed Hodge theory to higher homotopy types and nonabelian coefficients.
• To address the lack of a weight filtration in nonabelian homotopy theory, which has hindered the study of higher functoriality and variations in families.
• To construct a cohomology theory $ H = \mathrm{Hom}(X_M, V) $ that encodes higher homotopy data with mixed Hodge structure, generalizing classical cohomology.
• To prove that for a specific coefficient object $ \mathcal{V} $ of homotopy type $ S^2_\mathbb{C} $, the resulting cohomology is a namhs, providing a foundational example.

Proposed method

• Proposes a definition of nonabelian mixed Hodge structure using filtered $ n $-stacks and $ \mathbb{G}_m $-equivariant perfect complexes.
• Uses Dold-Puppe linearization to construct pre-namhs from filtered complexes and analyze weight filtrations.
• Applies the theory of mixed Hodge complexes and linearization to define the cohomology stack $ \underline{\mathrm{Hom}}(X_B, T) $ as a nonabelian cohomology object.
• Analyzes the structure of the cohomology stack via the zero-subscheme of a quadratic morphism $ Q'_{\mathcal{H},0} $, which arises from cup-product and weight shifts.
• Establishes flatness and ideal conditions (A1–A3) on the cohomology stack to verify it satisfies the axioms of a namhs.
• Reduces the problem to analyzing the scheme-theoretic union of a quadratic cone and a divisor, using trivialization over $ \mathbb{A}^1_{\text{hod}} $.

Experimental results

Research questions

• RQ1 Can a consistent notion of weight filtration be defined for nonabelian homotopy types, extending classical mixed Hodge theory?
• RQ2 Is the nonabelian cohomology $ \mathrm{Hom}(X_M, V) $ of a smooth projective variety with coefficients in a nonabelian mixed Hodge structure $ V $ itself a namhs?
• RQ3 How does the secondary Kodaira-Spencer map detect higher variation in Hodge structures when classical variations are constant?
• RQ4 What is the structure of the cohomology stack when $ V $ has the homotopy type of $ \mathbb{P}^1 $, and does it satisfy the axioms of a namhs?
• RQ5 Can the construction be generalized to other varieties such as $ \mathbb{P}^n $ or Grassmannians?

Key findings

• The cohomology stack $ \mathrm{Hom}(X_M, V) $ is a nonabelian mixed Hodge structure when $ V $ has the homotopy type of the complexified 2-sphere $ S^2_\mathbb{C} $, as shown via explicit computation.
• The resulting cohomology is a smooth morphism of linearized pre-namhs, and its structure is determined by the zero-subscheme of a quadratic morphism arising from cup-product.
• The cohomology stack $ \mathcal{H} $ is flat over the Hodge parameter space $ \mathbb{A}^1_{\text{hod}} $, and its ideal structure satisfies the axioms A1–A3 required for a namhs.
• The stack $ \mathcal{K} $, defined as the zero-subscheme of the quadratic morphism, is trivializable in the Hodge direction and decomposes as a union of a quadratic cone and the inverse image of a divisor.
• The annihilator ideal of the weight parameter $ t $ is isomorphic to the ideal defining the quadratic cone, and this ideal is a sub-mixed Hodge structure, satisfying condition A3.
• The entire construction is compatible with the Hodge decomposition and weight filtrations, confirming that the cohomology stack inherits a well-defined namhs structure.