[Paper Review] The big Chern classes and the Chern character

Ajay C. Ramadoss|ArXiv.org|Dec 5, 2005

Algebraic structures and combinatorial models6 references3 citations

TL;DR

This paper establishes a derived Lie algebra structure on the complex of polydifferential operators on a smooth scheme over a field of characteristic zero, showing that the tangent bundle shifted by -1 forms a Lie algebra object in the derived category. It proves that the symmetrization map fails to commute with multiplication precisely by a factor involving the Atiyah class, leading to an explicit formula for the big Chern classes in terms of the Chern character via the universal enveloping algebra construction.

ABSTRACT

Let $X$ be a smooth scheme over a field of characteristic 0. Let $\dd^{\bullet}(X)$ be the complex of polydifferential operators on $X$ equipped with Hochschild co-boundary. Let $L(\dd^1(X))$ be the free Lie algebra generated over $\strc$ by $\dd^1(X)$ concentrated in degree 1 equipped with Hochschild co-boundary. We have a symmetrization map $I: \oplus_k \sss^k(L(\dd^1(X))) ar \dd^{\bullet}(X)$. Theorem 1 of this paper measures how the map $I$ fails to commute with multiplication. A consequence of Theorem 1 and Theorem 2 is Corollary 1, a result "dual" to Theorem 1 of Markarian [3] that measures how the Hochschild-Kostant-Rosenberg quasi-isomorphism fails to commute with multiplication. In order to understand Theorem 1 conceptually, we prove a theorem (Theorem 3) stating that $\dd^{\bullet}(X)$ is the universal enveloping algebra of $T_X[-1]$ in $\dcat$. An easy consequence of Theorem 3 is Theorem 4, which interprets the Chern character $E$ as the "character of the representation $E$ of $T_X[-1]$" and gives a description of the big Chern classes of $E$. Finally, Theorem 4 along with Theorem 1 is used to give an explicit formula (Theorem 5) expressing the big Chern classes of $E$ in terms of the components of the Chern character of $E$.

Motivation & Objective

• To understand the conceptual mechanism behind the failure of the Hochschild-Kostant-Rosenberg quasi-isomorphism to commute with multiplication.
• To realize the Atiyah class of the tangent bundle as an honest map of complexes via a Lie algebra structure on polydifferential operators.
• To provide an explicit formula expressing big Chern classes in terms of the Chern character components.
• To interpret the Chern character as the character of a representation of the Lie algebra $ T_X[-1] $ in the derived category.
• To unify insights from Markarian [3] and Caldararu [10] into a coherent framework for Chern class computations.

Proposed method

• Constructs the free Lie algebra $ L( ext{D}_{ ext{poly}}^1(X)) $ over $ ext{D}_{ ext{poly}}^1(X) $, concentrated in degree 1, with Hochschild co-boundary.
• Introduces a symmetrization map $ I: igoplus_k ext{Sym}^k(L( ext{D}_{ ext{poly}}^1(X))) o ext{D}_{ ext{poly}}^ullet(X) $, which is an isomorphism of complexes.
• Defines a map $ ar{ ho} $ such that $ ext{ad} = ext{ad} imes rac{ ext{ad}}{1 - e^{- ext{ad}}} $, encoding the failure of $ I $ to commute with multiplication.
• Uses the universal enveloping algebra construction to show $ ext{D}_{ ext{poly}}^ullet(X) $ is the universal enveloping algebra of $ T_X[-1] $ in $ ext{D}^+(X) $.
• Establishes a quasi-isomorphism $ eta: T_X[-1] \to L( ext{D}_{ ext{poly}}^1(X)) $, realizing the Atiyah class as the Lie bracket on $ L( ext{D}_{ ext{poly}}^1(X)) $.
• Derives an explicit formula for big Chern classes using the Chern character via the $ \frac{\omega}{1 - e^{-\omega}} $-twisted product on symmetric powers.

Experimental results

Research questions

• RQ1 How does the symmetrization map $ I $ fail to commute with multiplication in the complex of polydifferential operators?
• RQ2 Can the Atiyah class of the tangent bundle be realized as an honest map of complexes via a Lie algebra structure?
• RQ3 Is there a conceptual explanation for the relation between the Chern character and big Chern classes in derived algebraic geometry?
• RQ4 Can the universal enveloping algebra of $ T_X[-1] $ be identified with $ \text{D}_{\text{poly}}^\bullet(X) $ in the derived category?
• RQ5 What explicit formula expresses the big Chern classes in terms of the components of the Chern character?

Key findings

• Theorem 1 establishes that the symmetrization map $ I $ fails to commute with multiplication by a factor involving the $ \frac{\omega}{1 - e^{-\omega}} $-twisted product, where $ \omega $ encodes the adjoint action.
• Theorem 2 proves that $ T_X[-1] $ is quasi-isomorphic to $ L( ext{D}_{\text{poly}}^1(X)) $, and the natural Lie bracket on the latter realizes the Atiyah class of $ T_X $ as a genuine map in $ \text{Ch}^+(\mathcal{O}_X\text{-mod}) $.
• Theorem 3 identifies $ \text{D}_{\text{poly}}^\bullet(X) $ as the universal enveloping algebra of $ T_X[-1] $ in $ \text{D}^+(X) $, providing a conceptual foundation for the Chern character construction.
• Theorem 4 interprets the Chern character of a vector bundle $ E $ as the character of its action as a representation of the Lie algebra $ T_X[-1] $, yielding a description of the big Chern classes.
• Theorem 5 gives an explicit formula for the big Chern classes of $ E $ in terms of the components of its Chern character, using the $ \frac{\omega}{1 - e^{-\omega}} $-twisted product on symmetric powers of $ L(\text{D}_{\text{poly}}^1(X)) $.
• The paper establishes a derived Lie algebra structure on $ \text{D}_{\text{poly}}^\bullet(X) $, unifying the Hochschild-Kostant-Rosenberg isomorphism with characteristic classes via representation theory.