[Paper Review] Derivative complex, BGG correspondence, and numerical inequalities for compact Kähler manifolds

Robert Lazarsfeld, Mihnea Popa|arXiv (Cornell University)|Jul 3, 2009

Geometry and complex manifolds17 references4 citations

TL;DR

This paper establishes a BGG correspondence between cohomology modules of compact Kähler manifolds and vector bundles on projective spaces, using the derivative complex to derive numerical inequalities and geometric invariants. It shows that the BGG complex resolves a sheaf encoding the infinitesimal structure of paracanonical divisors, and proves that the canonical series is exorbitant (an irreducible component of the paracanonical system) if and only if a certain Segre number vanishes, with this condition computable from Hodge numbers under mild hypotheses.

ABSTRACT

The cohomology algebra of the canonical bundle of a compact Kähler manifold is naturally viewed as a module over an exterior algebra. We use the Bernstein-Gel'fand-Gel'fand correspondence, together with Generic Vanishing theory, in order to understand the regularity properties of this module. We also relate it to the infinitesimal theory of the canonical linear series inside paracanonical space. Finally, we apply vector bundle methods on the polynomial ring side to obtain inequalities for the holomorphic Euler characteristic and the Hodge numbers of compact Kähler manifolds without irregular fibrations.

Motivation & Objective

• To establish a BGG correspondence between cohomology modules of compact Kähler manifolds and linear complexes over symmetric algebras.
• To show that the derivative complex governs the deformation theory of cohomology groups twisted by line bundles in Pic⁰(X).
• To use the BGG complex to derive numerical inequalities among Hodge numbers and invariants of the Albanese fibration.
• To characterize when the canonical linear series is an irreducible component of the space of paracanonical divisors using the BGG sheaf.

Proposed method

• Construct the BGG complex L_X as a linear complex of graded S-modules via cup product with elements of H¹(X, O_X).
• Define the BGG sheaf F_X as the cokernel of the final map in the complex L_X, which is a vector bundle when X has no irregular fibrations.
• Use the BGG correspondence to relate the E-module structure of H^i(X, ω_X) to the sheaf F_X on P(H¹(X, O_X)).
• Apply Grothendieck duality and spectral sequences to relate RHom complexes and establish isomorphisms involving duals of Fourier-Mukai transforms.
• Compute Segre numbers of the dual of F_X to determine the surjectivity of the map from P(F_X) to the canonical system |ω_X|.
• Use the vanishing of the codimension (p_g - χ) Segre number to characterize exorbitance of the canonical series.

Experimental results

Research questions

• RQ1 How does the BGG correspondence refine the structure of cohomology modules H^i(X, O_X) and H^i(X, ω_X) over an exterior algebra?
• RQ2 What geometric invariants of the Albanese map can be recovered from the BGG complex of a Kähler manifold?
• RQ3 Under what conditions is the canonical linear series |ω_X| an irreducible component of the space of paracanonical divisors?
• RQ4 How can numerical invariants of X be bounded using the BGG sheaf and its Segre numbers?
• RQ5 Can the exorbitance of the canonical series be determined purely from Hodge numbers under standard hypotheses?

Key findings

• The BGG complex L_X is exact in the first d−k terms, where k is the dimension of the general fiber of the Albanese map.
• When X has no irregular fibrations, the BGG sheaf F_X is a vector bundle of rank χ(ω_X) on P(H¹(X, O_X)).
• The canonical series |ω_X| is exorbitant if and only if the natural map P(F_X) → P(H^d(X, O_X)) fails to be surjective.
• The exorbitance of |ω_X| is equivalent to the vanishing of the Segre number s_{1^{×(p_g−χ)}}(γ_1,…,γ_{q−1}) for the dual of F_X.
• In the case of surfaces without irrational pencils, |ω_X| is exorbitant if and only if q is even, a result confirmed via explicit computation of the Segre number.
• Under the condition p_g − χ ≤ q − 1, exorbitance depends only on Hodge numbers, specifically the vanishing of a certain Schur function in the Chern roots of F_X∨.