[Paper Review] Postnikov-Stability for Complexes

Georg Hein, David Ploog|ArXiv.org|Apr 19, 2007

Topological and Geometric Data Analysis1 references4 citations

TL;DR

This paper introduces Postnikov-stability, a novel stability condition for complexes in the bounded derived category of coherent sheaves on a smooth projective variety. By leveraging Postnikov systems and convolutions, the authors construct a stability notion that compactifies moduli spaces of stable bundles and is preserved under Fourier-Mukai transforms, offering a derived-category framework for classical $μ$-semistability.

ABSTRACT

We present a novel notion of stable objects in the derived category of coherent sheaves on a smooth projective variety. As one application we compactify a moduli space of stable bundles using genuine complexes.

Motivation & Objective

• To define a new stability condition for complexes in the derived category of coherent sheaves on a smooth projective variety.
• To provide a derived-category framework that captures classical $μ$-semistability via Hom-conditions.
• To compactify non-complete moduli spaces of stable bundles using genuine complexes.
• To ensure stability is preserved under Fourier-Mukai transforms, linking to known equivalences in derived geometry.

Proposed method

• The authors define a P-datum as a finite collection of objects $C_i$ in a triangulated category $\mathcal{T}$, together with integers $N_i^j$ encoding homological constraints.
• They introduce P-stability via three conditions: (i) prescribed $\operatorname{Hom}^j(A, C_i) = N_i^j$, (ii) existence of a Postnikov system with morphisms $d_i: C_i \to C_{i-1}$, and (iii) $A$ orthogonal to the convolution $K$ of the system.
• The construction uses convolutions and Postnikov systems to generalize filtrations and total complexes in the derived category.
• The method relies on derived functors and homological algebra in $\mathrm{D^b}(X)$, with key tools including the generalized Euler sequence (Lemma 15) and the $S^m$-construction for symmetric powers of homomorphism spaces.
• The $S^m$-construction is defined as the cone of a morphism between symmetric powers of homomorphism spaces, ensuring functoriality in the homotopy category of injective complexes.
• The approach avoids reliance on Harder-Narasimhan filtrations, instead focusing on orthogonality and numerical constraints via $\operatorname{Hom}$-spaces.

Experimental results

Research questions

• RQ1 Can classical $μ$-semistability for sheaves be reformulated in terms of derived category Hom-conditions using Postnikov systems and convolutions?
• RQ2 How can a non-compact moduli space of stable bundles be compactified using genuine complexes rather than sheaves?
• RQ3 Is the proposed stability notion preserved under Fourier-Mukai transforms, and how does it relate to existing stability conditions like Bridgeland’s?
• RQ4 What are the homological constraints that ensure the existence and uniqueness of convolutions in derived categories?
• RQ5 Can the $S^m$-construction be made functorial in the derived category, and what are its geometric invariants?

Key findings

• The authors construct a well-defined stability notion—Postnikov-stability—for complexes in $\mathrm{D^b}(X)$, generalizing classical $μ$-semistability.
• The moduli space of stable bundles on a smooth projective curve is compactified by including genuine complexes, with the compactification arising from P-stable objects.
• The $S^m(V,a,b)$ construction yields a vector bundle with rank $\binom{m + \dim V - 1}{m+1}$ and determinant $L^{-\otimes \binom{m + \dim V - 1}{m}}$ when $V \subset H^0(L)$ is base-point-free.
• The $S^m$-construction becomes functorial in the homotopy category of injective complexes, resolving the non-functoriality of cones in triangulated categories.
• The $S^m$-functor $\mathcal{C}^m(\mathrm{D^b}(X)) \to \mathrm{D^b}(X)$ is not triangulated for $m \geq 1$, except in the trivial case $m=0$, highlighting a structural limitation.
• The method ensures that $\operatorname{Hom}^*(A,K) = 0$ for $K$ the convolution of the Postnikov system, which enforces orthogonality and stabilizes the construction.