# Complete complexes and spectral sequences

## TL;DR

This paper introduces the variety of complete complexes, linking spectral sequences to algebraic geometry, and demonstrates it as a smooth projective desingularization of the Buchsbaum-Eisenbud variety of complexes.

## Contribution

It defines the variety of complete complexes as an analogue to classical spaces and proves its structure as a smooth projective variety with a desingularization property.

## Key findings

- The set of equivalence classes forms a smooth projective variety.
- Provides a desingularization with normal crossings boundary of the Buchsbaum-Eisenbud variety.
- Establishes a geometric framework for spectral sequences.

## Abstract

By analogy with the classical (Chasles-Schubert-Semple-Tyrell) spaces of complete quadrics and complete collineations, we introduce the variety of complete complexes. Its points can be seen as equivalence classes of spectral sequences of a certain type. We prove that the set of such equivalence classes has a structure of a smooth projective variety. We show that it provides a desingularization, with normal crossings boundary, of the Buchsbaum-Eisenbud variety of complexes, i.e., a compactification of the union of its maximal strata.