Abstract
Building on the scaffolding constructed in the first two articles in this series, we now proceed to the geometric phase of our sheaf (‐complex) theoretic quasidualization of Kubota′s formalism for n‐Hilbert reciprocity. Employing recent work by Bridgeland on stability conditions, we extend our yoga of t‐structures situated above diagrams of specifically designed derived categories to arrangements of metric spaces or complex manifolds. This prepares the way for proving n‐Hilbert reciprocity by means of singularity analysis.
Highlights
After developing topological and derived sheaf-categorical aspects of our quasidualization of Kubota’s formalism 1 for n-Hilbert reciprocity, in 2, 3, we proceed to the geometric aspect of our construct
This marvelous state of affairs is the principal motivation for our shifting our focus from t-structures to stability conditions, in which context we presently delineate classes of the latter belonging to a single t-structure; the idea is to cap off the architecture of sheaf constructs we developed in Parts One and Two of the present series with an arrangement of spaces which permit a certain kind of singularity analysis
The definition of a stability condition in the sense of [4,5,6] has its immediate antecedents in an investigation by Douglas 19 in the area of D-branes and mirror symmetry situated at the intersection of physics and mathematics
Summary
After developing topological and derived sheaf-categorical aspects of our quasidualization of Kubota’s formalism 1 for n-Hilbert reciprocity, in 2, 3 , we proceed to the geometric aspect of our construct. A single t-structure can have any number of stability conditions associated to it by coupling it to certain C-valued homomorphisms from the Grothendieck group of the underlying derived category of coherent sheaves. The salient point here is that, qua data, a Bridgeland stability condition is a bounded t-structure together with a suitable HN “central charge.” Dealing these sets of stability conditions, the structure of a metric space is one of Bridgeland’s most exciting results 5. The projected final tactics will doubtlessly be informed by, for instance, homology perhaps even intersection homology as per Goresky-MacPherson 8–10 , suitable attendant cohomological approaches, Morse theory, or index theory These choices will be made in our paper; our present purpose is, so to speak, geometrical, what with Part One having a topological orientation and Part Two being concerned with homological algebra in the broad, modern sense
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