Previlleged role of H F 0 [04FX]
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Previlleged role of
We consider quantitative almost calibrated Lagrangians. We mentioned above that for is problematic, by analogy with singular cohomology. On the other hand, is much more robust compared to higher cohomologies, in the sense that the fundamental cycle of can deform in a continuous way, under topological changes such as the shrinking of a codimension two cycle. Continuing with the analogy, we expect the geometric information in behaves more continuously under current/varifold limits than the higher degree Floer groups. This is compatible with the fact that the bordism current between encodes the compositions and , with and , and we expect bordism currents have some continuity properties under varifold/current convergence.
Remark 5.21.
This previlleged role of is reflected in the usual Thomas-Yau argument (cf. section 2.2), which only makes use of , not the higher Floer cohomologies, nor full set of higher products.
Remark 5.22.
In the passage from the Fukaya category to the derived category, the morphism space only retains , not the full . The ususal way the derived category remembers higher Floer cohomology, is via the shift operator . However, in the Thomas-Yau-Joyce picture, working with the almost calibrated setting means conjecturally that we are picking out an abelian subcategory, which breaks the shift symmetry of the derived category. This gives a categorical explanation why may behave very differently from the higher Floer groups.