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Previlleged role of H ​ F 0 [04FX]

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Previlleged role of H​F0HF^{0}

We consider quantitative almost calibrated Lagrangians. We mentioned above that H​Fm​(L,L)HF^{m}(L,L) for m≥1m\geq 1 is problematic, by analogy with singular cohomology. On the other hand, H0​(L)≃Hn​(L)H^{0}(L)\simeq H_{n}(L) is much more robust compared to higher cohomologies, in the sense that the fundamental cycle of LL 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 H​F0HF^{0} behaves more continuously under current/varifold limits than the higher degree Floer groups. This is compatible with the fact that the bordism current 𝒞\mathcal{C} between L,L′L,L^{\prime} encodes the compositions α∘β=1L′\alpha\circ\beta=1_{L^{\prime}} and β∘α=1L\beta\circ\alpha=1_{L}, with α∈H​F0​(L,L′)\alpha\in HF^{0}(L,L^{\prime}) and β∈H​F0​(L′,L)\beta\in HF^{0}(L^{\prime},L), and we expect bordism currents have some continuity properties under varifold/current convergence.

Remark 5.21.

This previlleged role of H​F0HF^{0} is reflected in the usual Thomas-Yau argument (cf. section 2.2), which only makes use of H​F0HF^{0}, not the higher Floer cohomologies, nor full set of higher A∞A_{\infty} products.

Remark 5.22.

In the passage from the Fukaya category to the derived category, the morphism space only retains H​F0HF^{0}, not the full H​F∗HF^{*}. The ususal way the derived category remembers higher Floer cohomology, is via the shift operator [m][m]. 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 H​F0HF^{0} may behave very differently from the higher Floer groups.

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