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Let $X$ and $X^\vee$ be a mirror pair, homological mirror symmetry relates the symplectic geometry of $X$ to the complex geometry of $X^\vee$ via the equivalence of triangulated categories

$D^\pi\mathscr{W}(X)\cong D^b(X^\vee).$

Let's assume $X$ and $X^\vee$ are both affine, then we have a well-defined Liouville structure on $X$ which essentially doesn't depend on the compactification of $X$ by normal crossing divisors, see Seidel's work: http://arxiv.org/pdf/0704.2055v6.pdf. In particular, the wrapped Fukaya category $\mathscr{W}(X)$ is well-defined. $D^b(X^\vee)$ is the derived category of coherent algebraic sheaves.

Question Why don't we consider coherent analytic sheaves on $X^\vee$?

Since $X^\vee$ is assumed to be affine, the derived category of coherent analytic sheaves is in general not equivalent to $D^b(X^\vee)$. Recent work of Abouzaid, Auroux, etc. shows that there is an embedding

$D^b(X^\vee)\hookrightarrow D^\pi\mathscr{F}_A(X),$

where $\mathscr{F}_A(X)$ should be the partially wrapped Fukaya category introduced by Auroux in his study of Heegard-Floer homologies. This seems to suggest that we should add analytic sheaves to achieve an equivalence.

On the other hand, Seidel defined the growth rate for symplectic cohomology $SH^\ast(X)$ on the A-side

$\Gamma(X)=\overline{\lim}_\tau\frac{r(X,\tau)}{\log\tau},$

where $\tau$ is the slope of the Hamiltonian used to define $SH^\ast(X)^{<\tau}$ and $r(X,\tau)$ is the total dimension of the image of the map $SH^\ast(X)^{<\tau}\rightarrow SH^\ast(X)$ obtained by considering the continuation maps. $\Gamma(X)$ is an invariant of the Liouville structure. By the work of Seidel and Mclean, we have an upper bound:

$\Gamma(X)\leq m_X\leq\dim_\mathbb{C}(X),$

where $m_X$ is defined by looking at the boundary divisors $\overline{X}\setminus X$ in the compactification of $X$. One should be able to considering the open string analogue of Seidel's growth rate, i.e. one can analogously define the growth rate $\gamma(L)$ of the wrapped Floer cohomology $HW^\ast(L)$ of any $L\in\mathrm{Ob}\big(\mathscr{W}(X)\big)$. In his paper (http://arxiv.org/pdf/1011.2542v4.pdf), Mclean mentioned that we should expect the following upper bound:

$\gamma(L)\leq\dim_\mathbb{C}(X).$

Mirror to the growth rate $\gamma(L)$ of $L\in\mathrm{Ob}\big(\mathscr{W}(X)\big)$ we have Cornalba-Griffiths' transcendental cycle theory. This is a very old paper: http://publications.ias.edu/sites/default/files/analyticcycles.pdf. Roughly speaking, for an affine variety $X^\vee$ obtained by removing a smooth divisor from $\overline{X}^\vee$, Grauert proved that every cohomology class of $H^{2k}(X^\vee,\mathbb{Q})$ can be represented by an analytic cycle $Z$. However, such an analytic cycle $Z$ is algebraic if and only if its volume growth $n(Z,r)$ with respect to certain exhaustion $X^\vee[r]$ of $X^\vee$ remains finite. Cornalba-Griffiths refined the theory by showing that the order of the volume growth $n(Z,r)$ is related to the Hodge type of $Z$ in the compactification $\overline{X}^\vee$. More explicitly, let $\overline{Z}$ be the compactification of the analytic cycle $Z$ in $\overline{X}^\vee$, and the cohomology class $[\overline{Z}]\in H^{k+l,k-l}(\overline{X}^\vee)$ with $l\neq0$, then we have the following lower bound:

$n(Z,r)\geq C\cdot r^l.$

If one believes that the mirror symmetry between $X$ and $X^\vee$ relates non-compact exact Lagrangians in $X$ to analytic transcendental cycles in $X^\vee$, then the above discussions provide further evidences for enlarging $D^b(X^\vee)$ using analytic sheaves.

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  • $\begingroup$ Why should you only get an embedding of categories? In all examples that I know, the issue of proving generation is an issue of understanding things well enough to apply the generation criterion (rather than the embeddings which are produced not being equivalences). This is especially true I think in the ongoing work of Abouzaid, Auroux that I believe you are referencing, where the mirror is not actually of the form you suggest but is a pair $(X,W)$ where $X$ is a toric variety and $W$ is an algebraic function, where the generation criterion is not even formulated. $\endgroup$ Oct 15, 2014 at 13:37
  • $\begingroup$ a) I haven't looked at the growth rate you discuss in a ton of detail, but I don't see why it's the mirror of the McLean/Seidel growth rate. b) If you add analytic sheaves you would also have to consider their Exts in the analytic category, which doesn't match examples where we know HMS to hold e.g. for (C^*) $\endgroup$ Oct 15, 2014 at 13:41
  • $\begingroup$ @DanielPomerleano At least in Seidel's suspending Lefschetz fibration paper, he only gets an embedding. $\endgroup$
    – YHBKJ
    Oct 15, 2014 at 13:41
  • $\begingroup$ I think in the examples he is considering that is also probably just a question of someone not having proved that the objects he constructs generate the category of compact exact Lagrangians. There is no generation criterion for compact exact Lagrangians. $\endgroup$ Oct 15, 2014 at 13:47
  • $\begingroup$ @DanielPomerleano For local Calabi-Yau $K_Y$, its mirror is just an open Riemann surface, I think you can use an immersed Lagrangian to generate the Fukaya category, just like the case of pair of pants, but we don't have an equivalence in general. $\endgroup$
    – YHBKJ
    Oct 15, 2014 at 13:50

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