Question: Let's assume we have a pair $(X,\check{X})$ that are mirror dual to each other in the sense of Homological mirror symmetry (EDIT: this does not have to be CY n-folds, but can also be a Fano variety and a Landau-Ginzburg partner etc). Now take a subvariety $Y \subset X$ and consider $Z = Bl_Y X$.

  • Are there any examples where such mirror pairs is known?
  • Ideally: is there a general recipe for what should be the mirror $\check{Z}$?

Motivation: Derived categories behave nicely under blowup; In fact by Bondal-Orlov, we can construct the derived category of sheaves on $Z$ from that of $X$ and $Y$. Thus one should expect the same for Fukaya categories (e.g., Ivan Smith's Floer cohomology and pencils of quadrics provides some evidence in that direction). Now, such an equivalence should probably have some geometric meaning via SYZ perhaps and be explained in terms of a nice mirror recipe.

  • $\begingroup$ A minor comment, but since Kodaira dimension is a birational invariant, your $Z$ will never be either Fano or general type; since the canonical bundle is nontrivial, it will not be CY either. $\endgroup$ – Bertie May 16 '17 at 21:45
  • $\begingroup$ Bertie right. In fact if $\pi: \tilde X\to X$ be birational blow up then $K_{\tilde X}=\pi^*(K_X)+E$, so if $c_1(X)=0$ then $c_1(\tilde X)=E$ which is not CY $\endgroup$ – user21574 May 16 '17 at 21:49
  • 2
    $\begingroup$ A silly comment - since $X$ is a symplectic specialization of $Z$, $\check{X}$ should be complex specialization of $\check{Z}$. So you will need to understand how to extract a semi-orthogonal decomposition of the Fukaya-Seidel category from a specialization family. For example, the mirror of a del Pezzo $dP_{n}$ is a rational elliptic surface with an $I_{9-n}$ fiber at infinity. The blow-up of $dP_{n}$ at a point corresponds to morsifying one node of the fiber at infinity, i.e. to a splitting it as $I_{1}+I_{9-(n+1)}$ and morsification gives you a semi-orthogonal decomposition of FS. $\endgroup$ – Tony Pantev May 17 '17 at 12:55

You can find a lot of such examples in the paper of Abouzaid-Auroux-Katzarkov: https://link.springer.com/article/10.1007/s10240-016-0081-9.

Basically, they studied the case when $X$ is $(\mathbb{C}^\times)^{n-1}\times\mathbb{C}$, and $Y\subset X$ is codimension 2 and is a hypersurface in $(\mathbb{C}^\times)^{n-1}$.

For Fukaya categories, first one should expect a fully faithful embedding


Such an embedding has already been proved in the case when $X$ is a smooth even-dimensional quadric, and $Y\subset X$ is the base locus of a pencil which includes $X$ as one of its members. The functor $\Phi_\mathbb{L}$ should arise from a Lagrangian correspondence $\mathbb{L}\subset Y^-\times Z$. In the simplest case when $X=\mathbb{C}^2$ and $Y$ is a point, $\mathbb{L}$ is simply the Clifford torus $T\subset\mathbb{C}^2$, which arises as the composition of two correspondences $\mathbb{L}_1\subset\mathit{pt}\times E$ and $\mathbb{L}_2\subset E^-\times Z$. In this case, $\mathbb{L}_1$ is the large circle in the exceptional divisor $E\cong\mathbb{P}^1$, and $\mathbb{L}_2$ is the standard $S^3$ in $Z$, namely the boundary of the unit disc bundle associated to $\mathcal{O}(-1)$. The construction in the pencil of quadric case is just a family version of this. In general, the Lagrangian correspondence $\mathbb{L}_2\rightarrow E$ arises as a bundle over the exceptional divisor $E\subset Z$, whose fibers are Clifford tori in the projective space $\mathbb{P}^k$. You can find constructions of similar flavor in the paper of Abouzaid-Auroux-Katzarkov, in which case you will be forced to deal with non-compact Lagrangian correspondences.

It is expected by many people that the Fukaya category is well-behaved under birational maps, see the introduction of https://arxiv.org/abs/1508.01573. The main result of the paper referred above deals with the case when the center is trivial, namely $Y$ is a point. For related works in the case of Landau-Ginzburg models, see the paper of Kerr: https://arxiv.org/abs/1603.08074.

There are many difficulties in this direction. For example, when the center is not trivial, then you don't get a toric neighborhood in $Z$ after blowing up $Y$, and it's not known in general how to show that the blow-up creates new non-displaceable Lagrangian tori. In the toric case, of course, the existence problem reduces to the analysis of broken holomorphic discs, and this has been done in the paper of Charest-Woodward. In this particular case when the center is trivial, the understanding of $\mathcal{F}(Z)$ reduces essentially to the understanding of $\mathcal{F}(X)$ and $\mathcal{F}(\mathcal{O}(-1))$. The latter one is well-understood by the work of Ritter-Smith.

  • $\begingroup$ Thanks!! Abouzaid-Auroux-Katzarkov looks like what I was looking for. $\endgroup$ – Nati May 18 '17 at 20:29

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