\( \def\CC{{\mathbb C}} \def\RR{{\mathbb R}} \def\NN{{\mathbb N}} \def\ZZ{{\mathbb Z}} \def\TT{{\mathbb T}} \def\CF{{\operatorname{CF}^\bullet}} \def\HF{{\operatorname{HF}^\bullet}} \def\SH{{\operatorname{SH}^\bullet}} \def\ot{{\leftarrow}} \def\st{\;:\;} \def\Fuk{{\operatorname{Fuk}}} \def\emprod{m} \def\cone{\operatorname{Cone}} \def\Flux{\operatorname{Flux}} \def\li{i} \def\ev{\operatorname{ev}} \def\id{\operatorname{id}} \def\grad{\operatorname{grad}} \def\ind{\operatorname{ind}} \def\weight{\operatorname{wt}} \def\Sym{\operatorname{Sym}} \def\HeF{\widehat{CHF}^\bullet} \def\HHeF{\widehat{HHF}^\bullet} \def\Spinc{\operatorname{Spin}^c} \def\min{\operatorname{min}} \def\div{\operatorname{div}} \def\SH{{\operatorname{SH}^\bullet}} \def\CF{{\operatorname{CF}^\bullet}} \def\Tw{{\operatorname{Tw}}} \def\Log{{\operatorname{Log}}} \def\TropB{{\operatorname{TropB}}} \def\wt{{\operatorname{wt}}} \def\Span{{\operatorname{span}}} \def\Crit{\operatorname{Crit}} \def\CritVal{\operatorname{CritVal}} \def\FS{\operatorname{FS}} \def\Sing{\operatorname{Sing}} \def\Coh{\operatorname{Coh}} \def\Vect{\operatorname{Vect}} \def\into{\hookrightarrow} \def\tensor{\otimes} \def\CP{\mathbb{CP}} \def\eps{\varepsilon} \) SympSnip: HMS for Fanos

HMS for Fanos

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These are based on notes from a seminar talk given by Nick Sheridan In this reading group, we'll discuss mirror symmetry for Fano varieties. In mirror symmetry, we have pairs of Calabi-Yau manifolds \(X, Y\) whose symplectic and algebraic invariants are equivalent. A precise version of this statement is the homological mirror symmetry conjecture:
figure 0.0.1:Mirror symmetry exchanges the symplectic invariants (\(A\)-side) and complex invariants (\(B\)-side) on a pair of ``mirror'' spaces.
When \(X\) is Fano, the mirror \(Y = (Y, W)\) is a Landau-Ginzburg model, where In this case, our equivalence of invariants needs to be slightly modified:
figure 0.0.2:Mirror symmetry exchanges Fano varieties with Landau-Ginzburg models.
Where \(\FS(Y, W)\) is the Fukaya-Seidel category, and \(\Sing(Y, W)\) is the category of singularities. In the literature, there is more work in the direction of \(D\Fuk(X) \leftrightarrow D\Sing(Y, W)\), but in this reading group, we will focus on \(D\Coh(X)\leftrightarrow D\Sing(Y, W)\). We call \(D\Coh(X)\) the B-side, while \(D\FS(Y, W)\) is the A-side.

1: B-side invariants

definition 1.0.1

Let \(X\) be an algebraic variety. Denote by \(D\Coh(X)\) the dg-enhancement of the bounded derived category of coherent sheaves on \(X\).

definition 1.0.2

Let \(\mathcal{C}\) be a triangulated category. A semi-orthogonal decomposition of \(\mathcal{C}\) is the data \[\mathcal{C} = \langle \mathcal{C}_0, \ldots, \mathcal{C}_i\rangle\] where When \(\mathcal{C}_i = D\Coh(\bullet) = D\Vect\), we say that \(\mathcal{C}\) admits a full exceptional collection. When \(X\) is Fano, typically \(D\Coh(X)\) admits a semi-orthogonal decomposition definition 1.0.2 \[D\Coh(X) = \langle \mathcal{C}_1, \cdots \mathcal{C}_n\rangle\]

example 1.0.3

When \(X = \mathbb{P}^n\) or a toric Fano variety, then \(D^b\Coh(X)\) admits a full exceptional collection.

example 1.0.4

It's easy to find varieties without a full exceptional collection. Suppose that \(X\) admits a full exceptional collection. Then \(H^{p, q}(X) = 0\) for \(p \neq q\). When we don't have a full exceptional collection, we can try to build one as best as we can and look at what is left. Typically, we have a semi-orthogonal decomposition \[D\Coh(X) = \langle \text{Ku}(X), \mathcal{C}_1, \ldots, \mathcal{C}_n\rangle\] where \(\mathcal{C}_i = D\Coh(\bullet)\), and a larger part \(\text{Ku}(X)\), which is often called the Kuznetsov component of \(D\Coh(X)\). Often, this Kuznetsov component contains information about the birational geometry of \(X\). It can also reveal hidden relationships, e.g., it may be that \(D\Coh(X) \neq D\Coh(X')\), but we have \(\text{Ku}(X) = \text{Ku}(X')\).

2: \(A\)-side Landau-Ginzburg model

On this side of the mirror, \(Y\) will be a Kähler manifold, along with a holomorphic function \(W: Y \to \CC\) which is: The objects of the Fukaya-Seidel category \(\FS(Y, W)\) are Lagrangian submanifolds \(L \subset Y\) such that \(W|_L\) is a fibration over \(\RR_+\) outside of a compact set.
figure 2.0.1:A Lagrangian submanifold in a Fukaya-Seidel category needs to have prescribed behavior going off to infinity.
To take the Lagrangian intersection Floer cohomology between two such Lagrangians, we need to perturb the Lagrangians at infinity; the perturbation we take will rotate one Lagrangian at infinity. As a vector space, \(\hom(L, K) = \CC(\langle L \cap \phi(K)\rangle)\).
figure 2.0.2:To obtain transversality, the Lagrangian submanifolds in the Fukaya-Seidel category are pushed off one another with respect to the projection \(W: Y \to \CC\).
The differential and product structure on hom-spaces is given by counts of pseudo-holomorphic disks.

2.1: Comparison to \(B\)-side

Let \(\CritVal(W) \subset \CC\) be the subset of critical values of \(W\). For each \(c \in \CritVal(W)\), we have a ``smaller'' Landau-Ginzburg model \(Y_c\) by trivially extending \(W^{-1}(B(c, \epsilon)) \to B(c, \epsilon)\) to a fibration over \(\CC\).
figure 2.1.1:A small neighborhood in the base of a symplectic LG model can be used to build another LG model.
Define \(\FS_c = \FS(Y_c, W)\).

definition 2.1.2

Let \(c \in \CritVal(W)\) be a critical point. A vanishing path for \(c\) is a path \(\gamma: [0, \infty) \to \CC\) such that A vanishing path determines an embedding \(\FS_c \to \FS(Y, W)\) by extending Lagrangian submanifolds over the vanishing path using symplectic parallel transport.
figure 2.1.3:A vanishing path determines a method for extending Lagrangians belonging to the Fukaya-Seidel category near a critical value to the Fukaya-Seidel category of \((Y, W)\).
If we choose ``disjoint'' vanishing paths, we get a semi-orthogonal decomposition: \[\FS(Y, W) = \langle \FS_1, \ldots, \FS_n\rangle \]

example 2.1.4

Consider the symplectic Landau-Ginzburg model whose critical points are arranged as follows:
figure 2.1.5:A set of vanishing paths for the Lefschetz fibration \(W(z_1, z_2) = z_1 + z_2 + (z_1z_2)^{-1}\)
We obtain a semi-orthogonal decomposition \(\FS(Y, W) = \langle \FS_1, \FS_2, \FS_3 \rangle\).

2.2: Exceptional collections

Suppose additionally that \(W^{-1}(c)\) has an \(A_1\) singularity (a node) then \[\FS_c = D\Coh(\text{pt}) = D\Vect\] and the generating object is called the vanishing thimble associated with the point. So if all critical points of \(W\) are more (i.e., \(W\) is a Lefschetz fibration), then \(\FS(Y, W)\) admits a full exceptional collection. If \(W\) has isolated singularities, they can be ``Morsified'' by a small perturbation (in the symplectic category). From this, we obtain the following principle: if \((Y, W)\) is a symplectic Landau-Ginzburg model and \(W\) has isolated singularities, then \(\FS(Y, W)\) admits a full exceptional collection. This tells us that many examples arising from mirror symmetry will not have isolated singularities, as their mirror spaces will not have full exceptional collections! However, we still have the following expectations. When \(X\) is Fano, and \((Y, W)\) is its mirror, then we expect \(W\) will have some isolated singularities, and an additional critical value \(c_{\text{Ku}}\) with non-isolated singularities with \[\text{Ku}(X) = \FS_{c_{\text{Ku}}}.\]

References

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