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