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

B-side invariants

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When \(X\) is Fano, typically \(D\Coh(X)\) admits a semi-orthogonal decomposition Missing Label (def:SemiOrthogonalDecomposition)! \[D\Coh(X) = \langle \C_1, \cdots \mathcal C_n\rangle.\]

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When we don't have a full exceptional collection, we can try ``as hard as we can'' to build one, 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 bigger 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 \(X\). They also reveal hidden relationships, e.g. it could be that \(D\Coh(X)\neq D\Coh(X')\), but we have \(\text{Ku}(X) = \text{K}(X')\).