\(
\def\CC{{\mathbb C}}
\def\RR{{\mathbb R}}
\def\NN{{\mathbb N}}
\def\ZZ{{\mathbb Z}}
\def\TT{{\mathbb T}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\HF{{\operatorname{HF}^\bullet}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\ot{{\leftarrow}}
\def\st{\;:\;}
\def\Fuk{{\operatorname{Fuk}}}
\def\emprod{m}
\def\cone{\operatorname{Cone}}
\def\Flux{\operatorname{Flux}}
\def\li{i}
\def\ev{\operatorname{ev}}
\def\id{\operatorname{id}}
\def\grad{\operatorname{grad}}
\def\ind{\operatorname{ind}}
\def\weight{\operatorname{wt}}
\def\Sym{\operatorname{Sym}}
\def\HeF{\widehat{CHF}^\bullet}
\def\HHeF{\widehat{HHF}^\bullet}
\def\Spinc{\operatorname{Spin}^c}
\def\min{\operatorname{min}}
\def\div{\operatorname{div}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\Tw{{\operatorname{Tw}}}
\def\Log{{\operatorname{Log}}}
\def\TropB{{\operatorname{TropB}}}
\def\wt{{\operatorname{wt}}}
\def\Span{{\operatorname{span}}}
\def\Crit{\operatorname{Crit}}
\def\CritVal{\operatorname{CritVal}}
\def\FS{\operatorname{FS}}
\def\Sing{\operatorname{Sing}}
\def\Coh{\operatorname{Coh}}
\def\Vect{\operatorname{Vect}}
\def\into{\hookrightarrow}
\def\tensor{\otimes}
\def\CP{\mathbb{CP}}
\def\eps{\varepsilon}
\)
SympSnip: B-side invariants
definition 0.0.1
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 0.0.2
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
- \(\mathcal{C}_i\) are full triangulated subcategories
- \(\hom(\mathcal{C}_i, \mathcal{C}_j) = 0\) whenever \(C_i \in \mathcal{C}_i, C_j \in \mathcal{C}_j\) and \(j > i\)
- The \(\mathcal{C}_i\) generate the category.
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 0.0.2
\[D\Coh(X) = \langle \mathcal{C}_1, \cdots \mathcal{C}_n\rangle\]
example 0.0.3
When \(X = \mathbb{P}^n\) or a toric Fano variety, then \(D^b\Coh(X)\) admits a full exceptional collection.
example 0.0.4
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 \(\text{Ku}(X) = \text{Ku}(X')\).