For $\mathbb{P}^n_k$, the Beilinson exceptional collection is $\langle \mathcal{O}, \mathcal{O}(1), \ldots, \mathcal{O}(n) \rangle$. The DG category $\mathcal{E}$ on these objects has:

$\operatorname{Hom}(\mathcal{O}(i), \mathcal{O}(j)) = H^0(\mathcal{O}(j-i)) = S^{j-i}(k^{n+1})^*$

for $j \geq i$ (and $0$ otherwise), concentrated in degree $0$. The $A_\infty$-structure is formal (all $m_n = 0$ for $n \geq 3$). The twisted complex category gives:

$D^b(\operatorname{Coh}(\mathbb{P}^n)) \simeq H^0(\operatorname{Tw}(\mathcal{E}))$

Every coherent sheaf on $\mathbb{P}^n$ is built from line bundles $\mathcal{O}(i)$ via iterated extensions (cones) and shifts.

The Grassmannian $\operatorname{Gr}(k, n)$ has a full exceptional collection given by Kapranov, consisting of Schur functors applied to the tautological bundle. For $\operatorname{Gr}(2, 4)$:

$\langle \mathcal{O}, \mathcal{U}^*, \mathcal{O}(1), \Lambda^2 \mathcal{U}^*(1), \mathcal{U}^*(1), \mathcal{O}(2) \rangle$

where $\mathcal{U}$ is the tautological rank-$2$ bundle. The twisted complex enhancement gives an explicit DG model for $D^b(\operatorname{Coh}(\operatorname{Gr}(2,4)))$.

For a finite quiver $Q$ without oriented cycles, the category $D^b(\operatorname{rep}_k(Q))$ has a full exceptional collection given by the simple representations $S_1, \ldots, S_n$ (one per vertex). The DG category $\mathcal{E}$ is the path algebra $kQ$ (viewed as a DG algebra concentrated in degree $0$):

$D^b(\operatorname{rep}_k(Q)) \simeq H^0(\operatorname{Tw}(kQ))$

The twisted complexes are precisely the bounded complexes of projective $kQ$-modules.

A del Pezzo surface $S$ (blowup of $\mathbb{P}^2$ at $r \leq 8$ points) has a full exceptional collection. For $S = \operatorname{Bl}_p \mathbb{P}^2$:

$\langle \mathcal{O}, \mathcal{O}(E), \mathcal{O}(H-E), \mathcal{O}(H) \rangle$

where $H$ is the hyperplane class and $E$ the exceptional divisor. The Ext-algebra (with $A_\infty$-structure) and twisted complexes provide a complete DG model for $D^b(\operatorname{Coh}(S))$.

More generally, if $\mathcal{T}$ is a triangulated category with a tilting object $T$ (a compact generator with $\operatorname{Hom}(T, T[n]) = 0$ for $n \neq 0$), then $A = \operatorname{End}(T)$ is an ordinary algebra and:

$\mathcal{T} \simeq D^{\mathrm{perf}}(A) \simeq H^0(\operatorname{Tw}(A))$

The DG enhancement is the DG category of perfect DG $A$-modules. The Bondal--Kapranov construction provides the explicit model via twisted complexes.

For a Koszul algebra $A$ with Koszul dual $A^!$, the derived categories are related:

$D^b(\operatorname{mod}\text{-}A) \simeq D^b(\operatorname{mod}\text{-}A^!)$

At the DG level, this is realized by a DG quasi-equivalence $\operatorname{Tw}(\mathcal{E}_A) \simeq \operatorname{Tw}(\mathcal{E}_{A^!})$ where $\mathcal{E}_A$ and $\mathcal{E}_{A^!}$ are the DG categories of the respective simple modules. The Koszul duality functor is constructed explicitly via twisted complexes.

An object $E \in D^b(\operatorname{Coh}(X))$ is exceptional if $\operatorname{Hom}(E, E[n]) = 0$ for $n \neq 0$ and $\operatorname{Hom}(E, E) = k$. An exceptional collection $(E_1, \ldots, E_n)$ is a sequence where:

• Each $E_i$ is exceptional.
• $\operatorname{Hom}(E_j, E_i[m]) = 0$ for all $m$ when $j > i$ (semi-orthogonality).

The collection is full if $E_1, \ldots, E_n$ generate $D^b(\operatorname{Coh}(X))$ (no nonzero object is orthogonal to all $E_i$). Full exceptional collections exist for:

• Projective spaces $\mathbb{P}^n$.
• Grassmannians $\operatorname{Gr}(k, n)$.
• Flag varieties $G/P$ (in certain cases).
• Toric varieties (in many cases).
• Del Pezzo surfaces.

Given a full exceptional collection $\langle E_1, \ldots, E_n \rangle$, the $A_\infty$-algebra:

$\mathcal{E} = \bigoplus_{i,j} \operatorname{Ext}^*(E_i, E_j)$

(with its minimal $A_\infty$-structure obtained by transfer from the DG endomorphism algebra) completely determines $D^b(\operatorname{Coh}(X))$:

$D^b(\operatorname{Coh}(X)) \simeq H^0(\operatorname{Tw}(\mathcal{E})) \simeq D^{\mathrm{perf}}(\mathcal{E})$

This is one of the most powerful applications of the Bondal--Kapranov theorem: it reduces the study of derived categories to finite-dimensional $A_\infty$-algebra computations.

Given an exceptional pair $(E_1, E_2)$, the left mutation $L_{E_1} E_2$ is defined by the exact triangle:

$\operatorname{Hom}^*(E_1, E_2) \otimes E_1 \to E_2 \to L_{E_1} E_2$

Then $(L_{E_1} E_2, E_1)$ is again an exceptional pair. Mutations act transitively on the set of full exceptional collections (conjecturally, for most varieties). At the DG level, mutations correspond to changes of basis in the twisted complex category.

Not every smooth projective variety admits a full exceptional collection. For example:

• Curves of genus $g \geq 1$ have no exceptional objects at all.
• Abelian varieties, K3 surfaces, and Calabi--Yau manifolds of dimension $\geq 2$ have no full exceptional collections.

The Bondal--Kapranov enhancement via exceptional collections is specific to "rational-like" varieties. For general varieties, one needs other enhancement methods (injective/projective resolutions, the Drinfeld DG quotient, etc.).

The Bondal--Kapranov theorem is foundational for Kontsevich's program of noncommutative algebraic geometry. A "noncommutative variety" is a DG category $\mathcal{A}$ (smooth and proper), and morphisms are DG bimodules. The twisted complex construction shows that such categories are finitely generated: determined by finitely many objects (generators) and their $A_\infty$-Ext algebra.

This perspective unifies commutative and noncommutative geometry: a smooth projective variety $X$ corresponds to the DG category $\operatorname{Perf}_{\mathrm{dg}}(X)$, and a noncommutative deformation of $X$ is a deformation of this DG category.

In Kontsevich's homological mirror symmetry conjecture, the Bondal--Kapranov enhancement plays a key role. For a Fano variety $X$ with mirror Landau--Ginzburg model $(Y, W)$:

$D^b(\operatorname{Coh}(X)) \simeq D^b(\operatorname{Fuk}(Y, W))$

Both sides have DG (or $A_\infty$) enhancements. The exceptional collection on $X$ (Bondal--Kapranov side) corresponds to a collection of Lagrangian vanishing cycles in $(Y, W)$ (Fukaya side). The twisted complex construction on both sides provides the concrete identification.