Examples of tilting objects that don't come from exceptional sequences This is a question on geometric tilting theory. On smooth projective variety it is possible to define in general tilting object as perfect complex that satisfy some properties, but are there examples of tilting objects that are actually complexes? All examples I know are vector bundles obtained as sums of exceptional sequences. What would be examples of different nature?
Edit: if we denote variety $X$ then tilting object $T$ in $D(X)$ is a perfect complex (or from the point of view of abstract triangulated categories "compact object") such that $T$ generates $D(X)$ and $Ext^i(T,T)=0$ for $i \neq 0$.
 A: The answer to your more specific question is yes, there are tilting objects that involve complexes not quasi-isomorphic to any vector bundle. Consider the blowup $X$ of $\mathbb{P}^2$ at a single point $p$. Then, Orlov showed that there is a semiorthogonal decomposition $D^b(X)=\langle e,O_X,O_X(1),O_X(2)\rangle$, where $O_X(i)$ is the pullback of $O_{\mathbb{P}^2}(i)$, and where $e$ can be taken to be $i_*O_E(-1)$, where $i:E\rightarrow X$ is the inclusion of the exceptional divisor. The objects $e,O_X(i)$ are in fact all exceptional, so their direct sum is a tilting object for $D^b(X)$. However, $e$ is not quasi-isomorphic to a vector bundle, because it is supported along the exceptional divisor.
A: One trivial example is the derived category of a central simple algebra $D(A)$ where the class of $A$ in the Brauer group is non-zero. A non-zero simple left ideal $I$ of $A$ is a tilting complex, but the endomorphism algebra of $I$ is not the base field $k$ but rather $D$, the division algebra Morita equivalent to $A$. So, it is not an exceptional object. Similar examples arise in the derived categories of Severi-Brauer varieties.
