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Let $X$ be a smooth variety over a field, and $T\in D^b(Coh(X))$ be a tilting object, i.e. (1) $Ext^i(T,T)=0$ for all $ i\neq 0$; (2)Tilting algebra $A:=End(T)$ has finite global dimension; (3)$add$-$T$ generates $D^b(Coh(X))$, $add$-$T$ is the category of direct summands of finite direct sums of copies of $T$.

If $X$ is also quasi-projective Calabi-Yau (CY), Ginzburg(arXiv:math/0612139) proved that $A$ is also CY.

Bondal had considered for the full strongly exception collections, the tilting algebra is Koszul algebra (See "L. Hille: Consistent algebras and special titling sequence" for proof!)

My question is: If $X$ be a smooth quasi-projective CY variety, then $A$ is Koszul algebra and Artin-Schelter Gorenstein?

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I think this question has some serious confusion. Smooth \emph{compact} Calabi-Yau having a tilting object is not compatible with Serre duality... – Daniel Pomerleano Aug 29 '13 at 7:47
Thanks Daniel Pomerleano for comments! X can not compact! – S. Yang Aug 29 '13 at 10:10

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