Let $X,Y$ be varieties over $\mathbb{C}$, and $D(X),D(Y)$ be the derived categories of bounded complex of coherent sheaves. Let $U \subset X, V \subset Y$ be open subvarieties， and let $X-U, Y-V$ have codimensions $c\_1, c\_2$ respectively. Assume $U $ is isomorphism to $V$. I heard a result that says: when the codimension $c\_1, c\_2$ high enough, then $D(X)$ is derived equivalent to $D(Y)$.


Does anyone know any references or the precise statement of any results along this direction, I am especially interest in the upper bound of codimension.



Derived equivalence of two varieties which are isomorphism over certain open subvarieties
In your assumption the varieties $X$ and $Y$ are birational. In particular, there is a smooth variety $Z$ with two proper birational maps $p:Z\to X$ and $q:Z\to Y$ which are isomorphisms over $U$ and $V$ (for $Z$ you can take an appropriate resolution of singularities of the graph of your birational isomorphism).


There is a conjecture saying that if $p^\*K\_X = q^\*K\_Y$ then $D(X) \cong D(Y)$ (for short K-equivalence implies D-equivalence). Its condition holds if both $codim(Z\setminus p^{-1}(U))$
and $codim(Z\setminus q^{-1}(V))$ are at least 2. But I am afraid this cannot be formulated just in terms of $codim(X\setminus U)$ and $codim(Y\setminus V)$.


