The blow-up formula conjecture for Bott–Chern cohomology

Let $X$ be a compact complex manifold with complex dimension $n$, and let $Z\subseteq X$ be a closed complex submanifold of complex codimension $r\geq 2$. Suppose that $\pi:\tilde{X}\rightarrow X$ is the blow-up of $X$ along $Z$. The blow-up formula conjecture. There is a canonical isomorphism

$$H^{p,q}_{BC}(\tilde{X})\cong H^{p,q}_{BC}(X)\oplus\Big(\bigoplus_{i=1}^{r-1}H^{p-i,q-i}_{BC}(Z)\Big),$$

for any $0\leq p,q\leq n$.

This conjecture proposes a Bott–Chern cohomology analogue of the blow-up formulas for de Rham and Dolbeault cohomology on general complex manifolds. It concerns the behavior of Bott–Chern cohomology under blow-ups along arbitrary closed complex submanifolds.