# Analogue of Grauert's upper semi-continuity for Bott–Chern cohomology

In Coherent analytic sheaves, one has the following theorem due to Grauert:

Let $$f: X \rightarrow Y$$ be a holomorphic family of compact complex manifolds with connected complex manifolds $$X, Y$$ and $$V$$ a holomorphic vector bundle on $$X$$. Then for any integers $$q, d \geq 0$$, the set $$\left\{y \in Y: h^{q}\left(X_{y},\left.V\right|_{X_{y}}\right) \geq d\right\}$$ is an analytic subset of $$Y$$.

Hence, we can take $$V=\Omega^p$$ to get the hodge number. I wonder if the above assertion still holds for $$(p,q)$$-Bott–Chern cohomology? In other words, for the same conditions, is the set $$\left\{y \in Y: h^{p,q}_\text{BC}\left(X_{y}\right) \geq d\right\}$$a complex analytic subset of $$Y$$? Notice that one cannot apply Grauert's theorem to Bott–Chern cohomology, since now we just have the following isomorphism (one can refer to Demailly's book Basic results on sheaves and analytic sets for more details): $$H_\text{BC}^{p, q}(X) \cong \mathbb{H}^{p+q-1}\left(X, \mathscr{L}_{X}^{\bullet}\right).$$

• You referred to "Demailly's ebook". I wasn't sure, but guessed that you meant Basic results on sheaves and analytic sets. If that was not correct, please fix it to the correct reference. Nov 26 '21 at 19:43

The semi-continuity is true for elliptic complexes: if $$(C, d_t)$$ is a continuous family of elliptic complexes, parametrized by $$t\in \mathbb R$$, the cohomology of $$(C, d_t)$$ is semicontinuous in $$t$$. However, Bott–Chern cohomology are cohomology of an elliptic complex:
To see that the jumping loci are complex analytic, you could use an exact sequence $$H^{*}(\Lambda^{p, q-1}(M), \bar\partial)\oplus \overline{H^{*}(\Lambda^{q, p-1}(M), \bar\partial)} \rightarrow H_{BC}^{p,q}(M) \rightarrow H^{p+q}(M).$$ I don't have a citation for this, except my own paper with Ornea, Morse–Novikov cohomology of locally conformally Kähler manifolds, Theorem 4.7, where we proved it for BC-cohomology with coefficients in a local system; but I suppose it's rather well known. I don't expect that the theorem that you want (that all jumping loci are complex analytic) is published anywhere, but this exact sequence is a good starting point.
• Dear @Misha Verbitsky, many thanks for your answer. I still have some doubt looking forward to your reply. Assume the exact sequence you listed is correct, it seems that we still need some exact sequence of the following type ($V_0=V_{r+1}=0)$:$$0 \rightarrow V_{1} \rightarrow V_{2} \rightarrow \cdots \rightarrow V_{r} \rightarrow 0$$ to use the property $$\sum_{i=1}^{r}(-1)^{i} \operatorname{dim}\left(V_{i}\right)=0.$$ Am I right? Or perhaps you have some another way to go... Thanks again. Nov 27 '21 at 16:31
• Dear@Misha Verbitsky, yes, you're right, the only possible one is the short exact sequence, such that $$\text{dim}V_2=\text{dim}V_1+\text{dim}V_3,$$ which is not suitable for our case. Nov 27 '21 at 17:11