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):
$$
H_\text{BC}^{p, q}(X) \cong \mathbb{H}^{p+q-1}\left(X, \mathscr{L}_{X}^{\bullet}\right).
$$
 A: 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:
M. Schweitzer, Autour de la cohomologie de Bott–Chern,
arXiv:0709.3528.
This takes care of semicontinuity in the real analytic setting,
but I suppose you need your jumping loci to be complex analytic.
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.
