Dan Popovici (Toulouse) - “Deformation Limits of Compact Kaehler Manifolds.”

The speaker considered the following situation: suppose $\pi \colon X \to D$ is a proper holomorphic submersion. Let $n$ be the complex dimension of the fibre, and suppose fibres compact.

Example of Hironaka '62:

explicit example where fibres $X_t$, $t \ne 0$ are projective but $X_0$ is not Kähler.

Conjecture:

If $X_t$ Kähler, then $X_0$ should be a class $\mathcal C$ manifold.

Definition:

$X$ is class $\mathcal C$ if exists $\mu \colon \tilde X \to X$ proper, holomorphic, bimeromorphic and $\tilde X$ is Kähler.

i.e. should think of $X$ as blown up Kähler.

Theorem

(Demailly, Pali 2004??) Suppose $X$ compact, cplx. $X$ is of class $\mathcal C$ if and only if there is a Kähler current $T$. (i.e. $T$ is a $(1,1)$ current, $dT = 0$ and $T \ge \epsilon \omega$ where $\omega$ is a $C^\infty$ $(1,1)$ form on $X$.)
The goal is then to produce a Kähler current on $X_0$. Dan then showed the problem reduced to studying a conjecture of Demailly on "transcendental Morse inequalities".