# Extending polarization (or Kähler metric) from the central fiber to the nearby fibers (or the total space)

Let $\pi: \mathcal{X}\to B$ be a complex analytic family of compact complex manifolds, i.e. $\pi$ is a surjective, proper submersion between complex manifolds. For simplicity, we assume $B$ is the open unit disc $\Delta$ in the complex plane. Denote the fibers of $\pi$ by $X_t:=\pi^{-1}(t)$ and the central fiber is $X:=\pi^{-1}(0)$.

1. Let $[\omega]=c_1(L)\in H^2(X,\mathbb{Z})$ be the first Chern class of an ample line bundle, i.e. an integral Kähler class, on $X$. Is it possible to extend this class to the nearby fibers in the sense that there exist Kähler metrics $\omega_t\in A^{1,1}(X_t)$ on $X_t$ such that $[\omega_t]=[\omega]\in H^2(X,\mathbb{Z})$? Is it possible to extend this class to the total space $\mathcal{X}$ in the sense that there exist Kähler metric $\omega_{\mathcal{X}}\in A^{1,1}(\mathcal{X})$ on $\mathcal{X}$ such that $[\omega_{\mathcal{X}}\mid_{X}]=[\omega]\in H^2(X,\mathbb{Z})$?

2. Let $\omega$ be a Kähler metric on $X$. Is it possible to extend it to the total space $\mathcal{X}$ in the sense that there exist a Kähler metric $\omega_{\mathcal{X}}\in A^{1,1}(\mathcal{X})$ on $\mathcal{X}$ such that $\omega_{\mathcal{X}}\mid_{X}=\omega$?

3. Let $L$ be a holomorphic line bundle on $X$. Is it possible to extend it to the total space $\mathcal{X}$ in the sense that there exist a holomorphic line bundle $\mathcal{L}$ on $\mathcal{X}$ such that $\mathcal{L}\mid_{X}=L$?

These questions arise in the following situation:

Consider the Kuranishi family $\pi: (\mathfrak{X}, X)\to (\textrm{Def},0)$ of a polarised Calabi-Yau manifold $(X,[\omega])$. $\pi$ itself is not a polarized family. The problem is how to give polarizations on the other fibers of $\pi$ so that we have the period mapping of Griffiths: $\textrm{Def}\to D$. In the paper of Tian, define

$$H^1(X,\Theta_X)_\omega :=\{\theta\in H^1(X,\Theta_X) \mid \theta\lrcorner \omega=0 \in H^2(X,\mathcal{O}_X)\},$$

and $\textrm{Def}_\omega :=\textrm{Def} \cap H^1(X,\Theta_X)_\omega$, then he claims that "$\rm Def_\omega$ consists of those local deformations of $X$ preserving the polarization $[\omega]$" and that "each point $\theta\in \rm Def_\omega$ stands for a fiber of the Kuranishi family $\pi: (\mathfrak{X}, X)\to (\textrm{Def},0)$ with a polarization $[\omega_{\theta}]=[\omega]\in H^2(X,\mathbb{Z})$". I don't know why?

In page 95 of the book Algebraic Geometry III: Complex Algebraic Varieties Algebraic Curves and Their Jacobians, by Kulikov-Kurchanov-Shokurov, the above claims of Tian are made more precisely. According to them, $\textrm{Def}_{\omega}$ is just the submanifold of $\textrm{Def}$ on which the Kähler form $\omega$ has type $(1,1)$. By this, they probably mean that the Kähler form $\omega$ has type $(1,1)$ as a $2$-form on each fiber over points in $\textrm{Def}_{\omega}$. Again I don't know why?

In page 8 of this paper, Yoshikawa claims that every holomorphic line bundle $L$ on a Calabi-Yau threefold $X$ extends to a holomorphic line bundle $\mathcal{L}$ on the total space $\mathfrak{X}$ of the Kuranishi family of $X$. Why?

• The answer to the 3 questions is no. If $H^{2,0}(X)\neq 0$, for general $t$ the class $[\omega ]$ will no longer be of type $(1,1)$. This is very standard material, you should start reading basic books (e.g. Griffiths-Harris, or Voisin) before asking these questions here. – abx Jun 3 '18 at 13:48
• @abx: This is a well-posed question arising from reading papers or graduate level books, and so is clearly on topic according to the criteria specified in the help centre. – Neil Strickland Jun 4 '18 at 8:00