Let $X$ be a compact Kahler manifold with first Chern class $c_1(X)>0$ (i.e. positive). Consider a family $\pi\colon \mathcal{X} \to \mathcal{D}$ over the unit disc $\mathcal{D}$. \mathcal{D}$, with $X_0=X$. Do we know that $c_1(X_t)>0$ for $t \neq 0$?

Easy example: let $Y$ be a compact Kahler manifold with $H^{2,0}(Y)=0$, then "Deform projective Kahler to projective Kahler"!

Let X $X$ be a compact Kahler manifold with first Chern class c1(X)>0(i.e. $c_1(X)>0$ (i.e. positive). Consider a family π:X^-->D $\pi\colon \mathcal{X} \to \mathcal{D}$ over the unit disc D,Do $\mathcal{D}$. Do we know that the first Chern class c1(Xt)>0 ($c_1(X_t)>0$ for $t is not 0)\neq 0$?

Easy exxampleexample: Let Y let $Y$ be a compact Kahler manifold with H2,0(Y)=0,then $H^{2,0}(Y)=0$, then "Deform projective Kahler to projective Kahler"!

Let X be a compact Kahler manifold with first Chern class c1(X)>0(i.e. positive). Consider a family π:X^-->D over the unit disc D,Do we know that the first Chern class c1(Xt)>0 (t is not 0)?

Easy exxample: Let Y be a compact Kahler manifold with H2,0(Y)=0,then "Deform projective Kahler to projective Kahler"!

Let X be a compact Kahler manifold with first Chern class c1(X)>0(i.e. positive). Consider a family π:X^-->D over the unit disc D,Do we know that the first Chern class c1(Xt)>0?

Easy exxample: Let Y be a compact Kahler manifold with H2,0(Y)=0,then "Deform projective Kahler to projective Kahler"!