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$, then "Deform projective Kahler to projective Kahler"! 3 added 42 characters in body Let X$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"!