Let $k$ be an algebraically closed field of char $p\neq 0$, $W_2(k)$ the witt vector of length 2. $C_1$ a smooth projective curve over $W_2(k)$, and $H_1$ a vector bundle over $C_1$. We denote $C_0$ the smooth projective curve from mod $p$ reduction of $C_1$ , and $H_0$ the vector bundle from the reduction of $H_1$ . Is the following statment true?
(1) If $H^0(C_0,H_0)\neq 0$, then $H^0(C_1,H_1)\neq0$.
If (1) is ture, I may go on asking:
(2) The map $H^0(C_1,H_1)\to H^0(C_0,H_0)$ is surjective.