Let $X$ be a smooth complex projective variety and $L$ be some ample divisor. For a holomorphic map $u:\Sigma \to X$, we define its degree to be $deg(u^*L)$.

Question: For a given positive integer $M$, is there a positive integer $N$ such that for a generic smooth hypesurface $D$ in the linear system $|NL|$, there are no nontrivial holomorphic maps of degree and genus less than $M$ into $D$?

Remark1: for a given degree $d$ and genus $g$, virtual dimension of moduli spaces of genus $g$ degree $d$ maps into a hypersurface $D\in |NL|$ is $$ c_1^D(d)+(dim(D)-3)(1-g) = c_1^X(d)-Nd +(1-g)(dim(D)-3).$$ If $g$ and $d$ are bounded, then for $N$ big enough, virtual dimension will be negative.

Remark2: It seems that the answer to some similar question in symplectic setting is positive.

Let $X$ be a smooth complex projective variety and $L$ be some ample divisor. For a holomorphic map $u:\Sigma \to X$, we define its degree to be $deg(u^*L)$.

Question: For a given positive integer $M$, is there a positive integer $N$ such that for a generic smooth hypesurface $D$ in the linear system $|NL|$, there are no nontrivial holomorphic maps of both degree and genus less than $M$ into $D$?

Remark1: for a given degree $d$ and genus $g$, virtual dimension of moduli spaces of genus $g$ degree $d$ maps into a hypersurface $D\in |NL|$ is $$ c_1^D(d)+(dim(D)-3)(1-g) = c_1^X(d)-Nd +(1-g)(dim(D)-3).$$ If $g$ and $d$ are bounded, then for $N$ big enough, virtual dimension will be negative.

Remark2: It seems that the answer to some similar question in symplectic setting is positive.