Can all nonzero degree map between compact Riemann surfaces (both genus >1 ) be deformed to holomorphic maps, if we can change the conformal structures on them? The simplest case: does there exist holomorphic map of degree one from \Sigma_m to \Sigma_n (m>n>1)?

The basic obstruction here is the RiemannHurwitz formula: If there is a degree $d$ map from a Riemann surface of genus $g$ to one of genus $h$, with branch points of orders $e_1$, $e_2$, ..., $e_r$, then $$2(g1) = 2d (h1) + \sum (e_i 1).$$ As a corollary, $2(g1) \geq 2d (h1)$. This has the standard corollaries: One always has $g \geq h$. If $g=h \geq 2$ then $d=1$. And, combined with the fact that a map of degree $1$ can have no braching, if $d=1$ then $g=h$. In the positive direction, one has the Riemann existence theorem. Given a Riemann surface $X$ of genus $g$, fix a finite number of points $x_1$, ..., $x_r$ on $X$. Give a $d$fold covering of $X \setminus \{ x_1, \ldots, x_r \}$ which completes topologically to a branched covering of $X$ by a surface $Y$ of genus $h$. Then there is a Riemann structure on $Y$ such that $Y \to X$ is a map of Riemann surfaces. So, if you can make your map look like a branched covering, then it will come from an actual holomorphic map, and you can even specify the downstairs holomorphic stucture in advance. 

