TITLE: Is there an upper bound on dimension of kernel of elliptic operator for a fixed closed manifold M
QUESTION [7 upvotes]: Assume that $M$ is a smooth closed manifold and $E,F$ are fixed smooth vector bundles over $M.$

Is there a number $C,$ such that for any elliptic operator $\mathcal{D}:\Gamma(E)\to\Gamma(F)$
  $$\dim\ker\mathcal{D}\leqslant C.$$

For simplicity, we may assume that $\Gamma(E)=\Gamma(F)=C^\infty(M).$

REPLY [2 votes]: Here is another simple counterexample: every holomorphic vector bundle over a Riemann surface gives rise to an elliptic operator (the $\bar\partial$-operator) whose kernel is the space of holomorphic sections. Two vector bundles over the same compact surface are isomorphic as smooth complex vector bundles if and only if the have the same rank and the same degree. Now take the Riemann sphere $P^1$ and a line bundle $L\to P^1$ of degree $d\geq 1$ and the rank $2$ vector bundle $$E=L\oplus L^*\to P^1.$$ $E$ has degree $0$ for every $d$ but the dimension of holomorphic sections, i.e. the dimension of the kernel is $d+1$.
On the other hand, there is a class of elliptic operators for which such an estimate exists:
 every linear elliptic differential operator 
$$D\colon\Gamma(M,L)\to\Gamma(M,\tilde L)$$
of order 1 between complex line bundles $L,\tilde L$ over a surface $M$ is given  as the $\bar\partial$-operator of some holomorphic line bundle (with respect to an appropriate Riemann surface structure). The dimension of the kernel is then restricted by $$\dim\ker D\leq \mathrm{deg}(L)+1$$
because we can always produce a $(k-1)$th order zero at a point for some section in a $k$-dimensional space of holomorphic sections of a line bundle.