• geometric representation theory
• coherent sheaves
• moduli spaces
• Hall algebras
• Higgs bundles

The object of this work is to study two problems in geometry: one is a problem of enumerative nature over finite fields, and the other is the study of the moduli spaces of Higgs sheaves on curves over arbitrary fields.

Let $X$ be a smooth projective and geometrically connected curve of genus g over a finite field $k$. It can be shown that the number of isomorphism classes of coherent sheaves of fixed rank and degree on $X$ is finite. In particular, there are finitely many geometrically indecomposable coherent sheaves of rank $r$ and degree $d$. The main result of this thesis is a proof that this number is a polynomial $\mathcal{A}_{r,d,g}(X)$ (depending only on $r$, $d$ and $g$) in the Weil numbers of $X$. These polynomials can be thought as analogues of the Kac polynomials counting indecomposable representations of quivers.

A key tool in the proof of this result is the Harder-Narasimhan filtration of a coherent sheaf and the existence of an algebraic stack parametrizing flat families of sheaves with fixed slopes in the Harder-Narasimhan filtration.
Then, by introducing a suitable notion of volume for algebraic stacks, we reduce the problem of counting vector bundles to that of computing the volume of an algebraic stack of iterated extensions of coherent sheaves. To carry out the computation of the volume, we use the language of spherical Hall algebras and Eisenstein series, and we both show the existence of the polynomial and provide an explicit formula in terms of generating series.\\

In the second part of the thesis, we study the moduli stack Higgs sheaves on a curve $X$, defined over an arbitrary field $k$. Higgs sheaves are coherent sheaves $\mathcal{F}$ equipped with a morphism $\phi: \mathcal{F} \to \mathcal{F} \otimes \omega_X$ and, after briefly describing their geometry, we specialize to the case of stable Higgs bundles and construct their moduli space $\higgs^{st}_{r,d}(X)$. We then prove two results: the first is that, if $k = \mathbb{F}_q$, the cardinality of the $\mathbb{F}_q$-rational points of the moduli space of stable Higgs sheaves of fixed rank and degree on $X$ is explicitly determined by $\mathcal{A}_{r,d,g}(X)$. The second result is that, if $k$ is a finite field or an algebraically closed field of characteristic $0$, the polynomial $A_{g,r,d}$ also completely determines the generating series of the Betti numbers of $\higgs^{st}_{r,d}(X)(k)$.