In diophantine geometry, the Batyrev-Manin-Peyre conjecture originally concerns rational points on Fano varieties. It describes the asymptotic behaviour of the number of rational points of bounded height, when the bound becomes arbitrary large.
A geometric analogue of this conjecture deals with the asymptotic behaviour of the moduli space of rational curves on a complex Fano variety, when the 'degree' of the curves 'goes to infinity'. Various examples support the claim that, after renormalisation in a relevant ring of motivic integration, the class of this moduli space may converge to a constant which has an interpretation as a motivic Euler product.
In this talk, we will state this motivic version of the Batyrev-Manin-Peyre conjecture and give some examples for which it is known to hold : projective space, more generally toric varieties, and equivariant compactifications of vector spaces. In a second part we will introduce the notion of equidistribution of curves and show that it opens a path to new types of results.[-]