K\"ahler Geometry of Moduli of Representations of Quivers

Abstract:

In this talk, I shall discuss some holomorphic aspects of moduli spaces of
finite dimensional
semistable representations of a finite quiver. Namely, I shall describe
the construction
of a natural Hermitian holomorphic line bundle on the stratified moduli
space of semistable
representations, and show that the curvature of this line bundle on each
stratum of the
moduli space is a scalar multiple of the K\"ahler form of that stratum. I
shall then recall
the definition of the tensor product $Q \otimes Q'$ of two quivers $Q$ and
$Q'$, and define the tensor
product $V \otimes W$ of a representation $V$ of $Q$ with a representation
$W$ of $Q'$, and discuss
the semistability of $V \otimes W$. Moreover, I shall describe a relation
between the natural line
bundles on the moduli spaces of representations of $Q, Q'$ and $Q \otimes
Q'$.