On the Compactly Supported A¹-Euler Characteristic of Symmetric Powers and Geometrically Cyclic Branched Coverings

This thesis contains two parts. The first part consists of the computation of the compactly supported A1-Euler characteristic of SymnX for n = 2, 3 and X a connected, smooth, projective scheme over a perfect field k of characteristic not two if n = 2 and characteristic not two or three if n = 3. In order to do this, we develop an equivariant quadratic Euler characteristic and study its relation with the quadratic Euler characteristic of a smooth quotient by a finite group. We then partially confirm a question asked by Pajwani-Pál by checking that the power structure constructed by Pajwani-Pál computes compactly supported A1-Euler characteristic of SymnX for n = 2, 3 for X a smooth, projective variety if the characteristic of k is not two or three and for all varieties if k has characteristic zero. Furthermore, we compute the compactly supported A1-Euler characteristic of a split toric variety and verify that the power structure of Pajwani-Pál computes the compactly supported A1-Euler characteristic of all symmetric powers of split toric varieties. In the second part, we analyse the change of the quadratic Euler characteristic under taking a geometrically cyclic branched covering f: Y → X. For this, we choose a morphism ρ: X → C to a smooth, projective curve C possessing a half-canonical line bundle such that ρ satisfies some finiteness condition regarding the number of critical points. In this situation, we prove comparison results relating the local Euler classes of the twisted sheaf of Kähler differentials at the critical points on X, Y, and the branch locus. For odd-fold covers, we devise a general formula for the relation with the branch locus. Using a Riemann-Hurwitz formula of Levine, we can relate the local Euler class comparisons to the quadratic Euler characteristics of X, Y, and the branch locus. Here, we prove explicit comparison formulae for certain odd-fold covers. Furthermore, as a specific example, we examine the situation for 2-fold branched coverings of P2.