Torusactions, motives and graphhypersurfaces
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Abstract
This thesis is concerned with the study of the motives of graph hypersur-
faces arising from renormalization integrals in Quantum Field Theory. The
study of these motives is closely related to the geometric interpretation of
Feynman amplitudes as period integrals of projective varieties. The only
known approach to directly study these motives is in the work of Bloch, Es-
nault and Kreimer who computed the mixed Hodge-structure of the wheels
with spokes graphs and this has been subsequently generalized by Doryn to
so-called Zig-Zag graphs, which are still fairly close to the class of the wheels
with spokes graphs. The motivic interpretation of these results is problem-
atic since they use results from etale cohomology that have no generalization
in motivic cohomology, hence this computation is not strictly a computa-
tion of the motive. To overcome this lack of available methods I study the
use of Torus actions on graph hypersurfaces and Bialynicki-Birula decom-
positions, which is a well established method in the smooth case. In the
singular case there is no strong established generalization of this decomposi-
tion. Thus establishing Bialynicki-Birula decompositions entails essentially
three questions:
• What graph hypersurfaces admit a torus action?
• What is the maximal dimension of such an action?
• How can one reduce the computation of the motive of a graph hyper-
surface to the motives of the fixed point loci?
I give sufficient criteria for the first question by identifying a class of determi-
nantal hypersurfaces that allow actions of tori. I further describe explicitly
the class of graphs that yield maximal dimensions of tori acting on them.
This is important because the Bialynicki-Birula decomposition depends on
the explicit knowledge of the fixed point locus of a torus action and large
torus actions lead to small, and in the case of graph hypersurfaces even often
linear, components of the fixed point locus. I also exhibit an explicit exam-
ple of a graph hypersurface that does not admit any torus action. Finally it
is illustrated how one can obtain an explicit description of the motive of a
graph hypersurface if one is able to compute explicitly the fixed point loci of
an explicit equivariant resolution of singularities of the graph hypersurface.