Homological action of the modular group on some cubic moduli spaces

Abstract

Abstract. We describe the action of the automorphism group of the complex cubic x 2 + y 2 + z 2 − xyz − 2 on the homology of its fibers. This action includes the action of the mapping class group of a punctured torus on the subvarieties of its SL(2, C) character variety given by fixing the trace of the peripheral element (socalled "relative character varieties"). This mapping class group is isomorphic to PGL(2, Z).We also describe the corresponding mapping class group action for the four-holed sphere and its relative SL(2, C) character varieties, which are fibers of deformations x 2 + y 2 + z 2 − xyz − 2 − P x − Qy − Rz of the above cubic. The 2-congruence subgroup PGL(2, Z) (2) still acts on these cubics and is the full automorphism group when P, Q, R are distinct.

The Algebraic de Rham Cohomology of Representation Varieties

“…Theorem 2.1 only guarantees that the algebraic de Rham cohomologies agree with the smooth de Rham cohomologies. These two results show that this is also true for these two particular singular spaces (compare [10]).…”

“…In the cases of Σ 1,1 and Σ 0,4 , the M C 's are 2-dimensional. In these two cases, their homologies have been calculated via topological methods [10]. A remarkable theorem of Grothendieck [12] states that the hypercohomology of the algebraic de Rham complex of a smooth variety computes its smooth de Rham cohomology.…”

“…Schultz implemented his algorithm in Singular [5]. Notice also that this monodromy action does not arise from a Dehn twist action on Σ 1,1 because any Dehn twist induced monodromy action is the identity on the above 2-sphere [10].…”

Euler and algebraic geometry

“…There is a hyperkahler moduli space of 2-dimensional representations of the fundamental group of P 1 minus 4 points with given local monodromy and determinant 1, which can be viewed as a complex affine cubic surface described by Fricke [20]. What is the subspace consisting of complex variations of Hodge structure over P 1 minus 4 points?…”

The algebraic entropy of the special linear character automorphisms of a free group on two generators

“…The automorphism group of κ has been studied by Goldman-Neumann [14], Brown [5], Fried [11], and others in the context of the SL(2, C)-character variety of a torus with one boundary component S. In this context, the fundamental group π 1 (S) = F 2 , the free group on two generators. The SL(2, C)-character variety of S is known to be all of C 3 , and the real points correspond to either SL(R 2 ) or SU (2) characters of F 2 (Morgan-Shalen [21]).…”