On a conjecture of Magnus on the Hurwitz Monodromy group

1973 ◽  
Vol 132 (1) ◽  
pp. 45-50 ◽  
Author(s):  
Colin Maclachlan
Keyword(s):  
Monodromy Group

scholarly journals Arithmeticity of the monodromy of some Kodaira fibrations

2019 ◽  
Vol 156 (1) ◽  
pp. 114-157
Author(s):  
Nick Salter ◽  
Bena Tshishiku
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Monodromy Group ◽  
Algebraic Curves ◽  
Algebraic Surfaces ◽  
Mapping Class ◽  
Complex Varieties

A question of Griffiths–Schmid asks when the monodromy group of an algebraic family of complex varieties is arithmetic. We resolve this in the affirmative for a class of algebraic surfaces known as Atiyah–Kodaira manifolds, which have base and fibers equal to complete algebraic curves. Our methods are topological in nature and involve an analysis of the ‘geometric’ monodromy, valued in the mapping class group of the fiber.

scholarly journals SUPERPOTENTIAL OF THE M-THEORY CONIFOLD AND TYPE IIA STRING THEORY

2004 ◽  
Vol 19 (04) ◽  
pp. 521-555 ◽  
Author(s):  
GOTTFRIED CURIO
Keyword(s):  
String Theory ◽  
Heisenberg Group ◽  
Group Action ◽  
Moduli Space ◽  
Monodromy Group ◽  
Witten Theory ◽  
Flat Bundle ◽  
M Theory

The membrane instanton superpotential for M-theory on the G2 holonomy manifold given by the cone on S3×S3 is given by the dilogarithm and has Heisenberg monodromy group in the quantum moduli space. We compare this to a Heisenberg group action on the type IIA hypermultiplet moduli space for the universal hypermultiplet, to metric corrections from membrane instantons related to a twisted dilogarithm for the deformed conifold and to a flat bundle related to a conifold period, the Heisenberg group and the dilogarithm appearing in five-dimensional Seiberg/Witten theory.

scholarly journals INTEGRAL CONSTRAINTS ON THE MONODROMY GROUP OF THE HYPERKÄHLER RESOLUTION OF A SYMMETRIC PRODUCT OF A K3 SURFACE

2010 ◽  
Vol 21 (02) ◽  
pp. 169-223 ◽  
Author(s):  
EYAL MARKMAN
Keyword(s):  
Hilbert Scheme ◽  
Prime Power ◽  
Monodromy Group ◽  
Euler Number ◽  
Hodge Structure ◽  
K3 Surface ◽  
Monodromy Operator ◽  
First Case ◽  
Hyperkähler Varieties ◽  
Restriction Homomorphism

Let S[n]be the Hilbert scheme of length n subschemes of a K3 surface S. H2(S[n],ℤ) is endowed with the Beauville–Bogomolov bilinear form. Denote by Mon the subgroup of GL [H*(S[n],ℤ)] generated by monodromy operators, and let Mon2be its image in OH2(S[n],ℤ). We prove that Mon2is the subgroup generated by reflections with respect to +2 and -2 classes (Theorem 1.2). Thus Mon2does not surject onto OH2(S[n],ℤ)/(±1), when n - 1 is not a prime power.As a consequence, we get counterexamples to a version of the weight 2 Torelli question for hyperKähler varieties X deformation equivalent to S[n]. The weight 2 Hodge structure on H2(X,ℤ) does not determine the bimeromorphic class of X, whenever n - 1 is not a prime power (the first case being n = 7). There are at least 2ρ(n - 1) - 1distinct bimeromorphic classes of X with a given generic weight 2 Hodge structure, where ρ(n - 1) is the Euler number of n - 1.The second main result states, that if a monodromy operator acts as the identity on H2(S[n],ℤ), then it acts as the identity on Hk(S[n],ℤ), 0 ≤ k ≤ n + 2 (Theorem 1.5). We conclude the injectivity of the restriction homomorphism Mon → Mon2, if n ≡ 0 or n ≡ 1 modulo 4 (Corollary 1.6).

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