478 research outputs found

### Hyperbolic Structures and Root Systems

We discuss the construction of a one parameter family of complex hyperbolic
structures on the complement of a toric mirror arrangement associated with a
simply laced root system. Subsequently we find conditions for which parameter
values this leads to ball quotients

### Abelian covers of surfaces and the homology of the level L mapping class group

We calculate the first homology group of the mapping class group with
coefficients in the first rational homology group of the universal abelian $\Z
/ L \Z$-cover of the surface. If the surface has one marked point, then the
answer is \Q^{\tau(L)}, where $\tau(L)$ is the number of positive divisors of
$L$. If the surface instead has one boundary component, then the answer is
\Q. We also perform the same calculation for the level $L$ subgroup of the
mapping class group. Set $H_L = H_1(\Sigma_g;\Z/L\Z)$. If the surface has one
marked point, then the answer is \Q[H_L], the rational group ring of $H_L$.
If the surface instead has one boundary component, then the answer is \Q.Comment: 32 pages, 10 figures; numerous corrections and simplifications; to
appear in J. Topol. Ana

### Virasoro constraints and the Chern classes of the Hodge bundle

We analyse the consequences of the Virasoro conjecture of Eguchi, Hori and
Xiong for Gromov-Witten invariants, in the case of zero degree maps to the
manifolds CP^1 and CP^2 (or more generally, smooth projective curves and smooth
simply-connected projective surfaces). We obtain predictions involving
intersections of psi and lambda classes on the compactification of M_{g,n}. In
particular, we show that the Virasoro conjecture for CP^2 implies the numerical
part of Faber's conjecture on the tautological Chow ring of M_g.Comment: 12 pages, latex2

### Monodromy of Cyclic Coverings of the Projective Line

We show that the image of the pure braid group under the monodromy action on
the homology of a cyclic covering of degree d of the projective line is an
arithmetic group provided the number of branch points is sufficiently large
compared to the degree.Comment: 47 pages (to appear in Inventiones Mathematicae

### Motivic Milnor fibre for nondegenerate function germs on toric singularities

We study function germs on toric varieties which are nondegenerate for their
Newton diagram. We express their motivic Milnor fibre in terms of their Newton
diagram. We extend a formula for the motivic nearby fibre to the case of a
toroidal degeneration. We illustrate this by some examples.Comment: 14 page

### Computing top intersections in the tautological ring of $M_g$

We derive effective recursion formulae of top intersections in the
tautological ring $R^*(M_g)$ of the moduli space of curves of genus $g\geq 2$.
As an application, we prove a convolution-type tautological relation in
$R^{g-2}(M_g)$.Comment: 18 page

### Forgetful maps between Deligne-Mostow ball quotients

We study forgetful maps between Deligne-Mostow moduli spaces of weighted
points on P^1, and classify the forgetful maps that extend to a map of
orbifolds between the stable completions. The cases where this happens include
the Livn\'e fibrations and the Mostow/Toledo maps between complex hyperbolic
surfaces. They also include a retraction of a 3-dimensional ball quotient onto
one of its 1-dimensional totally geodesic complex submanifolds

### Ground states of supersymmetric Yang-Mills-Chern-Simons theory

We consider minimally supersymmetric Yang-Mills theory with a Chern-Simons
term on a flat spatial two-torus. The Witten index may be computed in the weak
coupling limit, where the ground state wave-functions localize on the moduli
space of flat gauge connections. We perform such computations by considering
this moduli space as an orbifold of a certain flat complex torus. Our results
agree with those obtained previously by instead considering the moduli space as
a complex projective space. An advantage of the present method is that it
allows for a more straightforward determination of the discrete electric 't
Hooft fluxes of the ground states in theories with non-simply connected gauge
groups. A consistency check is provided by the invariance of the results under
the mapping class group of a (Euclidean) three-torus.Comment: 18 page

### Algebraic entropy and the space of initial values for discrete dynamical systems

A method to calculate the algebraic entropy of a mapping which can be lifted
to an isomorphism of a suitable rational surfaces (the space of initial values)
are presented. It is shown that the degree of the $n$th iterate of such a
mapping is given by its action on the Picard group of the space of initial
values. It is also shown that the degree of the $n$th iterate of every
Painlev\'e equation in sakai's list is at most $O(n^2)$ and therefore its
algebraic entropy is zero.Comment: 10 pages, pLatex fil

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