181 research outputs found
Maximal surface group representations in isometry groups of classical Hermitian symmetric spaces
Higgs bundles and non-abelian Hodge theory provide holomorphic methods with
which to study the moduli spaces of surface group representations in a
reductive Lie group G. In this paper we survey the case in which G is the
isometry group of a classical Hermitian symmetric space of non-compact type.
Using Morse theory on the moduli spaces of Higgs bundles, we compute the number
of connected components of the moduli space of representations with maximal
Toledo invariant.Comment: v2: added due credits to the work of Burger, Iozzi and Wienhard. v3:
corrected count of connected components for G=SU(p,q) (p \neq q); added due
credits to the work of Xia and Markman-Xia; minor corrections and
clarifications. 31 page
On semistable principal bundles over a complex projective manifold, II
Let (X, \omega) be a compact connected Kaehler manifold of complex dimension
d and E_G a holomorphic principal G-bundle on X, where G is a connected
reductive linear algebraic group defined over C. Let Z (G) denote the center of
G. We prove that the following three statements are equivalent: (1) There is a
parabolic subgroup P of G and a holomorphic reduction of the structure group of
E_G to P (say, E_P) such that the bundle obtained by extending the structure
group of E_P to L(P)/Z(G) (where L(P) is the Levi quotient of P) admits a flat
connection; (2) The adjoint vector bundle ad(E_G) is numerically flat; (3) The
principal G-bundle E_G is pseudostable, and the degree of the charateristic
class c_2(ad(E_G) is zero.Comment: 15 page
Triplodon chodo, a new species of pearly fresh water mussel from the Amazon Basin (Mollusca: Bivalvia: Unionoida: Hyriidae)
Hyperkahler Metrics from Periodic Monopoles
Relative moduli spaces of periodic monopoles provide novel examples of
Asymptotically Locally Flat hyperkahler manifolds. By considering the
interactions between well-separated periodic monopoles, we infer the asymptotic
behavior of their metrics. When the monopole moduli space is four-dimensional,
this construction yields interesting examples of metrics with self-dual
curvature (gravitational instantons). We discuss their topology and complex
geometry. An alternative construction of these gravitational instantons using
moduli spaces of Hitchin equations is also described.Comment: 23 pages, latex. v2: an erroneous formula is corrected, and its
derivation is given. v3 (published version): references adde
Rank two quadratic pairs and surface group representations
Let be a compact Riemann surface. A quadratic pair on consists of a
holomorphic vector bundle with a quadratic form which takes values in fixed
line bundle. We show that the moduli spaces of quadratic pairs of rank 2 are
connected under some constraints on their topological invariants. As an
application of our results we determine the connected components of the
-character variety of .Comment: 37 pages, 1 figur
Hyperholomorpic connections on coherent sheaves and stability
Let be a hyperkaehler manifold, and a torsion-free and reflexive
coherent sheaf on . Assume that (outside of its singularities) admits a
connection with a curvature which is invariant under the standard SU(2)-action
on 2-forms. If the curvature is square-integrable, then is stable and its
singularities are hyperkaehler subvarieties in . Such sheaves (called
hyperholomorphic sheaves) are well understood. In the present paper, we study
sheaves admitting a connection with SU(2)-invariant curvature which is not
necessarily square-integrable. This situation arises often, for instance, when
one deals with higher direct images of holomorphic bundles. We show that such
sheaves are stable.Comment: 37 pages, version 11, reference updated, corrected many minor errors
and typos found by the refere
Quantum curves for Hitchin fibrations and the Eynard-Orantin theory
We generalize the topological recursion of Eynard-Orantin (2007) to the
family of spectral curves of Hitchin fibrations. A spectral curve in the
topological recursion, which is defined to be a complex plane curve, is
replaced with a generic curve in the cotangent bundle of an arbitrary
smooth base curve . We then prove that these spectral curves are
quantizable, using the new formalism. More precisely, we construct the
canonical generators of the formal -deformation family of -modules
over an arbitrary projective algebraic curve of genus greater than ,
from the geometry of a prescribed family of smooth Hitchin spectral curves
associated with the -character variety of the fundamental
group . We show that the semi-classical limit through the WKB
approximation of these -deformed -modules recovers the initial family
of Hitchin spectral curves.Comment: 34 page
L-VRAP-a lunar volatile resources analysis package for lunar exploration
The Lunar Volatile Resources Analysis Package (L-VRAP) has been conceived to deliver some of the objectives of the proposed Lunar Lander mission currently being studied by the European Space Agency. The purpose of the mission is to demonstrate and develop capability; the impetus is very much driven by a desire to lay the foundations for future human exploration of the Moon. Thus, LVRAP has design goals that consider lunar volatiles from the perspective of both their innate scientific interest and also their potential for in situ utilisation as a resource. The device is a dual mass spectrometer system and is capable of meeting the requirements of the mission with respect to detection, quantification and characterisation of volatiles. Through the use of appropriate sampling techniques, volatiles from either the regolith or atmosphere (exosphere) can be analysed. Furthermore, since L-VRAP has the capacity to determine isotopic compositions, it should be possible for the instrument to determine the sources of the volatiles that are found on the Moon (be they lunar per se, extra-lunar, or contaminants imparted by the mission itself
Extremal Bundles on Calabi-Yau Threefolds
We study constructions of stable holomorphic vector bundles on Calabi–Yau threefolds, especially those with exact anomaly cancellation which we call extremal. By going through the known databases we find that such examples are rare in general and can be ruled out for the spectral cover construction for all elliptic threefolds. We then introduce a general Hartshorne–Serre construction and use it to find extremal bundles of general ranks and study their stability, as well as computing their Chern numbers. Based on both existing and our new constructions, we revisit the DRY conjecture for the existence of stable sheaves on Calabi–threefolds, and provide theoretical and numerical evidence for its correctness. Our construction can be easily generalized to bundles with no extremal conditions imposed
Quasi-K\"ahler groups, 3-manifold groups, and formality
In this note, we address the following question: Which 1-formal groups occur
as fundamental groups of both quasi-K\"ahler manifolds and closed, connected,
orientable 3-manifolds. We classify all such groups, at the level of Malcev
completions, and compute their coranks. Dropping the assumption on
realizability by 3-manifolds, we show that the corank equals the isotropy index
of the cup-product map in degree one. Finally, we examine the formality
properties of smooth affine surfaces and quasi-homogeneous isolated surface
singularities. In the latter case, we describe explicitly the
positive-dimensional components of the first characteristic variety for the
associated singularity link.Comment: 18 pages; accepted for publication in Mathematische Zeitschrif
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