56 research outputs found
Seiberg-Witten invariants for manifolds with , and the universal wall crossing formula
In this paper we describe the Seiberg-Witten invariants, which have been
introduced by Witten, for manifolds with . In this case the invariants
depend on a chamber structure, and there exists a universal wall crossing
formula. For every K\"ahler surface with and =0, these invariants
are non-trivial for all -structures of non-negative index.Comment: LaTeX, 27 pages. To appear in Int. J. Mat
A wall crossing formula for degrees of real central projections
The main result is a wall crossing formula for central projections defined on
submanifolds of a real projective space. Our formula gives the jump of the
degree of such a projection when the center of the projection varies. The fact
that the degree depends on the projection is a new phenomenon, specific to real
algebraic geometry. We illustrate this phenomenon in many interesting
situations. The crucial assumption on the class of maps we consider is relative
orientability, a condition which allows us to define a -valued degree map
in a coherent way. We end the article with several examples, e.g. the pole
placement map associated with a quotient, the Wronski map, and a new version of
the real subspace problem.Comment: 29 pages. First revised version: The proof of the "wall-crossing
formula" is now more conceptional. We prove new general properties of the set
of values of the degree map on the set of central projections. Second revised
version: minor corrections. To appear in International Journal of Mathematic
Real determinant line bundles
This article is an expanded version of the talk given by Ch. O. at the Second
Latin Congress on "Symmetries in Geometry and Physics" in Curitiba, Brazil in
December 2010. In this version we explain the topological and gauge-theoretical
aspects of our paper "Abelian Yang-Mills theory on Real tori and Theta divisors
of Klein surfaces".Comment: LaTeX, 8 page
Intrinsic signs and lower bounds in real algebraic geometry
A classical result due to Segre states that on a real cubic surface in â â 3 there exist two kinds of real lines: elliptic and hyperbolic lines. These two kinds of real lines are defined in an intrinsic way, i.e., their definition does not depend on any choices of orientation data. Segre's classification of smooth real cubic surfaces also shows that any such surface contains at least 3 real lines. Starting from these remarks and inspired by the classical problem mentioned above, our article has the following goals: (a) We explain a general principle which leads to lower bounds in real algebraic geometry. (b) We explain the reason for the appearance of intrinsic signs in the classical problem treated by Segre, showing that the same phenomenon occurs in a large class of enumerative problems in real algebraic geometry. (c) We illustrate these principles in the enumerative problem for real lines in real hypersurfaces of degree 2m-3 in â â m $\mathbb {P}^m_\mathbb {R}
Relations for virtual fundamental classes of Hilbert schemes of curves on surfaces
In [DĂŒrr, Kabanov, Okonek, Topology 46: 225-294, 2007] we constructed virtual fundamental classes for Hilbert schemes of divisors of topological type m on a surface V, and used these classes to define the PoincarĂ© invariant of V: We conjecture that this invariant coincides with the full Seiberg-Witten invariant computed with respect to the canonical orientation data. In this note we prove that the existence of an integral curve C â V induces relations between some of these virtual fundamental classes . The corresponding relations for the PoincarĂ© invariant can be considered as algebraic analoga of the fundamental relations obtained in [OzsvĂĄth, SzabĂł, Ann. of Math. 151: 93-124, 2000
Quaternionic Monopoles
We present the simplest non-abelian version of Seiberg-Witten theory:
Quaternionic monopoles. These monopoles are associated with
Spin^h(4)-structures on 4-manifolds and form finite-dimensional moduli spaces.
On a Kahler surface the quaternionic monopole equations decouple and lead to
the projective vortex equation for holomorphic pairs. This vortex equation
comes from a moment map and gives rise to a new complex-geometric stability
concept. The moduli spaces of quaternionic monopoles on Kahler surfaces have
two closed subspaces, both naturally isomorphic with moduli spaces of
canonically stable holomorphic pairs. These components intersect along
Donaldsons instanton space and can be compactified with Seiberg-Witten moduli
spaces. This should provide a link between the two corresponding theories.
Notes: To appear in CMP The revised version contains more details concerning
the Uhlenbeck compactfication of the moduli space of quaternionic monopoles,
and possible applications are discussed. Attention ! Due to an ununderstandable
mistake, the duke server had replaced all the symbols "=" by "=3D" in the
tex-file of the revised version we sent on February, the 2-nd. The command
"\def{\ad}" had also been damaged !Comment: LaTeX, 35 page
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