593 research outputs found
Regular homotopy of Hurwitz curves
We prove that any two irreducible cuspidal Hurwitz curves and (or
more generally, curves with A-type singularities) in the Hirzebruch surface
with coinciding homology classes and sets of singularities are regular
homotopic; and symplectically regular homotopic if and are
symplectic with respect to a compatible symplectic form.Comment: 26 page
Suspending Lefschetz fibrations, with an application to Local Mirror Symmetry
We consider the suspension operation on Lefschetz fibrations, which takes
p(x) to p(x)-y^2. This leaves the Fukaya category of the fibration invariant,
and changes the category of the fibre (or more precisely, the subcategory
consisting of a basis of vanishing cycles) in a specific way. As an
application, we prove part of Homological Mirror Symmetry for the total spaces
of canonical bundles over toric del Pezzo surfaces.Comment: v2: slightly expanded expositio
Mirror symmetry for Del Pezzo surfaces: Vanishing cycles and coherent sheaves
We study homological mirror symmetry for Del Pezzo surfaces and their mirror
Landau-Ginzburg models. In particular, we show that the derived category of
coherent sheaves on a Del Pezzo surface X_k obtained by blowing up CP^2 at k
points is equivalent to the derived category of vanishing cycles of a certain
elliptic fibration W_k:M_k\to\C with k+3 singular fibers, equipped with a
suitable symplectic form. Moreover, we also show that this mirror
correspondence between derived categories can be extended to noncommutative
deformations of X_k, and give an explicit correspondence between the
deformation parameters for X_k and the cohomology class [B+i\omega]\in
H^2(M_k,C).Comment: 40 pages, 9 figure
Special lagrangian fibrations on flag variety
One constructs lagrangian fibrations on the flag variety and proves
that the fibrations are special.Comment: 19 page
Brieskorn manifolds as contact branched covers of spheres
We show that Brieskorn manifolds with their standard contact structures are
contact branched coverings of spheres. This covering maps a contact open book
decomposition of the Brieskorn manifold onto a Milnor open book of the sphere.Comment: 8 pages, 1 figur
Constructions of generalized complex structures in dimension four
Four-manifold theory is employed to study the existence of (twisted)
generalized complex structures. It is shown that there exist (twisted)
generalized complex structures that have more than one type change loci. In an
example-driven fashion, (twisted) generalized complex structures are
constructed on a myriad of four-manifolds, both simply and non-simply
connected, which are neither complex nor symplectic
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