677 research outputs found
The 3-fold vertex via stable pairs
The theory of stable pairs in the derived category yields an enumerative
geometry of curves in 3-folds. We evaluate the equivariant vertex for stable
pairs on toric 3-folds in terms of weighted box counting. In the toric
Calabi-Yau case, the result simplifies to a new form of pure box counting. The
conjectural equivalence with the DT vertex predicts remarkable identities. The
equivariant vertex governs primary insertions in the theory of stable pairs for
toric varieties. We consider also the descendent vertex and conjecture the
complete rationality of the descendent theory for stable pairs.Comment: Typos fixed. 40 pages, 8 figure
Trihyperkahler reduction and instanton bundles on CP^3
A trisymplectic structure on a complex 2n-manifold is a triple of holomorphic
symplectic forms such that any linear combination of these forms has constant
rank 2n, n or 0, and degenerate forms in belong to a non-degenerate
quadric hypersurface. We show that a trisymplectic manifold is equipped with a
holomorphic 3-web and the Chern connection of this 3-web is holomorphic,
torsion-free, and preserves the three symplectic forms. We construct a
trisymplectic structure on the moduli of regular rational curves in the twistor
space of a hyperkaehler manifold, and define a trisymplectic reduction of a
trisymplectic manifold, which is a complexified form of a hyperkaehler
reduction. We prove that the trisymplectic reduction in the space of regular
rational curves on the twistor space of a hyperkaehler manifold M is compatible
with the hyperkaehler reduction on M.
As an application of these geometric ideas, we consider the ADHM construction
of instantons and show that the moduli space of rank r, charge c framed
instanton bundles on CP^3 is a smooth, connected, trisymplectic manifold of
complex dimension 4rc. In particular, it follows that the moduli space of rank
2, charge c instanton bundles on CP^3 is a smooth complex manifold dimension
8c-3, thus settling a 30-year old conjecture.Comment: 42 pages, v. 3.2, changes in section 3.1: the notion of trisymplectic
structure stated differently, Clifford algebra action introduce
Sur le morphisme de Barth
Let be a rank-2 semi-stable sheaf on the projective plane, with
Chern classes . The curve of jumping lines of
, in the dual projective plane, has degree . Let be
the moduli space of equivalence classes of semi-stables sheaves of rank 2 and
Chern classes on the projective plane and be the
projective space of curves of degree in the dual projective plane. The
Barth morphism associates the
point to the class of the sheaf . We prove that this
morphism is generically injective for The image of is a
closed subvariety of dimension of ; as a consequence of
our result, the degree of this image is given by the Donaldson number of index
of the projective plane.Comment: Plain.tex, 54 pages. Uses diagrams.te
CONSTRUCTION AND CONVERGENCE STUDY OF SCHEMES PRESERVING THE ELLIPTIC LOCAL MAXIMUM PRINCIPLE
International audienceWe present a method to approximate (in any space dimension) diffusion equations with schemes having a specific structure; this structure ensures that the discrete local maximum and minimum principles are respected, and that no spurious oscillations appear in the solutions. When applied in a transient setting on models of concentration equations, it guaranties in particular that the approximate solutions stay between the physical bounds. We make a theoretical study of the constructed schemes, proving under a coercivity assumption that their solutions converge to the solution of the PDE. Several numerical results are also provided; they help us understand how the parameters of the method should be chosen. These results also show the practical efficiency of the method, even when applied to complex models
Monotone corrections for generic cell-centered Finite Volume approximations of anisotropic diffusion equations
We present a nonlinear technique to correct a general Finite Volume scheme for anisotropic diffusion problems, which provides a discrete maximum principle. We point out general properties satisfied by many Finite Volume schemes and prove the proposed corrections also preserve these properties. We then study two specific corrections proving, under numerical assumptions, that the corresponding solutions converge to the continuous one as the size of the mesh tends to 0. Finally we present numerical results showing these corrections suppress local minima produced by the initial Finite Volume scheme
Moduli spaces of framed perverse instantons on P^3
We study moduli spaces of framed perverse instantons on P^3. As an open
subset it contains the (set-theoretical) moduli space of framed instantons
studied by I. Frenkel and M. Jardim. We also construct a few counterexamples to
earlier conjectures and results concerning these moduli spaces.Comment: 50 page
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