1,110 research outputs found
Webs of Lagrangian Tori in Projective Symplectic Manifolds
For a Lagrangian torus A in a simply-connected projective symplectic manifold
M, we prove that M has a hypersurface disjoint from a deformation of A. This
implies that a Lagrangian torus in a compact hyperk\"ahler manifold is a fiber
of an almost holomorphic Lagrangian fibration, giving an affirmative answer to
a question of Beauville's. Our proof employs two different tools: the theory of
action-angle variables for algebraically completely integrable Hamiltonian
systems and Wielandt's theory of subnormal subgroups.Comment: 18 pages, minor latex problem fixe
Homological mirror symmetry for the quintic 3-fold
We prove homological mirror symmetry for the quintic Calabi-Yau 3-fold. The
proof follows that for the quartic surface by Seidel (arXiv:math/0310414)
closely, and uses a result of Sheridan (arXiv:1012.3238). In contrast to
Sheridan's approach (arXiv:1111.0632), our proof gives the compatibility of
homological mirror symmetry for the projective space and its Calabi-Yau
hypersurface.Comment: 29 pages, 6 figures. v2: revised following the suggestions of the
referee
Seiberg-Witten and Gromov invariants for self-dual harmonic 2-forms
This is the sequel to the author's previous paper which gives an extension of
Taubes' "SW=Gr" theorem to non-symplectic 4-manifolds. The main result of this
paper asserts the following. Whenever the Seiberg-Witten invariants are defined
over a closed minimal 4-manifold X, they are equivalent modulo 2 to
"near-symplectic" Gromov invariants in the presence of certain self-dual
harmonic 2-forms on X. A version for non-minimal 4-manifolds is also proved. A
corollary to circle-valued Morse theory on 3-manifolds is also announced,
recovering a result of Hutchings-Lee-Turaev about the 3-dimensional
Seiberg-Witten invariants.Comment: 41 pages. Comments desired; to be submitte
Relative Ruan and Gromov-Taubes Invariants of Symplectic 4-Manifolds
We define relative Ruan invariants that count embedded connected symplectic
submanifolds which contact a fixed stable symplectic hypersurface V in a
symplectic 4-manifold (X,w) at prescribed points with prescribed contact orders
(in addition to insertions on X\V) for stable V. We obtain invariants of the
deformation class of (X,V,w). Two large issues must be tackled to define such
invariants: (1) Curves lying in the hypersurface V and (2) genericity results
for almost complex structures constrained to make V pseudo-holomorphic (or
almost complex). Moreover, these invariants are refined to take into account
rim tori decompositions. In the latter part of the paper, we extend the
definition to disconnected submanifolds and construct relative Gromov-Taubes
invariants
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