217 research outputs found
Singular hypersurfaces characterizing the Lefschetz properties
In the paper untitled "Laplace equations and the Weak Lefschetz Property" the
authors highlight the link between rational varieties satisfying a Laplace
equation and artinian ideals that fail the Weak Lefschetz property. Continuing
their work we extend this link to the more general situation of artinian ideals
failing the Strong Lefschetz Property. We characterize the failure of SLP (that
includes WLP) by the existence of special singular hypersurfaces (cones for
WLP). This characterization allows us to solve three problems posed by Migliore
and Nagel and to give new examples of ideals failing the SLP. Finally, line
arrangements are related to artinian ideals and the unstability of the
associated derivation bundle is linked with the failure of SLP. Moreover we
reformulate the so-called Terao's conjecture for free line arrangements in
terms of artinian ideals failing the SLP
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
Monodromy invariants in symplectic topology
This text is a set of lecture notes for a series of four talks given at
I.P.A.M., Los Angeles, on March 18-20, 2003. The first lecture provides a quick
overview of symplectic topology and its main tools: symplectic manifolds,
almost-complex structures, pseudo-holomorphic curves, Gromov-Witten invariants
and Floer homology. The second and third lectures focus on symplectic Lefschetz
pencils: existence (following Donaldson), monodromy, and applications to
symplectic topology, in particular the connection to Gromov-Witten invariants
of symplectic 4-manifolds (following Smith) and to Fukaya categories (following
Seidel). In the last lecture, we offer an alternative description of symplectic
4-manifolds by viewing them as branched covers of the complex projective plane;
the corresponding monodromy invariants and their potential applications are
discussed.Comment: 42 pages, notes of lectures given at IPAM, Los Angele
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