This is the first article of the two papers in which we investigate the holomorphic and formal flattening problem for a codimension two real submanifold in C with n ≥ 3 near a non-degenerate CR singular point. The problem is motivated from the study of the complex Plateau problem that seeks for the Levi-flat hypersurface bounded by a given real submanifold and is motivated by the classical comp...

This is the second article of the two papers, in which we investigate the holomorphic and formal flattening problem of a non-degenerate CR singular point of a codimension two real submanifold in C with n ≥ 3. The problem is motivated from the study of the complex Plateau problem that looks for the Levi-flat hypersurface bounded by a given real submanifold and by the classical complex analysis p...

For an (m+1)-dimensional space-time (X, g), define a mapped null hypersurface to be a smooth map ν : N → X (that is not necessarily an immersion) such that there exists a smooth field of null lines along ν that are both tangent and g-orthogonal to ν. We study relations between mapped null hypersurfaces and Legendrian maps to the spherical cotangent bundle ST M of an immersed spacelike hypersurf...

A cochain complex associated with the vector 1-form determined by the first and second fundamental tensors of a hypersurface M in E"+l is introduced. Its cohomology groups HP(M), called curvature groups, are isomorphic with the cohomology groups of M with coefficients in a subsheaf %R of the sheaf S of closed vector fields on M. If M is a minimal variety, the same conclusion is valid with S^ re...

Suppose that f defines a singular, complex affine hypersurface. If the critical locus of f is onedimensional, we obtain new general bounds on the ranks of the homology groups of the Milnor fiber of f . This result has an interesting implication on the structure of the vanishing cycles in the category of perverse sheaves.

In this article, we prove that either there exists at least one periodic orbit of Hamilton vector field on a given energy hypersurface in R 2n or there exist at least two periodic orbits on the near-by energy hypersurface in R. The more general results are also obtained.

We prove that any arithmetically Gorenstein curve on a smooth, general hypersurface X ⊂ P of degree at least 6, is a complete intersection. This gives a characterisation of complete intersection curves on general type hypersurfaces in P. We also verify that certain 1-cycles on a general quintic hypersurface are non-trivial elements of the Griffiths group.

We introduce a class extending the notion of Chern-Mather class to possibly nonreduced schemes, and use it to express the di erence between SchwartzMacPherson's Chern class and the class of the virtual tangent bundle of a singular hypersurface of a nonsingular variety. Applications include constraints on the possible singularities of a hypersurface and on contacts of nonsingular hypersurfaces, ...

Using Gauss-Manin derivatives of generalized normal functions, we arrive at results on the non-triviality of the transcendental regulator for Km of a very general projective algebraic manifold. Our strongest results are for the transcendental regulator for K1 of a very general K3 surface and its self-product. We also construct an explicit family of K1 cycles on H ⊕ E8 ⊕ E8-polarized K3 surfaces...

Let XR be the zero locus in RP n of one or two independently and Weyl distributed random real quadratic forms. Denoting by XC the complex part in CP n of XR and by b(XR) and b(XC) the sums of their Betti numbers, we prove that: (1) lim n→∞ Eb(XR) n = 1. In particular for one quadric hypersurface asymptotically Smith’s inequality b(XR) ≤ b(XC) is expected to be sharp. The methods we use combine ...