On normal forms of complex points of codimension 2 submanifolds

Abstract

In this paper we present some linear algebra behind quadratic parts of quadratically flat complex points of codimension two real submanifold in a complex manifold. Assuming some extra nondegenericity and using the result of Hong, complete normal form descriptions can be given, and in low dimensions, we obtain a complete classification without any extra assumptions.

On normal forms of complex points of small C2-perturbations of real 4-manifolds embedded in a complex 3-manifold

“…(99) Clearly, (99) for δ 2 = 0 is equivalent to (98) for δ 1 = 0. Next, we observe the range of the function f (r, t, β) = |ar 2 e iβ + 2brt + dt 2 e −iβ | given with a constraint r 4 + 2r 2 t 2 cos θ + t 4 = 1 (see (15)). Provided that (R, T) = (r 2 , t 2 ) lies on on an ellipse R 2 + 2RT cos θ + T 2 = 1, we can further assume that either r/t or t/r is any real nonnegative number.…”

“…Since M is the maximum of f with respect to a constraint (15), it follows that the maximum of f on a compact domain given by…”

The stabilizers for the action of orthogonal similarity on symmetric matrices and orthogonal $*$-conjugacy on Hermitian matrices

“…where ǫ = {ǫ j ∈ {1, −1} | λ j 0 ∨ α j even } witn ǫ j = 1 for λ j = 0, α j odd. Note that trivially H 2n−1 (0) is orthogonally congruent to −H 2n−1 (0) (see [15,Remark 4.5]). The next theorem concludes the answer to the question concerning uniqueness of the normal form (1.15).…”