The problem of approximating triples of commuting $n \times n$ matrices by generic matrices is equivalent to the problem of whether the variety $C(3,n)$ of commuting triples is irreducible. The answer to the problem is known to be positive for dimension no greater than 5 and negative for dimension no smaller than 30. Using simultaneous commutative perturbations of pairs of matrices in the centralizer of the third matrix we are able to show that the answer is positive in dimensions 6, 7 and 8 as well. All the proofs are given over an arbitrary algebraically closed field of characteristic zero.
COBISS.SI-ID: 13037401
Nondegenerate mappings that preserve Jordan triple product on M_n(F) are characterized. Here, n ) 2 and F is an arbitrary field.
COBISS.SI-ID: 14131801
We prove an orbifold Riemann-Rouch formula for polarized complex 3-fold $(X,D)$. As an application some new families of projective Calabi-Yaus are constructed.
COBISS.SI-ID: 13815129
We obtain new bounds for the exponent of the Schur multiplier of a given p-group. We prove that the exponent of the Schur multiplier can be bounded by a function depending only on the exponent of a given group, confirming a conjecture posed by Schur in 1904. As a consequence we show that the exponent of the Schur multiplier of any group of exponent four divides eight, and that this bound is best possible. The notion of the exponential rank of a p-group is introduced.
COBISS.SI-ID: 14325337
We show that Connes' embedding conjecture on von Neumann algebras is equivalent to the existence of certain algebraic certificates for a polynomial in noncommuting variables to satisfy a certain nonnegativity condition. These algebraic certificates involve sums of hermitian squares and commutators. We prove that they always exist for a similar nonnegativity condition where elements of separable $II_1$-factors are considered instead of matrices. Under the presence of Connes' conjecture, we derive degree bounds for the certificates.
COBISS.SI-ID: 14569561