  Quasidefinite sesquilinear forms for Laurent polynomials in the complex plane
and corresponding CMV biorthogonal Laurent polynomial families are studied.
Bivariate linear functionals encompass large families of orthogonalities like
Sobolev and discrete Sobolev types. Two possible Christoffel transformations of
these linear functionals are discussed. Either the linear functionals are
multiplied by a Laurent polynomial, or are multiplied by the complex conjugate
of a Laurent polynomial. For the Geronimus transformation, the linear
functional is perturbed in two possible manners as well, by a division by a
Laurent polynomial or by a complex conjugate of a Laurent polynomial, in both
cases the addition of appropriate masses (linear functionals supported on the
zeros of the perturbing Laurent polynomial) is considered. The connection
formulas for the CMV biorthogonal Laurent polynomials, its norms, and
Christoffel-Darboux kernels, in all the four cases, are given. For the
Geronimus transformation, the connection formulas for the second kind functions
and mixed Christoffel-Darboux kernels are also given in the two possible cases.
For prepared Laurent polynomials, i.e. of the form
$L(z)=L_nz^n+\cdots+L_{-n}z^{-n}$, $L_nL_{-n}\neq 0$, these connection formulas
lead to quasideterminantal (quotient of determinants) Christoffel formulas for
all the four transformations, expressing an arbitrary degree perturbed
biorthogonal Laurent polynomial in terms of $2n$ unperturbed biorthogonal
Laurent polynomials, their second kind functions or Christoffel-Darboux kernels
and its mixed versions. The unit circle case, given its exceptional properties,
is discussed in more detail.
