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Quasidefinite sesquilinear forms for Laurent polynomials in the complex planeand corresponding CMV biorthogonal Laurent polynomial families are studied.Bivariate linear functionals encompass large families of orthogonalities likeSobolev and discrete Sobolev types. Two possible Christoffel transformations ofthese linear functionals are discussed. Either the linear functionals aremultiplied by a Laurent polynomial, or are multiplied by the complex conjugateof a Laurent polynomial. For the Geronimus transformation, the linearfunctional is perturbed in two possible manners as well, by a division by aLaurent polynomial or by a complex conjugate of a Laurent polynomial, in bothcases the addition of appropriate masses (linear functionals supported on thezeros of the perturbing Laurent polynomial) is considered. The connectionformulas for the CMV biorthogonal Laurent polynomials, its norms, andChristoffel-Darboux kernels, in all the four cases, are given. For theGeronimus transformation, the connection formulas for the second kind functionsand 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 formulaslead to quasideterminantal (quotient of determinants) Christoffel formulas forall the four transformations, expressing an arbitrary degree perturbedbiorthogonal Laurent polynomial in terms of $2n$ unperturbed biorthogonalLaurent polynomials, their second kind functions or Christoffel-Darboux kernelsand its mixed versions. The unit circle case, given its exceptional properties,is discussed in more detail.

CMV biorthogonal Laurent polynomials: Christoffel formulas for Christoffel and Geronimus perturbations