Glushkov Institute of Cybernetics NAS,

40 Glushkov prospect,

03680 Kiev, UKRAINE

E-mail: glanm@yahoo.com, glanm@d105.icyb.kiev.ua

**On computational aspects of the Fourier-Mukai transform
and another dualities**

**Abstract:**

It is well known that the solution of a linear differential equation with constant coefficients is related by Fourier transform to a solution of the polynomial equation. The Fourier-Mukai transform is a strong generalization of the mentioned approach. Let A be an abelian variety, A¢ its dual abelian variety and P the Poincaré divisor on A ×A¢. Let D^{b}(A) and D^{b}(A¢) be derived categories of bounded complexes of sheaves on A and A¢ respectively. A Fourier-Mukai transform was defined by Mukai as an exact equivalence F: D^{b}(A) ® D^{b}(A¢) between derived categories of above mentioned bounded complexes. For this transform analogies of the Fourier Inversion Theorem and the Parseval Theorem are valid. Works by B. Bartocci, U. Bruzzo, D. Ruipérez and A. Maciocia have generalized this approach to another classes of varieties. We investigate computational aspects of Fourier-Mukai transforms of derived categories of bounded complexes of sheaves on non-singular projective varieties and some another dualities.