K-theory of algebraic curves

There exists a duality between elliptic curves and noncommutative tori, i.e. C∗-algebras generated by the unitary operators u and v such that vu = euv. We show that this duality can be included into a general picture involving the algebraic curves of higher genus. In this way we prove that a big part of geometry of complex algebraic curves can be developed from the K-theory of a noncommutative C∗-algebra Oλ coming from measured foliations and interval exchange transformations. The known projective invariants (canonical, special divisors, linear series, etc.) are shown to be the Morita invariants of algebra Oλ. A new K-invariant called “projective curvature” is introduced.