Complex Surfaces With Many Algebraic Structures.

We find new examples of complex surfaces with countably many non-isomorphic algebraic structures. Here is one such example: take an elliptic curve |$E$| in |$\mathbb P^{2}$| and blow up nine general points on |$E$|⁠. Then the complement |$M$| of the strict transform of |$E$| in the blow-up has countably many algebraic structures. Moreover, each algebraic structure comes from an embedding of |$M$| into a blow-up of |$\mathbb P^{2}$| in nine points lying on an elliptic curve |$F\not \simeq E$|⁠. We classify algebraic structures on |$M$| using a Hopf transform : a way of constructing a new surface by cutting out an elliptic curve and pasting a different one. Next, we introduce the notion of an analytic K-theory of varieties. Manipulations with the example above lead us to prove that classes of all elliptic curves in this K-theory coincide. To put in another way, all motivic measures on complex algebraic varieties that take equal values on biholomorphic varieties do not distinguish elliptic curves. [ABSTRACT FROM AUTHOR]