**Preprint No.**
A-03-03

#
Georg Hein

## Generalized Albanese morphisms

**Abstract:**
We define generalizations of the Albanese variety
for a projective variety X.
The generalized Albanese morphisms
X ----> Alb_r(X)
contract those curves
C in X for which the induced morphism
Hom(\pi_1(X),U(r)) --> Hom(\pi_1(C),U(r))
has a finite image.
Thus, they may be interpreted as a U(r)-version of the
Shafarevich morphism.
**Keywords: **Moduli spaces, vector bundles,
Albanese variety, algebraic surfaces,
fundamental group,
Shafarevich morphism

**Mathematics Subject Classification (MSC2000):**
14D20, 14E05

**Language:** ENG
**Available:** Pr-A-03-03.ps

**Contact**: Georg Hein, Freie Universität Berlin, Fachbereich Mathematik und Informatik, Arnimallee 2-6, D-14195 Berlin, Germany (ghein@math.fu-berlin.de)

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