IJPAM: Volume 25, No. 3 (2005)

QUIVERS, DECORATED VECTOR BUNDLES
AND ÉTALE COVERINGS OF SMOOTH CURVES

E. Ballico
Department of Mathematics
University of Trento
380 50 Povo (Trento) - Via Sommarive, 14, ITALY
e-mail: ballico@science.unitn.it


Abstract.Here we consider quiver-type stability for vector bundles on a smooth projective curve. Let $f: X \to Y$ be an unramified Galois covering between smooth and connected projective curve. We fix a quiver-type numerical datum $\tilde{\Gamma}$ on $X$ and a geometric $\tilde{\Gamma}$-object $\underset{=}{\tilde{E}}$ on $X$. We define the quiver-type numerical datum $f_\ast (\tilde{\Gamma})$ on $Y$ and the geometric $f_\ast (\tilde{\Gamma})$-object $f_\ast (\underset{=}{\tilde{E}})$ on $Y$. Then:

(i)
$f_\ast (\underset{=}{\tilde{E}})$ is $f_\ast (\tilde{\Gamma})$-semistable if and only if $\underset{=}{\tilde{E}}$ is $\tilde{\Gamma}$-semistable;
(ii)
$f_\ast (\underset{=}{\tilde{E}})$ is $f_\ast (\tilde{\Gamma})$-stable if and only if $\underset{=}{\tilde{E}}$ is $\tilde{\Gamma}$-stableand no two of the $\tilde{\Gamma}$-objects $\alpha ^\ast (\underset{=}{\tilde{E}})$, $\beta ^\ast (\underset{=}{\tilde{E}})$, $\alpha ,\beta \in G$ and $\alpha \ne \beta$ are isomorphic.


Received: August 23, 2005

AMS Subject Classification: 14H60

Key Words and Phrases: holomorphic triples on curves, decorated vector bundle, vector bundles on curves, stable vector bundles, quiver

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2005
Volume: 25
Issue: 3