A Universal Construction for Moduli Spaces of Decorated Vector Bundles over Curves

By A. Schmitt. 2000

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AbstractLet $X$ be a smooth projective curve over the field of complex numbers, and fix a homogeneous representation $\rho\colon \mathop{\rm GL}(r)\rightarrow \mathop{\rm GL}(V)$. Then one can associate to every vector bundle $E$ of rank $r$ over $X$ a vector bundle $E_\rho$ with fibre $V$. We would like to study triples $(E,L,\phi)$ where $E$ is a vector bundle of rank $r$ over $X$, $L$ is a line bundle over $X$, and $\phi\colon E_\rho\rightarrow L$ is a nontrivial homomorphism. This setup comprises well known objects such as framed vector bundles, Higgs bundles, and conic bundles. In this paper, we will formulate a general (parameter dependent) semistability concept for such triples, which generalizes the classical Hilbert--Mumford criterion, and we establish the existence of moduli spaces for the semistable objects. In the examples which have been studied so far, our semistability concept reproduces the known ones. Therefore, our results give in particular a unified construction for many moduli spaces considered in the literature.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/08b54c37f61767cf9ded4b8409b76b12907efb9e

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Citations in the corpus

QUANTUM KIRWAN MORPHISM AND GROMOV-WITTEN INVARIANTS OF QUOTIENTS I (2012, background): “Here we restrict to the case that the target X is a projective G-variety and note that a pair (P → C, u : C → P (X)) is by definition a morphism from C to the quotient stack X/G introduced by Deligne-Mumford [13].”
Moduli for decorated tuples of sheaves and representation spaces for quivers (2004, unknown): “Recall that Mσ ,χ(E•,α) = s ∑ j=1 α j · ( Pσ ,χ(Ev,v∈V) · rkσ (E v j ,v∈V)−Pσ ,χ(E v j ,v∈V) · rkσ (Ev,v∈V) ) .”
Moduli spaces for principal bundles in arbitrary characteristic (2005, background): “he form Wa,b,c := W⊗a ⊕b ⊗ dim^k(W) W ⊗−c for non-negative integers a, b, and c. Remark 4.1.3. Note that this result implies that the theory of decorated vector bundles over curves developed in [37] extends to algebraically closed ground ﬁe lds of arbitrary characteristic. Proof. By [21], 5.3 Prop”