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Stable pairs, linear systems and the Verlinde formula. (English) Zbl 0882.14003
The present paper, published more than three years ago, is to be seen as one of the milestones in the development of the moduli theory of vector bundles on algebraic curves and its applications to conformal quantum field theories. The main result is a general dimension formula for spaces of sections of line bundles on certain moduli spaces, which provides, as a special case, the celebrated Verlinde formula for spaces of generalized theta functions on the moduli spaces of semistable rank-2 vector bundles with fixed determinant on a smooth projective curve. Back then, in the early 1990’s, Thaddeus’s proof of the Verlinde formula was among the very first mathematically rigorous, affirmative establishments of this (formerly conjectural) dimension formula. In the meantime, as it is well-known, several other approaches to (and proofs of) the Verlinde formula have been presented, and that by various authors and in varying degrees of generality.
However, Thaddeus’s paper was and remains noteworthy, just because of the original, ingenious and far-reaching method that the author has developed in order to obtain his main result. Namely, his basic idea of tackling the space \(H^0(N, {\mathcal O} (k\cdot \Theta))\) of sections of \(k\)-th order theta functions, on the moduli space \(N\) of semi-stable rank-2 bundles and fixed determinant bundle \(L\) over a smooth projective curve \(X\), is to relate this space to the space of sections of another line bundle, not defined over \(N\), but over some derived, more manageable moduli space. To this end, the author studies so-called stable pairs of rank-2 bundles with fixed determinant and fixed non-trivial section of such a vector bundle.
There are several possible concepts of stability for such pairs, out of which the author uses the one that was introduced by S. Bradlow in 1991 [cf. S. B. Bradlow, J. Differ. Geom. 33, No. 1, 169-213 (1991; Zbl 0697.32014)]. This definition of stability for pairs \((E, \varphi)\), \(\varphi \in H^0 (X,E)\), depends on the choice of a real parameter \(\sigma\), and a good part of the author’s investigations is devoted to both establishing the existence of moduli spaces for such stable pairs, which are denoted by \(M (\sigma, L)\), and studying their dependence on the stability parameter \(\sigma\). This is inspired and modeled after D. Gieseker’s construction of the moduli spaces of stable vector bundles via geometric invariant theory. A very subtle analysis of the \(\sigma\)-stability condition for pairs \((E, \varphi)\) then leads to a tree of moduli spaces, partially related to each other by birational morphisms (“flips” in the sense of Mori), which finally relates the original moduli space \(N\) of semistable rank-2 vector bundles with determinant \(L\) to the “initial” moduli space \(M_0\) of the whole tree of moduli spaces. It turns out that the Verlinde space \(H^0 (N, (k\cdot \Theta))\) can be identified with the space of sections of an appropriate line bundle over any of the moduli spaces occurring in the tree, and that, at least for one of these moduli spaces, the higher cohomology of its “appropriate” line bundle vanishes. Thus the Verlinde number \(h^0(N, (k\cdot \Theta))\) can be computed as the Euler characteristic of some other moduli space, which, in turn, appears as the Euler characteristic of a line bundle over a symmetric product of the base curve \(X\).
Apart from establishing the Verlinde formula, in this way, the author obtains a whole family of dimension formulas for spaces of sections of line bundles over various moduli spaces, which might be important and useful for further investigations in related contexts. In addition, the rigorous establishing of moduli spaces for \(\sigma\)-stable pairs of vector bundles over curves represents a highly valuable contribution towards the moduli theory of varieties and vector bundles in its full generality.

14D20 Algebraic moduli problems, moduli of vector bundles
14H60 Vector bundles on curves and their moduli
14C20 Divisors, linear systems, invertible sheaves
14F25 Classical real and complex (co)homology in algebraic geometry
32G08 Deformations of fiber bundles
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