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The stringy E-function of the moduli space of Higgs bundles with trivial determinant. (English) Zbl 1173.14024

The stringy E-function \(E_{st}(W;u,v)\) of a normal irreducible variety \(W\) with at worst log-terminal singularities was introduced by V. Batyrev [in: Integrable systems and algebraic geometry. Proceedings of the 41st Taniguchi symposium, Kobe, Japan, June 30-July 4, 1997, and in Kyoto, Japan, July 7-11 1997. Singapore: World Scientific. 1–32 (1998; Zbl 0963.14015)] and is defined in terms of the discrepancy divisor for a suitable resolution of singularities. In particular, if \(W\) has a crepant resolution \(V\), then \(E_{st}(W;u,v)\) is equal to the Hodge-Deligne polynomial \(E(V;u,v)\) of \(V\). In general, however, the stringy E-function is only a rational function.
In this paper the authors compute \(E_{st}(M)\) for the moduli space of semistable Higgs bundles of rank \(2\) with trivial determinant over a smooth projective curve \(X\) of genus \(g\geq2\). For \(g=2\), \(E_{st}(M)\) is a polynomial, but, for \(g\geq3\), this is not the case. It follows that, when \(g\geq3\), the moduli space \(M\) does not admit a crepant resolution.
The first step in the argument is to compute the Hodge-Deligne polynomial for the stable locus \(M^s\) of \(M\). The computation involves constructing a stratification of \(M^s\), based on whether the underlying bundle of a stable Higgs bundle is stable, strictly semistable or non-semistable; in the last two cases, there is a further subdivision required before one can obtain a stratification for which the Hodge-Deligne polynomials can be computed. Finally one uses the additive property of Hodge-Deligne polynomials to compute \(E(M^s)\).
The next step is to show that \(M\) can be desingularized by three blow-ups by F. C. Kirwan’s algorithm [Ann. Math. (2) 122, 41–85 (1985; Zbl 0592.14011)]. The argument is based on C. T. Simpson’s description of \(M\) [Publ. Math., Inst. Hautes Étud. Sci. 79, 47–129 (1994; Zbl 0891.14005)], leading to the fact that the singularities and stratification of \(M\) are identical with those obtained by K. G. O’Grady [J. Reine Angew. Math. 512, 49–117 (1999; Zbl 0928.14029)] for a different moduli space. O’Grady’s method therefore yields the required construction for the Kirwan desingularization \(\widehat{M}\) of \(M\) together with a description of the exceptional divisors and a formula for the discrepancy divisor. From this, it is straightforward to obtain the formula for \(E_{st}(M)\). Although the formula is complicated, one can take the limit as \(u,v\to 1\) to show that, for \(g\geq3\), the stringy Euler number \(e_{st}(M)\) is equal to \(2^{2g}{3g-3\over2g-3}\). For \(g\geq4\), this is not an integer, from which it follows, as indicated earlier, that \(M\) does not admit a crepant resolution. For \(g=3\), one can check directly that \(E_{st}(M)\) is not a polynomial.

MSC:

14H60 Vector bundles on curves and their moduli
14F25 Classical real and complex (co)homology in algebraic geometry
14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
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References:

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