Discriminant and Hodge classes on the space of Hitchin covers
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Discriminant and Hodge classes on the space of Hitchin covers Mikhail Basok1 Received: 2 April 2019 / Revised: 23 March 2020 / Accepted: 18 June 2020 © Springer Nature B.V. 2020
Abstract We continue the study of the rational Picard group of the moduli space of Hitchin spectral covers started in Korotkin and Zograf (J Math Phys 59(9):091412, 2018). In the first part of the paper we expand the “boundary”, “Maxwell stratum” and “caustic” divisors introduced in Korotkin and Zograf (2018) via the set of standard generators of the rational Picard group. This generalizes the result of Korotkin and Zograf (2018), where the expansion of the full discriminant divisor (which is a linear combination of the classes mentioned above) was obtained. In the second part of the paper we derive a formula that relates two Hodge classes in the rational Picard group of the moduli space of Hitchin spectral covers. Keywords Hitchin covers · Moduli spaces · Picard group · Discriminant locus Mathematics Subject Classification 14C20 · 14C22
1 Introduction Hitchin integrable systems arise as a result of dimensional reduction of the self-dual Yang–Mils equation, see [2,5,6]. The Hamiltonians of Hitchin system are encoded in the so-called spectral cover (see [3,4]), which is an n-sheeted cover of a (smooth or, more generally, stable) complex projective curve , defined as a subvariety of T ∗ : = {(x, v) ∈ T ∗ | P(v, x) = 0},
B 1
(1.1)
Mikhail Basok [email protected] Laboratory of Modern Algebra and Applications, Saint-Petersburg State University, 14th Line, 29b, Saint Petersburg, Russia 199178
123
M. Basok
where P(v, x) = v n + q1 (x)v n−1 + · · · + qn (x),
(1.2)
⊗j
q j is a j-differential on (i.e. a holomorphic section of K ). In the framework of [3], the equation defining is given by the characteristic polynomial P(v, x) = det((x) − v I ) of the so-called Higgs field on . (n) We consider the moduli space PMg of Hitchin spectral covers in the case of GL(n, C) Hitchin systems, when all differentials q j as assumed to be arbitrary. A (n)
point in PMg parametrizes a pair (, [P]), where is a genus g curve and P is a polynomial of the form (1.2) considered up to multiplication by a non-zero constant (n) ξ given by (ξ · P)(v, x) = ξ n P(ξ −1 v, x). As a space PMg is a bundle over the Deligne–Mumford compactification Mg of the moduli space of genus g curves. Fibers of this bundle are isomorphic to a weighted projective space, see Sect. 3 for details. (n) Notice that if n = 1, then PMg is just the total space of the projectivized Hodge bundle over Mg , which can be thought of as a closure of the moduli space of Abelian differentials (considered up to a multiplicative constant) on genus g smooth projective curves. Kokotov and Korotkin [7] introduced a tau function on this moduli space called the Bergman tau function. The Bergman tau function is a generalization of the Dedekind eta function (they coincide if g = 1) and can be interpreted as determinants of a family of Cauchy–Riemann operators in the spirit of [14]. Globa
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