Motives and the Hodge conjecture for moduli spaces of pairs

Abstract

Abstract. Let C be a smooth projective curve of genus g ≥ 2 over C. Fix n ≥ 1, d ∈ Z. A pair (E, φ) over C consists of an algebraic vector bundle E of rank n and degree d over C and a section φ ∈ H 0 (E). There is a concept of stability for pairs which depends on a real parameter τ . Let Mτ (n, d) be the moduli space of τ -polystable pairs of rank n and degree d over C. We prove that for a generic curve C, the moduli space Mτ (n, d) satisfies the Hodge Conjecture for n ≤ 4. For obtaining this, we prove first that Mτ (n, d) is motivated by C.

Moduli spaces of stable pairs and non-abelian zeta functions of curves via wall-crossing

“…This was conjectured in [14]. Using generalizations of Zagier's formula for the motive of the moduli stack of semistable bundles (to be discussed in an appendix which also contains a new proof of Zagier's original formula), this yields the following explicit formula for the motive [M τ (r, (1 − q rp+1+1 t)(1 − q −rp t) − q…”

“…Hodge polynomials were computed by Thaddeus [18] in the rank two case and by Muñoz [15] in the rank three case. For rank four it was proved [14], and conjectured for general rank, that the motive of the moduli space can be expressed in terms of the motive of the curve. We will compute the motives of these moduli spaces for arbitrary rank in terms of an explicit Zagier-type formula, and in particular confirm the above conjecture.…”

Variations of Hodge Structures of Rank Three k-Higgs Bundles and Moduli Spaces of Holomorphic Triples

“…The reader may consult the works of Bradlow and García-Prada [6]; and Bradlow, García-Prada and Gothen [7] for the formal construction of the moduli space of triples. Also can see Muñoz, Oliveira and Sánchez [39]; and Muñoz, Ortega and Vázquez-Gallo [40] for other details.…”

“…The stability for triples depends on a parameter σ ∈ R, which gives a collection of moduli spaces N σ (r 1 , r 2 , d 1 , d 2 ) widely worked by several authors: e.g. [6,7,8,16,39,40]. The range of σ is a closed interval I = [σ m , σ M ] ⊆ R split by a finite number of critical values σ c .…”

Hodge conjecture for the moduli space of semi-stable sheaves over a nodal curve

“…Using this, Balaji-King-Newstead in [5] proved the conjecture for the moduli space M C (2, L) of rank 2 semi-stable, locally free sheaves with determinant L over C, for an odd degree invertible sheaf L on C. For higher rank, this was proved by Biswas-Narasimhan [11] in the late 1990s. It has also been shown for the moduli space of stable pairs over a smooth, projective curve in [23]. However, nothing is known in the case the underlying curve is irreducible, nodal.…”