Chow rings of fine quiver moduli are tautologically presented

Abstract

A result of A. King and C. Walter asserts that the Chow ring of a fine quiver moduli space is generated by the Chern classes of universal bundles if the quiver is acyclic. We will show that defining relations between these Chern classes arise geometrically as degeneracy loci associated to the universal representation. R and let σ − J :Lemma 3.5. (i) For every J ∈ Φ θ , the cone σ − J is a simplex of dimension ♯J.Proof. (i) We have to show that the elements s * λ α with α ∈ J are linearly independent over the reals. Let us assume there were an element 0as λ + is supported in J. By stability of J c , we obtain that θ, λ > 0. But, on the other hand, we also get χ α , r * (−λ) = 0 and thus, θ, −λ > 0. A contradiction.Proposition 3.7. We obtainand let X := X Σ + be the toric variety associated to Σ + . Then X is an open subset of R which is defined as the union X = J∈Φ θ X σ + J . By definition,and thus, it is clear that R θ = X.toric varieties. This morphism is T + -equivariant via s : T + → T − and therefore, T -invariant with respect to the induced T -actions via r : T → T + . We know that a geometric T -quotient exists. Thus, it suffices to show that Y is a categorical T -quotient. Let f : R θ → Z be a T -invariant morphism of varieties. Let X J := X σ + J . This is an affine open subset of R. Lemma 3.6 shows thatand therefore, η J : X J → Y J := X σ − J is a universal categorical quotient (cf. [16, Theorem 1.1]). Note that Y J is an open subset of Y and η J coincides with the restriction of η. By the quotient property, we obtain that there exists a unique morphism g J :because η Jν is a universal categorical quotient. As g J is uniquely determined by the property g J η J = f J , we obtain g Jν |Y J = g J . This proves that the maps g J with J ∈ Φ θ glue together to a map g : Y → Z with gη = f . Conversely, every such map g has to fulfill g|Y J = g J .We have obtained an explicit description of the fan of M θ . This enables us to prove Proposition 3.2 with the help of Danilov's theorem (cf. [3, Theorem 10.8]).

Fano quiver moduli

“…In the (numerically characterized) case of fine quiver moduli spaces for acyclic quivers, which are smooth projective varieties (and rational by [27]), several favorable geometric properties of cohomological nature are well-established: Their singular cohomology is algebraic [17], they are polynomial-count [23], the Betti numbers (and the counting polynomial) can be determined explicitly [23], and their (torus-equivariant) Chow rings are tautologically generated and presented [7,8].…”

Fano quiver moduli

“…In the (numerically characterized) case of fine quiver moduli spaces for acyclic quivers, which are smooth projective varieties (and rational by [16]), several favourable geometric properties of cohomological nature are well-established: Their singular cohomology is algebraic [9], they are polynomial-count [12], the Betti numbers (and the counting polynomial) can be determined explicitely [12], and their (torus-equivariant) Chow rings are tautologically generated and presented [3,4].…”

Cohomology rings of moduli of point configurations on the projective line

“…In the disguise of polygon spaces, the rational cohomology ring is described by generators and relations in [4]. In the case of the symmetric stability, a description of the rational Chow ring is given in [1] using an interpretation as a moduli space of quiver representations.…”

“…Note that in[1] θ-stability of a representation M was defined by θ(M ′ ) > 0 for every subrepresentation M ′ . We use the opposite sign convention here.…”