Cohomology Of Quotients Research Articles

Cohomology of quotients in real symplectic geometry

Given a Hamiltonian system $ (M,\omega, G,\mu) $ where $(M,\omega)$ is a symplectic manifold, $G$ is a compact connected Lie group acting on $(M,\omega)$ with moment map $ \mu:M \rightarrow\mathfrak{g}^{*}$, then one may construct the symplectic quotient $(M//G, \omega_{red})$ where $M//G := \mu^{-1}(0)/G$. Kirwan used the norm-square of the moment map, $|\mu|^2$, as a G-equivariant Morse function on $M$ to derive formulas for the rational Betti numbers of $M//G$. A real Hamiltonian system $(M,\omega, G,\mu, \sigma, \phi) $ is a Hamiltonian system along with a pair of involutions $(\sigma:M \rightarrow M, \phi:G \rightarrow G) $ satisfying certain compatibility conditions. These imply that the fixed point set $M^{\sigma}$ is a Lagrangian submanifold of $(M,\omega)$ and that $M^{\sigma}//G^{\phi} := (\mu^{-1}(0) \cap M^{\sigma})/G^{\phi}$ is a Lagrangian submanifold of $(M//G, \omega_{red})$. In this paper we prove analogues of Kirwan's Theorems that can be used to calculate the $\mathbb{Z}_2$-Betti numbers of $M^{\sigma}//G^{\phi} $. In particular, we prove (under appropriate hypotheses) that $|\mu|^2$ restricts to a $G^{\phi}$-equivariantly perfect Morse-Kirwan function on $M^{\sigma}$ over $\mathbb{Z}_2$ coefficients, describe its critical set using explicit real Hamiltonian subsystems, prove equivariant formality for $G^{\phi}$ acting on $M^{\sigma}$, and combine these results to produce formulas for the $\mathbb{Z}_2$-Betti numbers of $M^{\sigma}//G^{\phi}$.

Integral cohomology of quotients via toric geometry

We describe the integral cohomology of $X/G$ where $X$ is a compact complex manifold and $G$ a cyclic group of prime order with only isolated fixed points. As a preliminary step, we investigate the integral cohomology of toric blow-ups of quotients of $\mathbb{C}^n$. We also provide necessary and sufficient conditions for the spectral sequence of equivariant cohomology of $(X,G)$ to degenerate at the second page. As an application, we compute the Beauville--Bogomolov form of $X/G$ when $X$ is a Hilbert scheme of points on a K3 surface and $G$ a symplectic automorphism group of orders 5 or 7.

On the integral cohomology of quotients of manifolds by cyclic groups

We propose new tools based on basic lattice theory to calculate the integral cohomology of the quotient of a manifold by an automorphism group of prime order. As examples of applications, we provide the Beauville–Bogomolov forms of some irreducible symplectic orbifolds; we also show a new expression for a basis of the integral cohomology of a Hilbert scheme of two points on a surface.

On the cohomology of quotients of moment-angle manifolds

On the cohomology of quotients of moment-angle manifolds

Critical points of the linear entropy for pure L-qubit states

We present a substantially improved version of the method proposed by Sawicki et al (2012 Phys. Rev. A 86 040304(R), 2014 Rev. Math. Phys. 26 1450004) for finding critical points of the linear entropy for an L-qubit system. The new approach is based on the correspondence between the momentum maps for abelian and non-abelian groups, as described in Kirwan (1984 Cohomology of Quotients in Symplectic and Algebraic Geometry (Mathematical Notes vol 31) (Princeton: Princeton Univ. Press)). The proposed method can be implemented numerically and is much easier than the previous method.

On the cohomology of quotients of moment-angle manifolds

On the cohomology of quotients of moment-angle manifolds

Finite generation of the cohomology of quotients of PBW algebras

In this article we prove finite generation of the cohomology of quotients of a PBW algebra A by relating it to the cohomology of quotients of a quantum symmetric algebra S which is isomorphic to the associated graded algebra of A. The proof uses a spectral sequence argument and a finite generation lemma adapted from Friedlander and Suslin.

Intersection cohomology of quotients of nonsingular varieties

Let M ⊂ P be a nonsingular projective variety acted on by a connected complex reductive group G = KC via a homomorphism G → GL(n + 1) which is the complexification of a homomorphism K → U(n+1). Geometric invariant theory (GIT) gives us a recipe to form a quotient φ : M → M//G of the set of semistable points and many interesting spaces in algebraic geometry are constructed in this manner [MFK94]. But often this quotient is singular and hence intersection cohomology with middle perversity is an important topological invariant. The purpose of this paper is to present a way to compute the middle perversity intersection cohomology of the singular quotients. The choice of an embedding M ⊂ P provides us with a moment map for the K-action and the Morse stratification of M with respect to its norm square is Kequivariantly perfect, with the unique open dense stratum M [Kir84]. Hence one can compute the Betti numbers for the equivariant cohomology H∗ K(M ) as well as the cup product, at least in principle. See also [Kir92]. When G acts locally freely on M, we get an orbifold M//G = M/G and H∗ K(M ) ∼= H∗(M//G) ∼= IH∗(M//G).

The Harder-Narasimhan system in quantum groups and cohomology of quiver moduli

Methods of Harder and Narasimhan from the theory of moduli of vector bundles are applied to moduli of quiver representations. Using the Hall algebra approach to quantum groups, an analog of the Harder-Narasimhan recursion is constructed inside the quantized enveloping algebra of a Kac-Moody algebra. This leads to a canonical orthogonal system, the HN system, in this algebra. Using a resolution of the recursion, an explicit formula for the HN system is given. As an application, explicit formulas for Betti numbers of the cohomology of quiver moduli are derived, generalizing several results on the cohomology of quotients in 'linear algebra type' situations.

The decomposition theorem and the intersection cohomology of quotients in algebraic geometry

Suppose a connected reductive complex algebraic group G acts linearly on an irreducible complex projective variety X. We prove that if 1→N→G→H→1 is a short exact sequence of connected reductive groups and X ss the open set of semistable points for the action of N on X then IH H ∗(X ssN) is (non-canonically) a direct summand of IH G ∗(X ss) . The inclusion is provided by the decomposition theorem and certain resolutions of the action allow us to define projections.

The cohomology of quotients of classical groups

THE HOMOLOGY and cohomology rings of the classical compact Lie groups so(n), SU(n), Sp(n) are well known (for example see Bore1 [3]). Most of these groups have non-trivial centers, and in [4], Bore1 investigated the quotients of these groups by central subgroups, in particular, calculating cohomology rings with Zp coefficients, p prime. In this paper we pursue the investigation of these quotients further, extending Borel’s results. We obtain extra information on the integral cohomology and we completely determine the diagonal maps in cohomology with 2, coefficients p prime, and the action of the Steenrod algebra in any quotient of one of these groups by a central subgroup. This information leads to some applications such as the fact that homotopy equivalent compact connected simple groups are isomorphic, and a technique to prove facts about vector fields on real projective spaces, (see $9). These results on vector fields may be applied to prove non-immersion theorems for real projective spaces. In particular, it is shown that if n = 2’ + 3, r 2 3, then P*-l and P” do not immerse in R2”-7. Mahowald [9] and Sanderson [12] have shown that P” does immerse in RZn-‘, so that this is the best possible immersion for P” and P”-‘. The diagonal maps in the cohomology of these quotient groups with Z, coefficients are usually non-commutative, where p divides the order of the central subgroup, the only exceptions being low dimensional and PSp(n) for odd n (mod 2), (where PG denotes the quotient of G by its center). In particular ifp is an odd prime and p divides the order of the central subgroup, then the diagonal map in cohomology modp is non-commutative. Araki [2] has shown that the exceptional groups which have 3-torsion in homology have non-commutative diagonal maps in cohomology mod 3. One may conjecture: If a Lie group G hasp-torsion in its homology (p an odd prime) then the diagonal map in H*(G; Z,) is not commutative.

Towards Non-Reductive Geometric Invariant Theory

We study linear actions of algebraic groups on smooth projective varieties X. A guiding goal for us is to understand the cohomology of quotients under such actions, by generalizing (from reductive to non-reductive group actions) existing methods involving Mumford's geometric invariant theory (GIT). We concentrate on actions of unipotent groups H, and define sets of stable points X^s and semistable points X^{ss}, often explicitly computable via the methods of reductive GIT, which reduce to the standard definitions due to Mumford in the case of reductive actions. We compare these with definitions in the literature. Results include (1) a geometric criterion determining whether or not a ring of invariants is finitely generated, (2) the existence of a geometric of X^s, and (3) the existence of a canonical enveloping quotient variety of X^{ss}, denoted X//H, which (4) has a projective completion given by a reductive GIT and (5) is itself projective and isomorphic to Proj(k[X]^H) when k[X]^H is finitely generated.