Rational Cohomology Research Articles

Motives of moduli spaces of bundles on curves via variation of stability and flips

AbstractWe study the rational Chow motives of certain moduli spaces of vector bundles on a smooth projective curve with additional structure (such as a parabolic structure or Higgs field). In the parabolic case, these moduli spaces depend on a choice of stability condition given by weights; our approach is to use explicit descriptions of variation of this stability condition in terms of simple birational transformations (standard flips/flops and Mukai flops) for which we understand the variation of the Chow motives. For moduli spaces of parabolic vector bundles, we describe the change in motive under wall‐crossings, and for moduli spaces of parabolic Higgs bundles, we show the motive does not change under wall‐crossings. Furthermore, we prove a motivic analogue of a classical theorem of Harder and Narasimhan relating the rational cohomology of moduli spaces of vector bundles with and without fixed determinant. For rank 2 vector bundles of odd degree, we obtain formulae for the rational Chow motives of moduli spaces of semistable vector bundles, moduli spaces of Higgs bundles and moduli spaces of parabolic (Higgs) bundles that are semistable with respect to a generic weight (all with and without fixed determinant).

Holomorphic polylogarithms and Bloch complexes

Abstract For an integer n ≥ 2 {n\geq 2} we define a polylogarithm ℒ ^ n {\widehat{\mathcal{L}}_{n}} , which is a holomorphic function on the universal abelian cover of ℂ ∖ { 0 , 1 } {\mathbb{C}\setminus\{0,1\}} defined modulo ( 2 ⁢ π ⁢ i ) n / ( n - 1 ) ! {(2\pi i)^{n}/(n-1)!} . We use the formal properties of its functional relations to define groups ℬ ^ k ⁢ ( F ^ ) {\widehat{\mathcal{B}}_{k}(\widehat{F})} lifting Goncharov’s Bloch groups ℬ k ⁢ ( F ) {\mathcal{B}_{k}(F)} of a field F, and show that they fit into a complex Γ ^ ⁢ ( F , n ) {\widehat{\Gamma}(F,n)} lifting Goncharov’s Bloch complex Γ ⁢ ( F , n ) {\Gamma(F,n)} . When F = ℂ {F=\mathbb{C}} , we show that the imaginary part (when n is even) or real part (when n is odd) of ℒ ^ n {\widehat{\mathcal{L}}_{n}} agrees with a real single valued polylogarithm ℒ n {\mathcal{L}_{n}} on the group H 1 ⁢ ( Γ ^ ⁢ ( ℂ , n ) ) {H^{1}(\widehat{\Gamma}(\mathbb{C},n))} . When n = 2 {n=2} , this group is Neumann’s extended Bloch group. Goncharov’s complex conjecturally computes the rational motivic cohomology of F, and one may speculate whether the extended complex computes the integral motivic cohomology. Finally, we use ℒ ^ 3 {\widehat{\mathcal{L}}_{3}} to construct a lift of Goncharov’s map H 5 ⁢ ( SL ⁡ ( 3 , ℂ ) ) → ℝ {H_{5}(\operatorname{SL}(3,\mathbb{C}))\to\mathbb{R}} to a complex valued map whose real part agrees with that of Goncharov. The lift makes use of the cluster ensemble structure on the Grassmannian Gr ⁡ ( 3 , 6 ) {\operatorname{Gr}(3,6)} .

Stable Cohomology of the Moduli Space of Trigonal Curves

Abstract We prove that the rational cohomology $H^{i}(\mathcal {T}_{g};\mathbf {Q})$ of the moduli space of trigonal curves of genus $g$ is independent of $g$ in degree $i<\lfloor g/4\rfloor .$ This makes possible to define the stable cohomology ring as $H^{\bullet }(\mathcal {T}_{g};\mathbf {Q})$ for a sufficiently large $g.$ We also compute the stable cohomology ring, which turns out to be isomorphic to the tautological ring. This is done by studying the embedding of trigonal curves in Hirzebruch surfaces and using Gorinov–Vassiliev’s method.

The rational homotopy type of homotopy fibrations over connected sums

AbstractWe provide a simple condition on rational cohomology for the total space of a pullback fibration over a connected sum to have the rational homotopy type of a connected sum after looping. This takes inspiration from a recent work of Jeffrey and Selick, in which they study pullback fibrations of this type but under stronger hypotheses compared to our result.

On generators and relations of the rational cohomology of Hilbert schemes

We consider for $$d\ge 1$$ the graded commutative $${\mathbb {Q}}$$ -algebra $${\mathcal {A}}(d):=H^*({\text {Hilb}}^d({\mathbb {C}}^2);{\mathbb {Q}})$$ , which is also connected to the study of generalised Hurwitz spaces by work of the first author. These Hurwitz spaces are in turn related to the moduli spaces of Riemann surfaces with boundary. We determine two distinct, minimal sets of $${\lfloor }d/2{\rfloor }$$ multiplicative generators of $${\mathcal {A}}(d)$$ . Additionally, we prove when the lowest degree generating relations occur. For small values of d, we also determine a minimal set of generating relations, which leads to several conjectures about the necessary generating relations for $${\mathcal {A}}(d)$$ .

The family signature theorem

We discuss several versions of the Family Signature Theorem: in rational cohomology using ideas of Meyer, in$KO[\tfrac {1}{2}]$-theory using ideas of Sullivan, and finally in symmetric$L$-theory using ideas of Ranicki. Employing recent developments in Grothendieck–Witt theory, we give a quite complete analysis of the resulting invariants. As an application we prove that the signature is multiplicative modulo 4 for fibrations of oriented Poincaré complexes, generalizing a result of Hambleton, Korzeniewski and Ranicki, and discuss the multiplicativity of the de Rham invariant.

Cohomology of configuration spaces of surfaces as mapping class group representations

We express the rational cohomology of the unordered configuration space of a compact oriented manifold as a representation of its mapping class group in terms of a weight-decomposition of the rational cohomology of the mapping space from the manifold to a sphere. We apply this to the case of a compact oriented surface with one boundary component and explicitly compute the rational cohomology of its unordered configuration space as a representation of its mapping class group. In particular, this representation is not symplectic, but has trivial action of the second Johnson filtration subgroup of the mapping class group.

Intersection theory and Chern classes in Bott–Chern cohomology

In this article, we investigate an axiomatic approach introduced by Grivaux for the study of rational Bott-Chern cohomology, and use it in that context to define Chern classes of coherent sheaves. This method also allows us to derive a Riemann-Roch-Grothendieck formula for a projective morphism between smooth complex compact manifolds. In the general case of complex spaces, the Poincar\'e and Dolbeault-Grothendieck lemmas are not always valid. For this reason, and to simplify the exposition, we only consider non singular complex spaces. The appendix presents a calculation of integral Bott-Chern cohomology in top degree for a connected compact manifold.

Extra structure on the cohomology of configuration spaces of closed orientable surfaces

The rational homology of unordered configuration spaces of points on any surface was studied by Drummond-Cole and Knudsen. We compute the rational cohomology of configuration spaces on a closed orientable surface, keeping track of the mixed Hodge numbers and the action of the symplectic group on the cohomology. We find a series with coefficients in the Grothendieck ring of sp(2g) that describes explicitly the decomposition of the cohomology into irreducible representations. From that we deduce the mixed Hodge numbers and the Betti numbers, obtaining a new formula without cancellations.

The eleventh cohomology group of

AbstractWe prove that the rational cohomology group$H^{11}(\overline {\mathcal {M}}_{g,n})$vanishes unless$g = 1$and$n \geq 11$. We show furthermore that$H^k(\overline {\mathcal {M}}_{g,n})$is pure Hodge–Tate for all even$k \leq 12$and deduce that$\# \overline {\mathcal {M}}_{g,n}(\mathbb {F}_q)$is surprisingly well approximated by a polynomial inq. In addition, we use$H^{11}(\overline {\mathcal {M}}_{1,11})$and its image under Gysin push-forward for tautological maps to produce many new examples of moduli spaces of stable curves with nonvanishing odd cohomology and nontautological algebraic cycle classes in Chow cohomology.

Cohomology of moduli spaces of Del Pezzo surfaces

AbstractWe compute the rational Betti cohomology groups of the coarse moduli spaces of geometrically marked Del Pezzo surfaces of degree 3 and 4 as representations of the Weyl groups of the corresponding root systems. The proof uses a blend of methods from point counting over finite fields and techniques from arrangement complements.

The Looijenga–Lunts–Verbitsky Algebra and Verbitsky’s Theorem

In these notes we review some basic facts about the LLV Lie algebra. It is a rational Lie algebra, introduced by Looijenga–Lunts and Verbitsky, acting on the rational cohomology of a compact Kähler manifold. We study its structure and describe one irreducible component of the rational cohomology in the case of a compact hyperkähler manifold.

A division theorem for nodal projective hypersurfaces

Let \(V_{n,d}\) be the variety of equations for hypersurfaces of degree d in \({\mathbb {P}}^n({\mathbb {C}})\) with singularities not worse than simple nodes. We prove that the orbit map \(G'=SL_{n+1}({\mathbb {C}}) \rightarrow V_{n,d}\), \(g\mapsto g\cdot s_0\), \(s_0\in V_{n,d}\) is surjective on the rational cohomology if \(n>1\), \(d\ge 3\), and \((n,d)\ne (2,3)\). As a result, the Leray–Serre spectral sequence of the map from \(V_{n,d}\) to the homotopy quotient \((V_{n,d})_{hG'}\) degenerates at \(E_2\), and so does the Leray spectral sequence of the quotient map \(V_{n,d}\rightarrow V_{n,d}/G'\) provided the geometric quotient \(V_{n,d}/G'\) exists. We show that the latter is the case when \(d>n+1\).

Reconstructing the orbit type stratification of a torus action from its equivariant cohomology

We investigate what information on the orbit type stratification of a torus action on a compact space is contained in its rational equivariant cohomology algebra. Regarding the (labelled) poset structure of the stratification, we show that equivariant cohomology encodes the subposet of ramified elements. For equivariantly formal actions, we also examine what cohomological information of the stratification is encoded. In the smooth setting, we show that under certain conditions—which in particular hold for a compact orientable manifold with discrete fixed point set—the equivariant cohomologies of the strata are encoded in the equivariant cohomology of the manifold.

K$K$‐Motives and Koszul duality

We construct an ungraded version of Beilinson–Ginzburg–Soergel's Koszul duality for Langlands dual flag varieties, inspired by Beilinson's construction of rational motivic cohomology in terms of K $K$ -theory. For this, we introduce and study categories DK S ( X ) $\hbox{DK}_\mathcal {S}(X)$ of S $\mathcal {S}$ -constructible K $K$ -motivic sheaves on varieties X $X$ with an affine stratification. We show that there is a natural and geometric functor, called Beilinson realisation, from S $\mathcal {S}$ -constructible mixed sheaves D S m i x ( X ) $\hbox{D}^{mix}_\mathcal {S}(X)$ to DK S ( X ) $\hbox{DK}_\mathcal {S}(X)$ . We then show that Koszul duality intertwines the Betti realisation and Beilinson realisation functors and descends to an equivalence of constructible sheaves and constructible K $K$ -motivic sheaves on Langlands dual flag varieties.

Representation stability in the level 4 braid group

We investigate the cohomology of the level 4 subgroup of the braid group, namely, the kernel of the mod 4 reduction of the Burau representation at $$t=-1$$ . This group is also equal to the kernel of the mod 2 abelianization of the pure braid group. We give an exact formula for the first Betti number; it is a quartic polynomial in the number of strands. We also show that, like the pure braid group, the first homology satisfies uniform representation stability in the sense of Church and Farb. Unlike the pure braid group, the group of symmetries—the quotient of the braid group by the level 4 subgroup—is one for which the representation theory has not been well studied; we develop its representation theory. This group is a non-split extension of the symmetric group. As applications of our main results, we show that the rational cohomology ring of the level 4 braid group is not generated in degree 1 when the number of strands is at least 15, and we compute all Betti numbers of the level 4 braid group when the number of strands is at most 4. We also derive a new lower bound on the first rational Betti number of the hyperelliptic Torelli group and on the top rational Betti number of the level 4 mapping class group in genus 2. Finally, we apply our results to locate all of the 2-torsion points on the characteristic varieties of the pure braid group.

Vector bundles of non-negative curvature over cohomogeneity one manifolds

We provide several results on the existence of metrics of non-negative sectional curvature on vector bundles over certain cohomogeneity one manifolds and homogeneous spaces up to suitable stabilization.Beside explicit constructions of the metrics, this is achieved by identifying equivariant structures upon these vector bundles via a comparison of their equivariant and non-equivariant K-theory. For this, in particular, we transcribe equivariant K-theory to equivariant rational cohomology and investigate surjectivity properties of induced maps in the Borel fibration via rational homotopy theory.

Relative Gottlieb groups of mapping spaces and their rational cohomology

Let f:X →Y be a map of simply connected CW-complexes of finite type. Put maxπ★(Y)⊗Q = max{ i | πi(Y)⊗Q≠0 }. In this paper we compute the relative Gottlieb groups of f when X is an F0-space and Y is a product of odd spheres. Also, under reasonable hypothesis, we determine these groups when X is a product of odd spheres and Y is an F0-space. As a consequence, we show that the rationalized G-sequence associated to f splits into a short exact sequence. Finally, we prove that the rational cohomology of map(X,Y;f) is infinite dimensional whenever maxπ★(Y)⊗Q is even.

Invariance of the Goresky–Hingston algebra on reduced Hochschild homology

We prove that two quasi-isomorphic simply connected differential graded associative Frobenius algebras have isomorphic Goresky–Hingston algebras on their reduced Hochschild homology. Our proof is based on relating the Goresky–Hingston algebra on reduced Hochschild homology to the singular Hochschild cohomology algebra. For any simply connected oriented closed manifold M $M$ of dimension k $k$ , the Goresky–Hingston algebra on reduced Hochschild homology induces an algebra structure of degree k − 1 $k-1$ on H ¯ ∗ ( L M ; Q ) $\overline{\mathrm{H}}^*(LM;\mathbb {Q})$ , the reduced rational cohomology of the free loop space of M $M$ . As a consequence of our algebraic result, we deduce that the isomorphism class of the induced algebra structure on H ¯ ∗ ( L M ; Q ) $\overline{\mathrm{H}}^*(LM;\mathbb {Q})$ is an invariant of the homotopy type of M $M$ .

Halving spaces and lower bounds in real enumerative geometry

We develop the theory of halving spaces to obtain lower bounds in real enumerative geometry. Halving spaces are topological spaces with an action of a Lie group $\Gamma$ with additional cohomological properties. For $\Gamma=\mathbb{Z}_2$ we recover the conjugation spaces of Hausmann, Holm and Puppe. For $\Gamma=\mathrm{U}(1)$ we obtain the circle spaces. We show that real even and quaternionic partial flag manifolds are circle spaces leading to non-trivial lower bounds for even real and quaternionic Schubert problems. To prove that a given space is a halving space, we generalize results of Borel and Haefliger on the cohomology classes of real subvarieties and their complexifications. The novelty is that we are able to obtain results in rational cohomology instead of modulo 2. The equivariant extension of the theory of circle spaces leads to generalizations of the results of Borel and Haefliger on Thom polynomials.