Why is the Hodge Conjecture so important?

The Hodge Conjecture states that every Hodge class of a non singular projective variety over $\mathbf{C}$ is a rational linear combination of cohomology classes of algebraic cycles: Even though I’m able to understand what it says, and at first glance I do find it a very nice assertion, I cannot grasp yet why it is … Read more

Cohomology and deformations of moduli of vector bundles

I believe that the following is well-known, but I cannot find a reference in the literature… Let X be a smooth variety (in our case X=Ms(r) coarse moduli space of stable rank r vector bundles with trivial determinant over a curve C , so X is not compact !) and let L be a line … Read more

References for bilinear forms on chain complexes?

I am looking for references that include general results and theorems for bilinear forms defined on chain complexes. That is, bilinear forms ⟨⋅,⋅⟩i:Ci×Ci→K defined for all i on chain complexes ⋯→Ci+1→Ci→Ci−1→⋯ of K-modules over a principal ideal domain K. I am particularly interested in how these descend to homology groups and what invariants/classifications can be … Read more

A dimension condition on the cohomology of a homogeneous space

The rational cohomology of a homogeneous space G/K admits a homomorphism from H∗(BK) induced from the classifying map G/K→BK of the principal K-bundle G→G/K. Assume the Lie group is K connected, so that π1(BK)=0; then the space G/K is formal in the sense of rational homotopy theory if and only if H∗(G/K) is a free … Read more

Obstructions to symplectically embedding compact manifolds of dimension 44 or higher

It is known in Li’s paper (http://arxiv.org/pdf/0812.4929v1.pdf) that in compact symplectic manifolds (X2n,ω) of dimension at least 2n≥4, an immersed symplectic surface represents a 2-homology class as long as that homology class has positive symplectic area. When 2n≥6, this immersion may even be taken to be an embedding. My question regards finding symplectically embedded submanifolds … Read more

Reference request: local cohomology in disjoint union

I have a topological space X and two disjoint, closed subspaces Y and Z of X. I believe that in this situation, for any abelian sheaf F on X and any p∈N, there is a natural isomorphism HpY(X,F)⊕HpZ(X,F)→HpY∪Z(X,F) between local cohomology groups. I can obtain this by taking an injective resolution 0→F→I∙ of F, and … Read more

When is the restriction map $res:H^2(G,U(1))\to H^2(Z_p\times Z_p,U(1))$ not the zero map?

Consider $G$ to be a finite group with non-trivial Schur Multipler $H^2(G,U(1))$, where $G$ acts trivially on the circle group $U(1)$. By Example of a Schur-nontrivial group with no abelian subgroup of the form $H\times H$?, $G$ must have a subgroup of the form $Z_p\times Z_p$ for some prime $p$. My question is: under what … Read more

Pairing in Group Cohomology [closed]

Closed. This question is off-topic. It is not currently accepting answers. Want to improve this question? Update the question so it’s on-topic for MathOverflow. Closed 5 years ago. Improve this question I am following Ararat Babakhanian’s Cohomological Methods in Group theory. Let A,B,C be G modules then we have a G module structre on HomZ(B,C) … Read more

Cohomology class of a non torsion point

Let k be a finitely generated fields of positive characteristic p>0. Let E be an ordinary elliptic curve over k with a non torsion, non zero, rational point x. By the kummer sequence we get a map: ϕ:E(k)→lim←nE(k)[pn]E(k)→lim←nH1flat(k,E[pn])→H1(Γk,Tp(E)) where Tp(E) is lim←nE(ksep)[pn]. Is it possible to say that ϕ(x) is non zero and non torsion? … Read more