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Aug
28
comment Are linear algebraic groups rigid?
The group law on a unipotent algebraic group deforms non-trivially in general. For instance by scaling the symplectic form you get a one parameter deformation of the three dimennsional Heisenberg group (upper triangular $3\times 3$ matrices with ones on the diagonal) to the additive group of a $3$ dimensional vector space.
Aug
17
comment FIltrations on a vector bundle on a curve
I must be missing something, we have rank two (semi-)stable bundles of arbitrary negative degree. So you can not bound the degree of the line sub-bundles from below universally. Do you want a bound that depends on $g$, $n$, and the degree?
Jun
4
comment Is $\pi_2$ algebraic?
In general it is neither. Serre's goodness says we have an isomorphism of cohomologies of a discrete group with coefficients in all modules and the cohomologies of the pro-finite completion. Our condition is the same but we have the pro-algebraic completion and coefficients which are locally compact vector spaces. I am pretty sure that if a group $\Gamma$ is an arithmetic lattice in a semisimple group of rank $\geq 1$, then the super-rigidity theorem will imply that $\Gamma$ is Serre's good if and only it is algebraically good. In particular $Sp_{4}(\mathbb{Z})$ is not algebraically good.
Jun
3
comment Is $\pi_2$ algebraic?
Fundamental groups of algebraic surfaces are completely general. By Lefschetz hyperplane section theorem every fundamental group of a smooth projective variety is the fundamental group of a smooth algebraic surface. So asking if the fundamental groups of algebraic surfaces have a certain property is the same as asking if all fundamental groups of projective varieties have that property. But you are right: fundamental groups of algebraic surfaces are not good in general and it is hard to decide when they are.
Jun
3
awarded  Nice Answer
Jun
3
comment Is $\pi_2$ algebraic?
Now given a finitely generated group $\pi$, we say that $\pi$ is algebraically good if the schematization of the Eilenberg-Mclane space $K(\pi,1)$ is $K(\pi^{alg},1)$, i.e. if the schematization has no higher homotopy groups. Algebraically good groups are not easy to come by. In the paper we prove that free groups, fundamental groups of compact surfaces, and fundamental groups of Artin neighborhoods are algebraically good. We also give a lenghty discussion and characterization of goodness.
Jun
3
comment Is $\pi_2$ algebraic?
It is a tricky notion. Given a homotopy type $X$ one can construct its complex schematization $(X\otimes \mathbb{C})^{sch}$. By definition this is a higher stack on the etale site of schemes, which has the property that its fundamental group is the pro-algebraic completion of $\pi_{1}(X)$ and every finite dimensional representation $V$ of $\pi_{1}(X)$ gives rise to a coherent sheaf on the schematization, so that the cohomology of the local system $V$ on $X$ is naturally isomorphic to the cohomology of the coherent sheaf on the schematization. The schematization exists by a theorem of Toen.
Jun
3
answered Is $\pi_2$ algebraic?
Apr
6
awarded  Enlightened
Apr
6
awarded  Nice Answer
Apr
3
answered On a question motivating Lurie's treatment of formal moduli problems
Dec
4
revised Branch loci of Ramified covers
deleted 6 characters in body
Dec
3
awarded  Nice Answer
Dec
3
revised Branch loci of Ramified covers
added 8 characters in body
Nov
27
revised Branch loci of Ramified covers
edited body
Nov
27
answered Branch loci of Ramified covers
Oct
16
awarded  Yearling
Oct
10
awarded  Constituent
Oct
1
awarded  Caucus
Sep
27
comment Representation variety vs. space of flat connections
The holonomy does not provide such a bijection. It only provides a bijection between the space of flat connections on the trivial bundle and a connected component of of the representation variety. Often the representation variety will have more components. If $M$ is a compact manifold and $G$ is a complex reductive group, then this bijection is a homeomorphism. In fact if $M$ is a complex projective algebraic variety, the bijection is a complex analytic isomorphism.