Dimension of the singular locus of $\mathcal M_X(r,d)$

Let $X$ be an algebraic smooth curve of genus $g$ over $\mathbb C$, and let $\mathcal M_X(r,d)$ (resp $\mathcal M_X^0(r)$) be the moduli space of vector bundle of rank $r$ and degree $d$ (resp. with trivial determinant) over $X$, we know that when $g=r=2$ and $d\equiv 0 \mod 2$ this space is smooth. So we … Read more

Higher tangent bundles of manifolds with non integer dimension

One way to define the tangent space of a manifold at a point p∈M is the following: We define an equivalent relation on the space of curves passing p as follows: Two curves α,β are equivalents iff they have tangencity of order at least one that is ∥α(t)−β(t)∥=o(|t|), in a local smooth coordinate. Then the … Read more

Proving that an EE-oriented manifold has an EE-oriented normal bundle

This is the setting we are working in: M is a closed, smooth n-manifold embedded in Rn+k with a chosen embedding e:Mn→Rn+k. It is E-oriented, for E a connected ring spectrum, i.e. there exists a fundamental class z∈En(M) which is mapped, for every m∈M to the generator of En(M,M∖{m}) via the inclusion im:M→M,M∖{m}. What I … Read more

A trivialization of an almost complex structure

Recently, I have been studying the Carleman Similiarity Principle, which is used to study the regularity and unique continuation of J-holomorphic curves. Roughly, one takes a solution $ u $ of a certain PDE (a la Cauchy-Riemann) and tries to find a tranformation $ \Phi $ and a holomorphic $ \sigma $ with $ u … Read more

Do we have classical Riemann-Hilbert correspondence for infinite dimensional flat vector bundles?

Let $E$ be an $n$-dimensional vector bundle on a manifold $M$ and $\nabla: \Gamma(E)\to \Omega^1(M,E)$ be a flat connection on $E$. Classical Riemann-Hilbert correspondence tells us that ker$\nabla$ is locally an $n$-dimensional vector space and it gives a local system on $M$. Now if we drop the condition that $E$ is finite dimensional, do we … Read more

Obstruction to the existence of lifting of the classifying map

Let E be an n-plane bundle over CW complex X. Then E is a pullback of tautological bundle γn over BO(n) i.e. E=f∗(γn). This f is called classyfing map. One can show that the universal covering of BO(n) is BSO(n) and f can be lifted to BSO(n) (meaning that there is ˜f:X→BSO(n) such that p∘˜f=f … Read more

Splitting principle for real vector bundles with $w_i=0$, $0

Is the following true/known? Let $E$ be a (real) vector bundle over a compact CW-complex $X$. Suppose that $w_i(E)=0$ for $0<i<2^r$. Then there exist a space $Y$ and a map $f\colon Y\to X$ such that (1) $f^*\colon H^*(X;\mathbb F_2)\to H^*(Y;\mathbb F_2)$ is injective; (2) for some $k\geqslant0$, the vector bundle $f^*(E)\oplus(k)$ over $Y$ is isomorphic … Read more

Spectrum of the hypoelliptic transverse signature operator

Let D be the transverse signature operator constructed by Connes and Moscovici in the paper “Local index formula in Noncommutative Geometry”:this is first order hypoelliptic pseudodifferential operator D defined by the equality D|D|=Q where Q=(dVd∗V−d∗VdV)⊕(dH+d∗H) where dV,dH are vertical and horizontal exterior derivative. It acts on the sections of the bundle Λ(V∗)⊗Λ(p∗(T∗M)) over P:=GL+(M)/SO(n) (the … Read more