Does a compact Lie group action on a family of compact manifolds have...
Let $\pi: M\to B$ be a fiber bundle of smooth manifolds with $B$ connected and each fiber of $\pi$ is a compact manifold. Let $G$ be a compact Lie group acting smoothly on $M$ such that$\pi(g\cdot...
View ArticleHow to see $\delta_2(\hat{\chi}(V))=\chi(V)$ in differential cohomology?
I'm reading the paper "Differential Characters and Geometric Invariants" by Cheeger and Simons. In Page 62 the authors defined the differential Euler character $\hat{\chi}(V)\in \hat{H}^{2n-1}(M,...
View ArticleDo we have an equivariant version of integrability theorem of flat connections?
I am reading Donaldson and Kronheimer's book The Geometry of Four-Manifolds. In page 48, I found Theorem 2.2.1:Let $H$ be the hypercube $H=\{\mathbf{x}\in \mathbb{R}^d|~|x_i|<1\}$. If $E$ is a...
View ArticleCan we define $\partial\bar{\partial}(\log|z_1|^2)\wedge...
In complex analysis, by Poincare-Lelong theorem, we have$$\frac{\sqrt{-1}}{\pi}\partial\bar{\partial}(\log|z|^2)=T_{z=0}$$as currents, where$$T_{z=0}(\eta)=\int_{z=0}\eta.$$Now suppose we have two...
View ArticleDo we have $d(P_{+}(\omega\wedge \theta))=d_{+}\omega\wedge...
Let $(X, g)$ be a smooth, oriented, Riemannian 4-diemnsional manifold. Let $\Lambda^2$ denote the bundle of 2-forms on $X$. Then the Hodge-star operator decomposes $\Lambda^2$ into the space of...
View ArticleIs there any "deep" relation between the localization theorem of equivariant...
First let's consider equivariant cohomology: if a compact Lie group $G$ acts on a compact manifold $M$. We have the equivariant cohomology $ H_G(M)$ defined as the cohomology of the cochain complex...
View ArticleIs any deformation of an acyclic complex gauge equivalent to a trivial one?
This question is related to this question. Let $(C^{\bullet},d)$ be a cochain complex over a field $k$ with char$k$=0. We fix an Artin local $k$-algebra $(A,m)$. A deformation of $d$ is a new operator...
View ArticleIs an isomorphism between holomorphic vector bundles still holomorphic with...
Let $X$ be a compact complex manifold and $E$ be a finite dimensional holomorphic vector bundle on $X$ with a fixed $\bar{\partial}$-connection $\bar{\partial}_E$.Now we consider a small neighborhood...
View ArticleDo we have a Grauert-Fischer theorem for non-trivial families?
This question is related to my previous question. Let $X$ be a compact complex manifolds and $\Delta\in \mathbb{C}^n$ be a small neighborhood of $0$. A family of deformations of $X$ over $\Delta$ is a...
View ArticleIs an infinite direct sum of quasi-coherent $\mathcal{O}_X$-modules...
On any ringed space $(X,\mathcal{O}_X)$ we can define quasi-coherent $\mathcal{O}_X$-modules: A sheaf of $\mathcal{O}_X$-modules $\mathcal{F}$ is quasi-coherent if for every point $x\in X$ there exists...
View ArticleHas anyone studied the derived category of Higgs sheaves?
Let $X$ be a complex manifold and $\Omega^1_X$ be the sheaf of holomorphic $1$-forms on $X$. A Higgs bundle on $X$ is a holomorphic vector bundle $E$ together with a morphism of $\mathcal{O}_X$-modules...
View ArticleAnswer by Zhaoting Wei for Proof of the Hirzebruch-Riemann-Roch theorem using...
I think the top of page 148 computes the case where $W$ is trivial, whose second term $\frac{1}{2}\sum_k(Rw_k,\bar{w}_k)$ is exactly $\frac{1}{2}\text{Tr}_{T^{1,0}M}(R^+)$. The general case then follows.
View ArticleAnswer by Zhaoting Wei for Does there exist a GRR-like generalization of the...
I'm sorry for the self-citation. But your question is largely answered in the monograph Coherent Sheaves, Superconnections, and Riemann-Roch-Grothendieck, or the arxiv version, joint work of...
View ArticleWhat are the norms of the generators of the standard Podleś sphere?
Fix a real number $0<q<1$. We consider the standard Podles sphere $A_q$ as the universal unit $C^*$-algebra generated by $a$ and $b$ with relations\begin{equation*}\begin{split}&a=a^*,~...
View ArticleHow obtain the right definition of smooth elements in a $C^*$-algebra?
In Alain Connes'$C^*$-algèbres et géométrie différentielle (an English translation is here,), for a $C^*$-algebra $A$, we consider a $C^*$-dynamic system $(A,G,\alpha)$, where $G$ is a Lie group and...
View ArticleCan we have a nontrivial division of a irreducible root system as the union...
The question is related to this MO question. Let $(\Phi, E)$ be a irreducible crystallographic root system where $\Phi$ is the set of all roots and $E$ is the $\mathbb{R}$-span of $\Phi$. As in the...
View ArticleComment by Zhaoting Wei on Is the deformation of a $C^{\infty}$-manifold over...
@Z.M It is just a intuitive definition. We should use the ringed space definition as the true definition.
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