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1

If $M$ is even-dimensional, Clifford multiplication with the Clifford volume element is an isomorphism between the spaces of Killing spinors for $\alpha$ and for $-\alpha$. If you change the orientation, you can actually leave the Clifford multiplication $c\colon TM\to\operatorname{End}\Sigma(M,g,o)$ untouched, or you replace it by $-c$. Then either the ...

5

The theorem is the following: (from 1.15 of here) Theorem: Let $M$ be a connected manifold and suppose that $f:M\to M$ is smooth with $f\circ f= f$. Then the image $f(M)$ of $f$ is a submanifold of $M$. Proof: We claim that there is an open neighborhood $U$ of $f(M)$ in $M$ such that the rank of $T_yf$ is constant for $y\in U$. Then by the constant rank ...

0

In topological K-Theory, there is such an expression: For $E=\bigoplus L_i$, $\psi^n E = \bigoplus L_i^n$. Now $\Lambda_k E$ is the $k$-th elementary symmetric polynomial in the $L_i$. Since any symmetric polynomial is expressible in terms of the elementary ones, a description follows. Note that it involves subtraction, so this is genuinely a stable thing. ...

6

Theorem 7.4 of J. F. Adams, Vector fields on spheres, Ann. of Math. 75 (1962), 603–632 says that $$\tilde{KO}({\mathbb R} P^n)=\mathbb Z\,/\,2^{\phi(n)},$$ generated by $\xi-1$, where $\phi(n)$ is the number of integers $s$ such that $0 < s\le n$ and $s$ is congruent to 0,1, 2 or 4 modulo 8, and $\xi$ is the canonical line bundle over $\mathbb RP^n$ given ...

8

$BO(n)$ is the infinite-dimensional Grassmannian $Gr(n,\infty)$ of $n$-planes in ${\mathbf R}^\infty$. There is a natural direct sum operation $$\oplus\colon Gr(n,\infty)\times Gr(m,\infty)\to Gr(n+m,\infty)$$ (just taking the direct sum of linear subspaces) and it gives you the desired map $$BO(n)\times BO(m)\to BO(n+m).$$

4

You are right, the condition that $L$ be ample can be weakened. In fact, on an $n$-dimensional normal projective variety $X$ one can measure (semi-)stability with respect to an arbitrary movable curve class $\alpha \in N_1(X)_{\mathbb R}$. A numerical curve class $\alpha$ is said to be movable if $D \cdot \alpha \ge 0$ for all effective divisors $D$ on $X$. ...

3

You get this by picking a frame $(e_1,\dots,e_r)$ of $E|_U$ for a neighbourhood $U$ of each point $x$ such that $(e_i,e_j)=\delta_{ij}$. As pointed out by t3suji, these neighbourhoods are étale in general, because they have to carry some square roots of regular functions. If you use such frames to trivialise your bundle, you get transition functions as in ...

2

Consider the following conditions: $z\in Z(s)$ $s(z)=0$ $s\in\mathfrak{m}_zM$ the image of the map $A(V)\stackrel{\varphi_s}{\longrightarrow}M$ defined by $1\mapsto s$, is in $\mathfrak{m}_zM$ $\forall f\in M^{\vee}$ we have $(f\circ\varphi_s)(1)\in\mathfrak{m}_z$ $\forall f\in M^{\vee}$ we have $f(s)\in \mathfrak{m}_z$ $I_s\subseteq\mathfrak{m}_z$. ...

3

The following paper contains a proof (Corollary 1) of the identification of Ramanathan-semistability for principal $GL_n$-bundles and Mumford-semistability for the associated vector bundles: D. Hyeon and D. Murphy. Note on the stability for principal bundles. Proc. Amer. Math. Soc. 132 (2004), 2205-2213. The basic point is that there is a direct ...

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