6 votes

Are Chern classes always vertical?

abx's counterexample is correct. It might be worth remembering the splitting principle, though: Let $E$ be any rank $n$ vector bundle on $M$, and let $F(E)$ be the bundle of complete flags in $E$, so $...
David E Speyer's user avatar
5 votes

Are Chern classes always vertical?

For a counter-example (with real coefficients), take for $M$ the Grassmannian $\mathbb{G}(p,p+q)$ with $p\neq q$, and $p,q\geq 2$. If I computed correctly: $$c_2(M) = \frac{1}{2}\left[(p-q)^2-(p-q)+2\...
abx's user avatar
  • 37.3k
4 votes

A paper of Borel (in German) on compact homogeneous Kähler manifolds

An earlier source than Borel-Remmert (1962) is Matsushima (1957), who writes in French: Tout espace homogène kählérien compact est produit kählérien d'un tore complexe et d'un espace homogène ...
Carlo Beenakker's user avatar
3 votes

Question about Neumann eigenvalues on manifolds

Let $\Omega$ be a connected domain and denote its the Neumann eigenvalues by $ 0 = \mu_0(\Omega) < \mu_1(\Omega) \leq \cdots $ Let $\mathbb{S}_+^2 = \{(x,y,z)\in\mathbb{S}^2\ |\ y \geq 0\}$ be the ...
Neal's user avatar
  • 881
2 votes

Continuity of the volume function

I think you may have more restrictive hypotheses in mind, because in general this will be false due to behavior that may be obviously absent in the cases you want to consider. Anyway, here are two ...
Pierre PC's user avatar
  • 3,049
2 votes

Approximation of triply periodic minimal surfaces with trigonometric level sets

I believe these equations first appear in Ian Barnes' 1990 PhD thesis Microstructure of bicontinuous phases in surfactant systems. In section 6.2.1 he derives the Fourier representation of periodic ...
Greg Hurst's user avatar
1 vote

Taubes' SW$\Rightarrow$ Gr

$2d^*d|\mu|\le\frac1{|\mu|}\langle\nabla^*\nabla\mu,\mu\rangle$ (Rohil Prasad reminded me to use Leibniz and Cauchy-Schwarz), hopefully that's sufficient for you to reproduce and then complete the ...
Chris Gerig's user avatar
  • 17.2k

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