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 Yearling
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Dec
20
comment Vector bundle $\gamma^1$ over $\textbf{P}^\infty$ does not have finite type?
My point was that the term 'finite type' needs clarification, but I think most likely what is meant is that the base space admits a finite open cover such that restrictions of the bundle to its members are trivial (I googled).
Dec
20
comment Vector bundle $\gamma^1$ over $\textbf{P}^\infty$ does not have finite type?
Terminating prolongation series of the $G$-structure to which $\gamma^1$ is associated?
Dec
16
comment Why should the Kaehler form be closed?
$J$ is an endomorphism of the (real) tangent space that makes it into a complex space, whence it corresponds to multiplication by $i$. I think you'll get a better reception for these kind of questions over at math.stackexchange and probably there are some questions already there of interest to you.
Dec
16
comment Why should the Kaehler form be closed?
@physicsoutsideborders, note $\mathcal{K}(u, v) = g(u, Jv)$ where $J$ is the complex structure.
Dec
16
comment Why should the Kaehler form be closed?
You don't have to demand the form be closed, in fact there are several generalisations of Kahler geometry where it isn't. But if it is closed, many nice things happen, and we call it a Kahler manifold. Why do you need more than that?
Oct
28
comment List of generic properties of Riemannian metrics
Does the perturbation have to be generic? That seems to be the difference between you meaning "dense" and meaning "generic" in the sense of the above comments.
Sep
20
comment Examples of holomorphic Killing vector fields on compact Kahler manifolds
In case it helps you to know, any Killing vector field on a compact Kahler manifold is automatically holomorphic.
Sep
18
comment Finding the octonionic analog of the K3 surface, via (almost) hyperkahler geometry?
Yes, I meant $Spin_7$-structure in the sense of $G$-structure, not $Spin_7$-metric (just one of the types).
Sep
17
comment Finding the octonionic analog of the K3 surface, via (almost) hyperkahler geometry?
I haven't thought about the details, but are you sure you want almost-hyper-Kahler and not almost-quaternionic-Kahler geometry (or a Spin7-structure) a priori for condition 2 (it might turn out that it reduces further afterwards, as it does for K3)?
Sep
15
comment Sobolev Multiplication theorem for Fibre bundles
If $M$ is just a manifold admitting an action, how are you defining a metric on the total space $E(M)$ for your embedding? Also, you seem to be treating $\Gamma(X, E(M))$ as if it is a vector space.
Aug
27
comment Asymptotic $\int_M \mathrm{exp}[\mathbf{e}\left(n -\frac{t}{2\pi i}\right)] \left( 1 + \frac{5}{6} \mathbf{e}^2 \right)^{1/2} $ on quintic Calabi-Yau
I got something nearly identical to the RHS by just expanding the integrand as suggested, and I probably made a small mistake somewhere. Given that your manifold is Calabi-Yau, an example of such an $\mathbf{e}$ is a multiple of the Kahler form itself, that's how you're assured there is one.
Aug
22
comment When is a conformal class equal to a conformal orbit?
Isn't the first space finite dimensional for $n > 2$ and the second not?
Jul
21
awarded  Yearling
Jul
17
comment Oriented volume and determinants: Circularity
Are you really using determinants? Why not define an orientation as the choice of a connected component of the torsor of real frames?
Jun
5
comment What is the relation between two Riemannian metrics with the same Riemannian curvatures?
Relevant :mathoverflow.net/questions/100281/…
Mar
21
awarded  Nice Answer
Mar
12
comment Complex symplectic reduction
Shouldn't this be similar to hyper-Kahler reduction?
Mar
12
comment Bispinors, polyforms, bilinears and supersymmetric manifolds
The first isomorphism is one of vector spaces, not algebras. The second just says that a spinor 'squares' to a form. This squaring map uses a bilinear or sesquilinear form on the space of spinors. There may be several admissible such forms, depending on the dimension or ground field. These give different isomorphisms. There are references but I'm stuck here right now.
Mar
5
comment $Spin(7)$ as stabilizer of a $4$-form revisited
Right, thanks, of course, e.g. the volume form on any generic holonomy space. I have never really understood what's particularly special about the forms that do come from spinors vs those that don't, but I guess that's another question.
Mar
5
comment $Spin(7)$ as stabilizer of a $4$-form revisited
"and conversely": in general, must every invariant form be the square of some invariant spinor (say, wrt some invariant bilinear/sesquilinear form on complex spinors)?