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Apr
8
comment Weird definition of Kodaira-Spencer map (What's a relative Kähler differential on a manifold?)
Correct, "dga" means differential graded algebra. This is really a mild Lie algebroid, the point is that when you have some $\mathcal O_X$ module with a Lie bracket (in this case the usual Lie bracket of vector fields) and some compatible action on $\mathcal O_X$ (in this case differentiation of holomorphic functions by holomorphic vector fields), the exterior algebra on the dual has a canonical differential.
Apr
8
answered Weird definition of Kodaira-Spencer map (What's a relative Kähler differential on a manifold?)
Apr
8
comment Weird definition of Kodaira-Spencer map (What's a relative Kähler differential on a manifold?)
The definition of $\Theta_\mathcal{V/W}$? That's eq (1.5) on p. 6 of Advances of in Moduli Theory.
Apr
7
comment Weird definition of Kodaira-Spencer map (What's a relative Kähler differential on a manifold?)
Unfortunately I know hardly any scheme theory so I'm not sure what the corresponding differential geometric notion is. Is there a part on "Advances in Moduli Theory" that references this?
Apr
7
comment Weird definition of Kodaira-Spencer map (What's a relative Kähler differential on a manifold?)
$\Theta_\mathcal{V/W}$ is the sheaf of holomorphic sections of the holomorphic vector bundle that is the kernel of the differential of the projection $T\mathcal V \to T\mathcal W$. What relative differential are you asking about?
Apr
6
comment Working out an example of a Chern class
If you're not set on using the definition of Chern class you mention (as Poincare dual to a zero locus), you can try using the Chern-Weil description (which will give you the rational Chern class, which in your specific example gives you the $\mathbb Z$ Chern class since I'm pretty sure flag manifolds have no torsion in their cohomology).
Apr
5
comment Composition of Irreducible Representation and Surjective Homomorphism
This is more easily proved using the definition of irreducibility.
Mar
26
awarded  Informed
Mar
26
comment Is the determinant the “only” group homomorphism from $\mathrm{GL}_n(\mathbb R)$ to $\mathbb R^\times$?
I'm not sure I see how knowing that $SL$ is the commutator of $GL$ tells you that the trace is the unique form that vanishes on commutators. You could probably prove this though using uniqueness of the Killing form on $\mathfrak{sl}_n(\mathbb R)$.
Mar
26
answered Is the determinant the “only” group homomorphism from $\mathrm{GL}_n(\mathbb R)$ to $\mathbb R^\times$?
Mar
26
comment Is the determinant the “only” group homomorphism from $\mathrm{GL}_n(\mathbb R)$ to $\mathbb R^\times$?
a Lie group homomorphism is determined by the corresponding Lie algebra homomorphism. So what are all of the Lie algebra homomorphisms $\mathfrak{gl}_n(\mathbb R) \to \mathbb R$? These are the elements of the dual space of $\mathfrak{gl}_n(\mathbb R)$ that vanish on commutators. Are multiples of the trace the only ones (the trace is the derivative of $\det$)?
Mar
22
comment Prove the Curvature Tensor is a Tensor
@user13223423 if $T$ is a tensor and $D$ is some differential operator, $T = T+D - D$ is a tensor but neither $T+D$ nor $D$ are tensors. Of course, the sum of tensors is again a tensor. With regarding your other question, tensors can be characterized as those linear operators on sections of a vector bundle that are linear over $C^\infty(M$).
Mar
22
revised Prove the Curvature Tensor is a Tensor
added 1 characters in body
Mar
22
comment Prove the Curvature Tensor is a Tensor
Each of those terms is not a tensor on their own. You also had a typo in the definition of $R$, which I edited.
Mar
21
comment Time-dependent flat line bundle
@JoshBurby yep that's correct
Mar
20
comment Time-dependent flat line bundle
@JoshBurby see my edit. hope this helps.
Mar
20
revised Time-dependent flat line bundle
added 1097 characters in body
Mar
20
revised determinating the signature of a bilinear form
edited body
Mar
20
comment determinating the signature of a bilinear form
It looks like you have a typo, what should $y_n + 1 - i$ be? Should it be $y_{n+1-i}$ and then the sum should really start at $i=1$ instead of $0$?
Mar
19
answered Time-dependent flat line bundle