Eric O. Korman
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 Apr11 comment What is the kernel of a Maurer-Cartan form? But do you have an example of where this appears in the foliation literature you mention? Apr11 comment What is the kernel of a Maurer-Cartan form? For 1) It is a vector space valued 1-form so it gives a linear map from vector fields to the Lie algebra. Do you have more context for 2)? I don't see it can have a kernel (viewing it as a map as in 1)) since it takes a tangent vector $v$ at a point $g$ to ${L_{g^{-1}}}_* v$ which is non-zero if $v$ is non-zero since left multiplicaiton by $g$ is a diffeomorphism. Apr11 answered Isogenies and dimensions Apr10 comment Isogenies and dimensions Or is your question if $g = g'$? Apr10 comment Isogenies and dimensions Do you mean the linear map $\mathbb C^g \to \mathbb C^{g'}$ is surjective? Apr10 comment Can a ring isomorphism change the structure of a module? Yes that should be true. Though the Clifford algebra is a simpler object then the group algebra of $SO(2n)$. Apr10 comment Can a ring isomorphism change the structure of a module? Ok, I guess you can turn that into an example for a ring if you know about Clifford algebras: take the even Clifford algebra on $\mathbb R^{2n}$. Then the even and odd spinors are two non-isomorphic representations. Apr10 comment Can a ring isomorphism change the structure of a module? Note that if $\phi$ is an inner automorphism then you get an isomorphic module, but in general you don't. I can't think of an immediate example for rings but for groups (where modules are representations) an example is that $Spin(2n)$ has two non-isomorphic spin representations but are related by the outer automorphism of $Spin(2n)$. Apr10 comment Notation for the Covariant Derivative of a smooth section in the direction of a tangent vector X If $\alpha$ is a 1-form and $v$ a vector field, then does the notation $\langle \alpha, v\rangle$ make sense to you? Apr10 comment Is the space of smooth sections of a vector bundle finitely generated as a $C^\infty$-module? I'm guessing you want $M$ to be connected (or at least have finitely many connected components)? Apr9 comment Complex line bundle at symplectic manifold A line bundle and connection with curvature $\omega$ exist if and only if $[\omega/2\pi] \in H^2(M;\mathbb Z)$. Apr9 comment Explicit form of the foliation associated to a differential one-form The leaves are the submanifolds of the form $\{(x^0_0 \exp(t), x^1_0 \exp(t), x^2, x^3) \mid t, x^2,x^3 \in \mathbb R\}$ for some constants $x^0_0, x^1_0$. Apr9 comment Explicit form of the foliation associated to a differential one-form Right, I was being a little sloppy since integral curves usually mean a specific paramterization but for the foliation we just care about the image of the curve in the manifold (as per Lee's comment) Apr9 comment Explicit form of the foliation associated to a differential one-form Correct, the leaves are the integral curves of $x^0\partial_0 + x^1\partial_1$. So explicitly the leaves are parameterized by $x^0 = v^0 t, y^0 = v^1 t$, where $v^0,v^1$ are constants. Apr9 answered Explicit form of the foliation associated to a differential one-form Apr8 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. Apr8 answered Weird definition of Kodaira-Spencer map (What's a relative Kähler differential on a manifold?) Apr8 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. Apr7 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? Apr7 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?