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12

The pairs (U,X) are called Runge pairs. The homology version of your statement is proved in the paper of Andreotti and Narasimhan Annals of Math vol 76 no 3 (1962) 499-509 using Morse Theory.The title of the paper is "A Topological property of Runge pairs" The paper by Coltoiu Mihalache titled On the Homology Groups of Stein spaces and Runge pairs, ...

5

By the classification theorem of cubic surfaces (p.6 in this paper), a cubic surface belongs to the following classes Has at worst ADE singularities. Has an elliptic singularity, i.e., the surface is cone over a smooth cubic curve. Non-normal or non-integral, and singular along a curve. So if $X$ has two singularities, $X$ belongs to case 1 and all ...

5

Sorry, I just saw this question. One can consider Deligne cohomology for simplicial manifolds in the rather obvious way, namely by adding an additional "simplicial manifold" direction to the ususal double complex, and then take the total cohomology. As resolutions one can use certain hypercovers. For the simplicial manifold $BG$, I have described ...

2

It is not necessarily true. I can provide a counterexample but don't know any general statement about when this is true. Consider $$A=k[x,dx,y]/\langle x^2y, ydx\rangle$$ with $x$ in degree $-2$ and $y$ in degree $3$. The obvious map from $k[x,dx]$ to $A$ is a dg-algebra map which induces an isomorphism below degree $1$, which implies that $H^0(A)\cong k$ ...

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