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About Abhyankar's conjecture
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1h

answered  About Abhyankar's conjecture 
21h

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Geometric generic fibre
This looks right, even if counterintuitive. It may help to consider the following example. Let $E_t= \{y^2=x(x1)(xt)\}$. Then $E_\pi$ and $E_e$ are not isomorphic as $\mathbb{C}$schemes. However, there exists a field automorphism of $\mathbb{C}$ taking $\pi$ to $e$, and thus the first curve to the second. 
1d

answered  Is the Gysin morphism equivariant? 
1d

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Is the Gysin morphism equivariant?
Alternatively, you can argue that both the restriction $j^*$ and Poincare duality isomorphisms are equivariant (by naturality). 
1d

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Idea of using etale site
Will, I would argue that Deligne in his original proof of Riemann does use analogues of certain constructions in the topology of complex varieties; namely Lefschetz pencils and monodromy. 
Jun 18 
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Hodge numbers and weight filtration
Given a complex, $(A[m])^n= A^{nm}$ (and sometimes the differential gets multiplied by $(1)^{m}$). This implies that equality you ask about is true. 
Jun 4 
answered  Intuition for thinking about $R$module of Kähler differentials, universal receptacles, derivations? 
Jun 2 
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A question on an set of 8 matrices related to the SU(3) generators
Your last question is answered by Tanja, who gives a name to this set of matrices. So presumably, it comes up in the physics literature. For mathematician, however, $\{g_m\}$ is just a basis for the Lie algebra $su(3)$, and $\{F_m\}$ is just a subset of $SU(3)$ with no particular structure as far as I can tell. 
May 28 
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canonical divisors of a resolution of a normal surface singularity
(1) is the trace map in Grothendieck duality. There are various references for this. (2) is false. The first condition of (2) is the condition for $X$ to have rational singularities. It does not imply that $X$ is Gorenstein ($K_X$ Cartier). 
May 27 
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Singularities of the product of a $(\mathbb{C}^*)$surface with $\mathbb{C}$
It would help if you told us what a $\mathbb{C}^*$surface was, but fortunately I can answer this without knowing the definition. Claim: If $X$ is a normal surface with non rational singularities, then $\mathbb{C}\times X$ also has non rational singularities. Proof: If $p:Y\to X$ is a resolution of $X$, then $P:\mathbb{C}\times Y\to \mathbb{C}\times X$ is a resolution also. Notice that $R^1P_*O$ is nonzero because it is the pullback of $R^1p_*O$. 
May 10 
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Finite generation and profinite completion
Wouldn't $\hat{\mathbb{Z}}$ be a counterexample? Or am I missing something? 
May 8 
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Do I know what “coherent sheaf” means if I know what it means on locally Noetherian schemes?
I want to point out that the notion of coherence originated in complex analysis rather than algebraic geometry. Specifically, Oka says that the sheaf of holomorphic functions on a complex manifold is coherent, although it would not be "approximated" by (locally noetherian) schemes in any reasonable sense. This doesn't answer you asked, but it does suggest that maybe it isn't the right question. 
May 3 
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Characterizations of regular holonomic Dmodules
Two suggestions: (1) Interlibrary loan, or (2) ask the obvious person if you can borrow his copy. 
Apr 29 
answered  What are the easiest examples of irreducible, but not big, monodromy representations 
Apr 29 
answered  References for the moduli space of complex structures 
Apr 24 
answered  Variation of Hodge structures associated to a hermitian symmetric domain 
Apr 21 
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Why write GRR with the relative tangent sheaf?
The second version is also more general. If $X$ or $Y$ are singular, then the terms in first statement are undefined, but $\mathcal{T}_f$ may still be defined, e.g. when $f$ is smooth and proper. 
Apr 17 
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Spectrum of the Laplacian on pforms on the sphere
Good point. The other, more relevant, reference that I forgot to mention is Folland, Harmonic Analysis of the de Rham complex of the sphere, Crelles 1989 
Apr 16 
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Spectrum of the Laplacian on pforms on the sphere
Ref: Berger, Gauduchon, Mazet, Le Spectre d'un variete Riemannienne 