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1d

comment 
are extensions of flat connections flat?
If by "flat" you mean that $\nabla_X^2=0$, then yes. The curvature is locally a matrix of (log) $1$forms, so if it vanishes on $U$ it vanishes on $X$. 
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 23 
comment 
CohenMacaulay rings and Normal rings
While I agree with abx, I will leave a few key hints: use Serre criterion for normality to see that normality and the CohenMacaulay condition do not imply each other in either direction (a cusp is CM but not normal…) 
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 
comment 
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 
Apr 12 
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Understanding Faltings's Theorem
No, I actually meant that it might be a bit too ambitious at your stage… But anyway, for Faltings you would need an understanding of abelian varieties, moduli spaces, heights etc. None of these topics are covered in Hartshorne. 
Apr 12 
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Understanding Faltings's Theorem
(I mean for someone at your stage. Falting's proof requires more…) 
Apr 12 
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Understanding Faltings's Theorem
Not to discourage you or anything, but if you just go through Hartshorne or one of the other sources you mention carefully, that seems more than enough. 
Apr 7 
comment 
Picard Group Can Contain rational curve?
No, because $Pic^0(X)$ is an abelian variety. 
Apr 4 
answered  CohenMacaulayness of the direct image of the canonical sheaf 
Apr 3 
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higher direct images of O(E)
Here is one case: $Rf_*O(E)=O_X$. You have a triangle $O_X\to O_X(E)\to O_E$, and the same result is well known for direct image of the first sheaf, and direct image of third should vanish (reduce to cohomology of projective space). It's too late here to write more... 
Mar 26 
awarded  Enlightened 
Mar 26 
awarded  Nice Answer 
Mar 16 
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Mixed Hodge structure and cup product
I had similar thoughts. I wonder how many people who voted to close knew the answer. (Test: Is the category of mixed Hodge structures Tannakian, true of false?) 
Mar 14 
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Reference Request: Fundamental Group Scheme
I agree that the question seems reasonable and doesn't deserve to be closed. However, my suggestion to Priyankur would be to learn a bit more algebraic geometry before jumping into this topic. At the very least, first learn a bit about the etale fundamental group from Murre's "Lectures on Grothendieck's fundamental group" TIFR. 
Mar 9 
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Terminology regarding divisor on a curve
The "support of $D$" sounds good to me. 