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This question is about the existence of a definition. I'm far from being an expert in the field in question I apologize in advance for any inaccuracies or stupid and wrong assumptions. Let $X$ be a s …

Let $X$ be a finite type scheme over $\mathbb{C}$ and let $ \mathcal{X} \to Spf(\mathbb{C}[[x]])$ be a formal deformation of $X$. Which of the following assumptions (or combinations thereof) are suffi …

Let $H < G$ be a subgroup of a finite group $G$. Let $X:=G/H$ and $\mathcal{F} \in Sh_G(X)$ be an equivariant sheaf on $X$ (w.r.t. left multiplication) associated to a finite dimensional representatio …

This is inevitably an imprecise question, but there are already several questions like this on the site so I thought i'd try anyway. If I understand correctly, for any reductive algebraic group $G$ th …

I'll be using homological grading throughout this question. Let $G$ be a compact connected lie group. The following isomorphisms are classical and can be proven using several methods: $$H^{\bullet}( …

There's a very nice topological description of blow ups of complex manifolds at a point as connected sum with projective space. The following is an attmept to understand whether there's a higher dimen …

Let $X$ be a projective algebraic variety over a (perfect) field $k$. Let $Aut(X):k \text{-}Alg \to Grp$ be the functor of points defined by $$Aut(X) : A \mapsto Aut_{Spec (A)}(X \times_{k} Spec …

Let $X$ be an algebraic variety (separated quasi-compact scheme of finite type) over a field $k$. One of the possible definitions of an ample line bundle goes as follows: Def 1: A line bundle $\ …

While reading these notes by Victor Ginzburg on $D$-modules I found a certain construction of Microlocailzation in the algebraic setting which unfortunately doesn't seem to be elaborated on a lot in t …

Let $f: X \to Y$ be a morphism of schemes. I'd like to have a completely algebraic description of the belian category of descent data for modules along $f$. Here's my attempt: The category of quasi-c …

A Hodge structure can be defined as a real, algebraic representation of the Deligne torus ${Res}^\mathbb{C}_{\mathbb{R}}\mathbb{G}_m$. Coming from Kahler manifolds the intuition for this is clear. The …

Let $X$ be a smooth algebraic (/projective if it simplifies things considerably) variety over $\mathbb{C}$ and consider the derived category $\mathcal{C}=D_h^b(\mathcal{D}_X)$ of bounded complexes of …

Purity theorem: A map between smooth complex algebraic manifolds of the same dimension has ramification locus of pure codimension 1. One can prove this by a clever induction on dimension using punctu …

Let $X$ be a smooth affine variety over $\mathbb{C}$ and let $\mathcal{D}_X$ be its algebra of differential operators. Consider $\mathcal{C}=\mathcal{D}_X$-$\text{mod}$, the stable $\infty$ category …

I intentionally phrased the title to match a different question which is almost identical to the one i'm asking. However similar, the answer there, which uses commutative algebraic geometry, is not di …