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Please excuse me if this question is too naive. I know very little about the nonabelian Hodge correspondence but I am trying to understand how the correspondence works in the simplest case of the grou …

If I understood the question correctly, the answer is yes and pretty straightforward. The dual of the lattice of open sets is the lattice of closed sets. You can recover the notion of irreducible clos …

Let $X$ be a smooth projective variety over an algebraically closed field $K$ of characteristic zero and fix a point $x\in X(K)$. We can associate to $X$ two Tannakian categories: the category of Higg …

You can define $\pi_1^\text{pro-alg}(X,x)$ with no properness assumption as the Tannakian group associated to the Tannakian category of algebraic vector bundles with flat connection, regular at infini …

If $G$ is a unimodular locally compact group and $K$ is a compact subgroup, then the Hecke algebra of the pair $(G,K)$ is the algebra of compactly supported functions on $K\backslash G/K$ with a convo …

If $k$ is a field of characteristic zero and $X$ is a smooth irreducible projective variety over $k$, then $X$ satisfy Hodge symmetry, meaning that $$\dim H^p(X, \Omega_{X/k}^q) = \dim H^q(X, \Omega_{ …

Let $X$ be a smooth projective toric variety over $\mathbb{C}$. Is there a good presentation for the K-theory ring $K_0(X)$ in terms of the corresponding fan, analogous to the presentation of the Chow …

Let $X$ be a complex algebraic variety with an action of a connected algebraic group $G$. The forgetful functor from the category of $G$-equivariant perverse sheaves on $X$ to the category of perverse …

Here is a counterexample for surjectivity: $$B=k[x,y], A=B[X,Y]/(xX+yY-1)$$ It is easy to see that the image of the induced morphism is the plane minus the origin. Now suppose $a\in A$ is a root of a …