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It is the complement $\mathbb{P}^{n+1} - \{x\}$ of a point in a projective space.

Charlie, it is funny answering this way but here it is. The criterion you are thinking about is a criterion that is relative to an embedding. It says that if $X$ is a quasi-affine complex normal var …

The branch locus in $Y$ need not be a normal crossings divisor even when $Y$ is a projective space. Suppose $X$ is a smooth complex projective surface with non-abelian fundamental group. By Noether no …

Yes, this has been achieved in some sense. There is a (unpublished and possibly not yet written) work of Gaitsgory and Lurie where they propose an answer to this question. Given a split reductive grou …

This intuition seems to be only loosely right. There are many smooth compact CY threefolds with large fundamental groups. For instance $\mathbb{Z}/3\times \mathbb{Z}/3$, $\mathbb{Z}/8\times \mathbb{Z …

It is indeed true that rational equivalence gives bigger groups of cycles than say algebraic equivalence. However algebraic equivalence is also far away from homological equivalence. In complex geomet …

A small clarification on bhargav's answer: in algebraic geometry we only have quasi-unipotency of the local monodromy in one-parameter families (which is what bhargav is talking about); or in multi-pa …

The construction of an Albanese scheme and an Albanese map for proper and geometrically irreducible schemes over a perfect field goes back to the work of Chevalley, to this talk of Serre, and to Groth …

You have to be a bit careful here. Over $\mathbb{C}$ the stack of representations of $\pi_{1}(X)$ in $G$ and the stack of flat algebraic $G$-bundles on $X$ are isomorphic as complex analytic stacks b …

A slightly better variant of this question is to ask: is the Hurewitz image of $\pi_{2}(X)$ in $H_{2}(X)$ a sub Hodge structure? This is in fact an old question of Philippe Eyssidieux. In section 4.3 …

Such hyper Kaehler metrics do exist near the zero section, e.g. in a formal or an analytic tubular neighborhood of the zero section. After that one can use some homogeneity to spread them on the whol …

Quasi-unipotency is a well defined notion at any point of the discriminant. If we have a proper family $f : X \to S$ of varieties with a smooth total space and a smooth base, and if $p \in D \subset S …

In general it could happen that the Albanese variety does not admit a principal polarization at all. For instance the Albanese variety of an abelian variety is the Abelian variety itself. So choose $X …

The physicists (see e.g. this paper of Aganagic and Vafa) will write the mirror as a threefold $X$ which is an affine conic bundle over the holomorphic symplectic surface $\mathbb{C}^{\times}\times \ …

I am not sure if you will count this but you have the examples from the other side of geometric Langlands. On any smooth variety the skyscraper sheaves of points are eigensheaves for the tensorization …