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Yes, you can. This follows from the see-saw principle for instance. You can also argue directly as follows. The spaces of global sections $H^{0}(Y,f_*L)$ and $H^{0}(X,L)$ are naturally isomorphic. Sin …

Why not just take a class that is orthogonal to all the algebraic classes on $X$? For instance you can take $Y$ to be a point, and take $X$ to be a generic abelian surface over $\mathbb{C}$, i.e. and …

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 …

EDIT: This is a small correction on the previous argument that I posted here, prompted by a comment of Steven Sam. The question is stable under any base change. Let $\nu : \widetilde{T} \to T$ be the …

I don't think the modified question works either. Let $E$ be a general elliptic curve. Take $X$ to be the quotient of $E\times \mathbb{P}^{1}$ by an involution which is a translation by a point of or …

If $c$ is the codimension of $Z$ in $Y_{2}$, then $$\mathcal{E}xt^{i}_{X}(\mathcal{O}_{Y_{1}},\mathcal{O}_{Y_{2}}) = \wedge^{c} N_{Z/Y_{2}}\otimes \wedge^{i -c} \left( N_{Z/X}/N_{Z/Y_{2}}\right).$$ …

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 …

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 …

I must be missing something here. Why not use the standard Baum-Fulton-MacPherson $\tau$ map? This is the map one needs for Riemann-Roch to work anyway as in sections 18.2 and 18.3 of Fulton's "Inters …

There is a slightly bigger compactification of the configuration space constructed by Ulyanov in http://arxiv.org/pdf/math/9904049v2. It dominates the Fulton-MacPherson compactification and it is agai …

Charlie, as Dmitri pointed out there is a big difference between compact Kaehler and non-Kaehler manifolds as far as the structure of the representation varieties of their fundamental groups are conce …

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 …

Something is wrong with this statement: your map to $\mathbb{P}^{1}$ is given by a pencil in a linear system $|L|$ on $A$. Since the vector space dimension of the linear system is at least two it foll …

It appears that you are assuming that your varieties $C_{i}$ are smooth (you seem to assume that since you are talking about the tangent bundle). In this case each $C_{i}$ is an elliptic curve (I gues …

Proposition 3.4 in Orlov's paper Quasicoherent sheaves in commutative and noncommutative geometry. (Russian) Izv. Ross. Akad. Nauk Ser. Mat. 67 (2003), no. 3, 119--138; translation in Izv. Mat …