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This question is simple, and with some suggestion I can answer it myself. As abx pointed out, for a simply connected surface $h$ is an isomorphism, by the Hurewicz theorem. There are simply connecte …

EDIT: this is a stupid question (see the comments and the answer). Let $X$ be a projective complex manifold. Consider two different irreducible hypersurfaces $Y,Z\subset X$, with cohomology classes …

Yes, the formula is true in this case, too. As abx says, I've made a mistake in my computation (what a shame!). We have $Y\cap Z=P\cup Q$, and the irreducible components $P$ and $Q$ have different co …

Hi everyone. I need the following statement: For a Kahler manifold $X$, the natural map $H^n(X,\mathbb{C})\to H^n(X,\mathcal{O})$ (from the sheaf extension) coincides with the Hodge projection $\P …

Hi everyone. My question is about the absolute Galois group action on the set of the Grothendieck dessins. The dessins I am interested in are trees with only one vertex of valency more then 2. (I d …

Let ${\mathcal M}_S$ be the moduli space of (a family of) minimal smooth algebraic surfaces. (A precise definition does not matter here.) Denote $M=\dim {\mathcal M}_S$. Is it true that $$M\le b_ …

This question was inspired by [ Has the following problem posed by Deligne in the official description of the Hodge conjecture been solved? ] (which did not get any reply). I am curious if testing (no …

Let $X$ be a compact complex manifold. By definition, $Pic(X)={\rm H^1}(X,\mathcal{O}^\times)$. We know a lot about this group. What is known about the groups ${\rm H^n}(X,\mathcal{O}^\times)$ for $n …

Let $X$ be a projective complex manifold and $Y\subset X$ be an irreducible hypersurface. If $Y$ is smooth, there is a well known Gysin sequence. However, even if $Y$ is not smooth, a kind of Gysin …

In the preprint arXiv:math/0505432v1 by Batyrev and Kreuser I have found (on pages 2 and 10) the claim that "by a recent result of Kresch and Vistoli [arXiv:math/0301249]" the (usual) Brauer group of …

Let $X/k$ be a smooth curve of genus $g>1$ over a number field $k$. By the Faltings theorem (nee Mordell's conjecture), the set of $k$ - rational points $X(k)$ is finite. Due to the Mordell-Weil theo …

Let $X$ be a smooth complex projective surface. Is the Hurewicz image $h(\alpha)\in H_2(X)$ of a homotopy class $\alpha\in\pi_2(X)$ algebraic?

(A related question is this On the fundamental group of hypersurfaces). Let $X$ be a simply connected projective complex manifold of dimension at least 3. Let $Y\subset X$ be a smooth hypersurface. …

Let $X$ be a complex manifold (not necessarily Kahler or even compact). For the first Chern class $c_1(E)\in H^2(X,Z)$ of a holomorphic vector bundle $E\to X$ there is an obvious lift to $H^1(X, O^* …

When is the Jacobian of a hyperelliptic curve $$y^2=x(x-1)(x-a)(x-b)(x-c)$$ a product of two elliptic curves? (This is a sort of reverse to When is a product of elliptic curves isogenous to the Jaco …