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The Hodge conjecture seems to me the most mysterious among the Millennium problems (and many others). In particular, I am not sure about its logical complexity. It is not difficult to see that the …

If I got it right, there is a theorem due to M. Artin that on a projective complex manifold any point has a Zariski open neighbourhood which is a $K(\pi, 1)$ space. I have two questions about it. I …

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 …

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 …

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?

Let $X$ be a projective complex manifold of dimension $n$. Are torsion cohomology classes in $H^{2n-2}(X,\mathbb{Z})$ algebraic? (We may assume, without loss of generality, that $n=3$, because of the …

Let $Y$ be a complex manifold, $X\subset Y$ a compact submanifold, and $E\to X$ a holomorphic vector bundle. Can $E$ be extended to a bundle over an open neighborhood of $X$ in $Y$? (Four years a …

Let $X$ be a smooth projective variety of dimension $n=\dim X$ over a finite field $\mathbb{F}_q$. As is well known, its zeta function satisfies a functional equation of the form $$Z(X,q^{-n}T^{-1})=\ …

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 …

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 …

Is very well known that the only algebraic curve which admits no deformations is the projective line. Q. What are "rigid" smooth algebraic surfaces? Is there a sensible classification?

In the (very nice) book "Diophantine geometry" by Hindry and Silverman there is a proof of the Appell-Humbert theorem in Exercise A.5.5. I believe that it contains a serious mistake and I want to find …

This is not really an answer, but hopefully may be of help. As David Speyer already pointed out, the problem is basically about a matrix equation $$(X+Y+Z)^{-1}=X^{-1}+Y^{-1}+Z^{-1}.$$ Three symmetri …

(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. …