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Here is a proof of the formula $$\sum_{j>0} \frac{1}{j} w(q)^j P_qw(q)^{-j}=2\log \frac{w(q)}{q}-\log w'(q).$$ (The notation is the same as in the Timothy's answer except I prefer $P_qf(q)$ to $[q^{> …

Let $E\to X$ be a holomorphic vector bundle on a projective complex manifold. If $Y$ is another projective manifold diffeomorphic to $X$ then we are free to consider $E$ a smooth bundle over $ …

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 projective complex manifold and $E\to X$ a holomorphic vector bundle. When $E=X\times\mathbb{C}$ there is an injection (for any $n$) $$H^n(X, \mathcal{O}(E))=H^{0,n}(X)\to H^n(X,\mathbb{C …

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 …

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 …

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_ …

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})=\ …

In the van den Essen's paper, a polynomial map $t\mapsto (x(t),y(t))$ is called an embedding if there is a polynomial $F$ such that $F(x(t),y(t))=t$. There is no problem with his proof: Theorem 1 is t …

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 …

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 …

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 …

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?

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 …

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^* …