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7 votes
0 answers
204 views

Global generation of $S^n \Omega_X$ for a fake projective plane

Let $X$ be a fake projective plane, namely, a compact complex surface with $$p_g(X)=q(X)=0, \quad K_X^2=9$$ and $K_X$ ample. Since $K_X^2=9 \chi(\mathcal{O}_X)$, Yau's celebrated proof of the Calabi ...
Francesco Polizzi's user avatar
6 votes
0 answers
175 views

Lower bound for $h^0(X, \operatorname{Sym}^n \Omega_X)$

This is a weaker version of my previous (unanswered) question MO429574. Let us start with a smooth, ample divisor $X$ in an abelian threefold $A$. It is a surface of general type such that $\Omega_X$ ...
Francesco Polizzi's user avatar
6 votes
0 answers
185 views

Arnold's theorem on small denominators and holomorphic tubular neighborhoods

By a theorem of Grauert, along a curve with negative self-intersection a complex surface is locally biholomorphic to a neighborhood of the zero section of that curve inside its normal bundle. For ...
Rodion N. Déev's user avatar
6 votes
0 answers
218 views

Can the base of an elliptically fibered Calabi-Yau threefold be an Enriques surface?

For this question, a Calabi-Yau manifold or variety of dimension $n$ is defined as a non-singular projective variety with trivial canonical bundle and $h^{i,0} = 0$ unless $i = 0$ or $i = n$. If ...
doetoe's user avatar
  • 515
5 votes
0 answers
229 views

Computation of $H^i(X, \, \operatorname{Sym}^n \Omega_X)$ for a surface of general type $(i=0, \, 1)$

Let $X$ be a smooth, complex surface of general type such that $\Omega_X$ is globally generated, and let $n \geq 2$ be a natural number. Question. Is there a way to compute $h^i(X, \, \operatorname{...
Francesco Polizzi's user avatar
5 votes
0 answers
218 views

Reducible surface as a degeneration

I am interested in the following situation. If $S_1\cup_D S_2$ is a union of two irreducible smooth projective surfaces over $k=\overline{k}$(over $k=\mathbb{C}$ is enough, if it's relevant) glued ...
Sungwoo's user avatar
  • 71
4 votes
0 answers
160 views

Surface with $\Omega_X$ globally generated and singular Albanese image

This question is inspired by abx's comment to my previous question MO430933. Let $X$ be a complex surface of general type, and denote by $$a \colon X \to \operatorname{Alb}(X)$$ the Albanese map of $X$...
Francesco Polizzi's user avatar
3 votes
0 answers
221 views

Historical proof of Leschetz Hyperplane Theorem

I browse in Phillip Griffiths' Slides on historical development of Hodge-theory and these include a sketch of the original approach with Lefschetz used to study complex surfaces in his famous ...
user267839's user avatar
  • 6,028
3 votes
0 answers
176 views

Elliptic fibration arising from a higher genus linear system

Let $H$ be a very ample linear system on a smooth compact complex surface $X$ whose Kodaira dimension is $\geq 0$. A general element of $H$ is smooth and has genus $\geq 2$. Let $L\subset H$ be a ...
Alan Muniz's user avatar
3 votes
0 answers
260 views

Contracting rational curves on surfaces and getting something non-algebraic

Recently I posted an "announcement" on arxiv where I said something to the effect of "this is the first example we know where contracting (a tree of) rational curves from a non-singular algebraic ...
pinaki's user avatar
  • 5,339
2 votes
0 answers
643 views

canonical bundle of Abelian surface fibrations

For minimal surfaces admitting an elliptic fibration over a smooth curve, there is a famous analysis of possible singular fibers and a canonical bundle formula due to Kodaira. There are two papers of ...
Mohammad Farajzadeh-Tehrani's user avatar
2 votes
0 answers
330 views

surfaces with effective first Chern class

Let $S$ be a smooth complex surface, If $c_1(S) \in N_1(S)$ is nef and non-torsion, then we know that this would imply some restrictions on the cone of effective curves (and surface itself)--see the ...
Mohammad Farajzadeh-Tehrani's user avatar
1 vote
0 answers
47 views

Positivity of self-intersection of dicisor associated to meromorphic function

In the book "Holomorphic Vector Bundles over Compact Complex Surfaces" by Vasile Brînzănescu, in the proof of theorem 2.13 there is the following claim Let $X$ be a compact non-algebraic ...
JerryCastilla's user avatar
1 vote
0 answers
116 views

Abelian subvarieties corresponding to vector subspaces

Let $S$ be a connected smooth projective surface. Let $C$ a smooth curve on $S$ In page 9 of the paper "https://arxiv.org/abs/1704.04187v1" a read the following: Let \begin{equation*} r: ...
Roxana's user avatar
  • 519
1 vote
0 answers
168 views

Rational classes of $(-2)$-curves in a minimal surface of general type

Let $X$ be a minimal surface of general type over $\mathbb{C}$. One can show that if for any set of $(-2)$-curves $C_1,\cdots,C_l$ on $X$, there exists $k$, $1\le k\le l$ such that $$\sum_{i=1}^k\...
astana's user avatar
  • 41
0 votes
0 answers
110 views

Lefschetz Theorem in Dolgachev's On automorphisms of Enriques Surfaces

Let $F$ be a Enriques surface over $\Bbb C$. I have a question about a detail in the proof of Proposition 2.1. from Dolgachev's On automorphisms of Enriques surfaces. This 2.1. Proposition. states ...
user267839's user avatar
  • 6,028
0 votes
0 answers
265 views

Explicit adjunction formula and local top form

I am reading the section 4.2 in Kollar-Mori, where they provide the explicit equations for Du Val Singularities. In the course of the proof, they reduce to studying the equation $x^2+f(x,y)=0$ in a ...
Stefano's user avatar
  • 625