All Questions
20 questions
22
votes
1
answer
883
views
Is being of general type stable under generization
This question is about how varieties of general type over an algebraically closed field of characteristic zero $k$ behave under generization in families.
Definition. An integral projective ...
- 9,497
12
votes
3
answers
4k
views
Does negative Kodaira dimension imply uniruled?
There is a conjecture (often attributed to Mumford) I believe which states that if, on a smooth proper variety $X$ (over an algebraically closed field of characteristic zero), there are no ...
- 153
9
votes
1
answer
502
views
Is the number of minimal models finite
Let $X$ be a variety of general type.
Assume that $\dim X = 3$. In https://eudml.org/doc/164223 it is proven that $X$ has only finitely many minimal models (i.e., only $\mathbb Q$-factorial terminal ...
- 133
7
votes
3
answers
970
views
Basepoints in the canonical system of algebraic surfaces
Let $X$ be a smooth projective variety defined over $\mathbb{C}$. In the context of the minimal model program it is often important to understand the geometry of the maps defined by the complete ...
- 2,097
7
votes
0
answers
571
views
Pseudo-effective divisor which is not nef in any birational model
Let $X$ be a smooth complex projective algebraic variety and let $D$ be a $\mathbb{Q}$-Cartier pseudo-effective divisor on $X$. Lets say that $D$ is birationally nef
if there exists a birational ...
- 1,595
6
votes
0
answers
564
views
Bogomolov–Miyaoka–Yau inequality for minimal varieties with intermediate Kodaira dimension $0<\kappa (X)<\dim X$
From the differential geometric proof of Yau and the algebraic proof of Miyaoka for minimal varieties of general type $\kappa (X)=\dim X$, we know that $$(-1)^nc_1^n(X)\leq (-1)^n\frac{2(n+1)}{n} c_1^...
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6
votes
0
answers
590
views
Semistable minimal model of a $K3$-surface and the special fibre
Suppose that $K$ is a $p$-adic field, that is a field of characteristic $0$ whose ring of integers is a complete discrete valuation ring $\mathcal O_K$ and with residue field $k$ (algebraic closed) of ...
4
votes
0
answers
221
views
A question about the dimension of a relatively ample divisor
Suppose $f: Y \to Z$ is a projective morphism of smooth varieties with connected fibers. If an effective divisor $H$ on $Y$ is relatively ample over $Z$, and $\dim Y >\dim Z$, is $h^0(Y, mH)>1$ ...
- 3,472
4
votes
0
answers
218
views
Example of a non-algebraic singularity II
In an answer of this MO question, Frank Loray constructed an example of analytic singularity which is not algebraic. On the other hand, as I learned from one of Joël's comments in that question, ...
- 1,829
4
votes
0
answers
235
views
Some examples where the plurigenera are nonconstant, when the fibres have worse singularities than canonical
Let start with a definition
Invariance of plurigenera: Choose $m$ large enough so that $mK_F$ has a non-zero global section for some fibre $F$. For any fibre $F$, we have $K_F = K_{X/D}~_{|F}$. So ...
user21574
3
votes
1
answer
291
views
Movable divisor with base locus on a hyperkahler variety
I'm looking for an example of the following:
$X$ is a hyperkahler fourfold (deformation equivalent to $K3^{[2]}$);
$D$ is a movable divisor on $X$ with $D^4=0$; and the base locus of
$D$ is a ...
- 549
3
votes
0
answers
290
views
Nefness property for symplectic equivalency of Moishezon manifolds
Definition:Two symplectic manifolds $(X,\omega_X)$ and $(Y,\omega_Y)$ are defined to be symplectically equivalent if there exists a diffeomorphism $\phi:X\to Y$ such that $\phi^∗ω_Y$ is in the same ...
user21574
3
votes
0
answers
198
views
$L^2$ extension theorem
Is there an Ohsawa-Takagushi $L^2$-Extension theorem for Kahler manifolds? For projective varieties Siu-Paun proved:
Let $\pi \colon X \to \mathbb D$ be a projective family of $n$-folds and $X_0$ be ...
- 99
3
votes
0
answers
214
views
local description of $\mathbb{P}^2$-fibrations over $\mathbb{P}^1$
Let $X$ be a rational threefold (over the field of complex numbers) with terminal singularities. It is well-known that $X$ has only finitely many singular points $x_1,x_2, \ldots,x_n$.
To be more ...
- 543
2
votes
1
answer
3k
views
Trivial canonical bundle
Let $X$ be a complex compact Kähler minimal surface with zero algebraic dimension and $H^2(X,\mathcal{O}_X) \ne 0$. We know that according to Enriques–Kodaira classification, $X$ is either a torus or ...
- 1,829
2
votes
0
answers
441
views
Moishezon projectivity criterion for Moishezon spaces with canonical singularites
A Moishezon manifold is projective if and only if it is Kähler. This is no longer true for a singular Moishezon space. Moishezon proved a projectivity criterion for Moishezon spaces with isolated ...
user21574
1
vote
1
answer
193
views
Two morphisms possess the same Viehweg's variation
Recall the definition of Viehweg's variation intimated from Weak Positivity and the Additivity of the Kodaira Dimension for Certain Fibre Spaces,
E. Viehweg
Let $f: V\rightarrow W$ be a fiber space (...
- 295
1
vote
0
answers
523
views
Quasi-projectivity of the moduli space of Kahler-Einstein Fano varities and vanishing Lelong number
Chi Li, Xiaowei Wang, Chenyang Xu proved the Quasi-projectivity of the moduli space of smooth Kahler-Einstein Fano manifolds. My question is about when central fibre $X_0$ along Kahler-Einstein Fano ...
user21574
0
votes
1
answer
406
views
$m$-th root of holomorphic section of direct image of relative line bundle
Question edited after the answer of Sándor Kovács:
Let $f:X\to B$ be a holomorphic fibre space of smooth projective
varieties which $f$ is relatively semi-ample and take $\mu$ as $m$-th root of ...
user21574
0
votes
0
answers
152
views
$C^\infty$-curvature of Kawamata's singular hermitian metric
Let $X,Y$ are two projective varieties and $f:X\to Y$ is an Iitaka fibration. Consider the following singular hermitian metric $$h(\sigma,\sigma)=\left(\int_{X_y}|\sigma|^{\frac{2}{m!}}\right)^{m!}$$ ...
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