All Questions
Tagged with complex-geometry birational-geometry
53
questions
3
votes
1
answer
176
views
Negative definite of exceptional curve in higher dimension
One direction of the Grauert's contractibility theorem shows
Let $f:S\rightarrow T$ be a surjective holomophic map where $S$ is a compact holomorphic surface. If $C$ is a reduced connected effective ...
- 361
1
vote
0
answers
57
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Seeking for bridges to connect K-stability and GIT-stability
We consider the variety $\Sigma_{m}$ := {($p$, $X$) : $X$ is a degree $n$ + 1 hypersurface over $\mathbb{C}$ with mult$_{p}(X) \geq m$} $\subseteq$ $\mathbb{P}$$^{n}$ $\times$ $\mathbb{P}$$^{N}$, ...
- 41
1
vote
0
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79
views
Birational deformations of holomorphic symplectic manifolds
Let $X$ and $X'$ be birational holomorphic symplectic manifolds. Then the birational morphism between them identifies $H^2(X)$ with $H^2(Y)$. The period space of $X$ is defined to be a subset of $\...
- 153
1
vote
1
answer
108
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Nonequidimensional birational Mori contractions
I have been looking for an excplicit example of a birational, divisorial Mori contraction such that the exceptional locus is not equidimensional onto its image.
To agree with the setup I like, the ...
- 11
1
vote
0
answers
161
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Proofs for if $\widetilde X\rightarrow X$ is a modification & $\widetilde X$ is a $\partial\bar{\partial}$-manifold, then so is $X$
A celebrated result due to Deligne--Griffiths--Morgan--Sullivan (see Theorem 5.22 in Real Homotopy Theory of Kaehler Manifolds) says that:
Consider a proper modification $f:\widetilde M\rightarrow M$ ...
- 295
2
votes
1
answer
163
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Explicit functor from Kuznetsov component to derived category of K3 for rational cubic fourfolds
Let $X \subset \mathbb{P}^5$ be a Pfaffian cubic fourfold (or one of the other known rational cubic fourfolds). It is known by Kuznetsov's Homological Projective Duality that $\mathcal{K}u(X) \simeq D^...
- 305
8
votes
1
answer
329
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Alterations and smooth complete intersections
Let $k$ be an algebraically closed field, and $X$ a projective variety over $k$. Let $i : X\subset \mathbf{P}^d_k$ be a closed immersion into a projective space of high enough dimension.
Is there a ...
user501361
2
votes
1
answer
219
views
Inclusion of (pulling back of) dualizing sheaves under normalization
I am reading O. Fujino's book Iitaka Conjecture. In page 42, Lemma 3.1.19 he restated one result due to Viehweg to use the base change arguments.
There exists some details in the proof of Step 2 in ...
- 295
1
vote
0
answers
105
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Does nefness in analytic setting depend on Hermitian metric?
I'm reading Demailly's book 'Analytic Methods in Algebraic Geometry'.
Let $X$ be a compact complex manifold with a Hermitian metric. A line bundle $L$ is said to be nef if for every $\epsilon>0$, ...
- 361
1
vote
0
answers
140
views
Do we have a simple proof for this criterion for basepoint freeness?
The following is a criterion by Fujita (On the structure of polarized varieties with $\Delta$-genera zero).
Consider a complex smooth projective variety $X$ of dimension $n$ and an ample divisor $H$, ...
user497488
1
vote
1
answer
446
views
Determine the coefficient of the exceptional divisor
Consider the following setting: suppose that $X$ is a smooth variety and let $f:X\rightarrow \Delta$ be a smooth morphism outside the origin $0$. Let the central fiber $X_0$ be a reduced (Cartier) ...
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3
votes
1
answer
411
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Current progress on rationality problem for complex hypersurfaces
How is the current progress on rationality problem for complex hypersurfaces $X\subset\mathbb{P}^{n+1}$ with $n\geq 3$?
There are many hypersurfaces are shown to be unrational, such as smooth cubic ...
user493108
0
votes
0
answers
102
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Every locally free sheaf is Cohen-Macaulay on complex variety with at canonical singualities?
Suppose $X$ is a normal complex space with at most singularities, can we say any locally free sheaf on it is Cohen-Macauly?
Recall that a coherent sheaf $\mathcal{F}$ over $X$ is called (maximal) ...
- 295
1
vote
0
answers
89
views
In complex analytic category, is the pluricanonical sheaf Cohen--Macaulay?
We adopt the following definition of canonical singularities in complex analytic category.
Let $X$ be a normal complex space of dimension $n$, and let $j:X_{\text{reg}}\rightarrow X$ be the open ...
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1
vote
1
answer
187
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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 (...
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