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# Questions tagged [kahler-differentials]

Kähler differentials provide an adaptation of differential forms to arbitrary commutative rings or schemes.

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### Universal derivations in complex analytic geometry

In algebraic geometry, we are used to the following universal property of K"ahler differential forms: Let $f: X \rightarrow S$ be a morphism of (Noetherian) schemes. Then there is a relative ...
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### When is the module of Kahler differentials free?

As the title says, when is the module of Kahler differentials a free module? In particular, are there known conditions or criterions that could be met that ensures that it will be free? For example, ...
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### Flatness of sheaf of relative Kahler differentials

Suppose we have a projective flat non-smooth morphism of Noetherian schemes $g: X \rightarrow S$. My question regards when the sheaf of relative Kahler differentials $\Omega_{X/S}$ is flat over $S$. ...
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### wedge product of second chern class and kahler form on Calabi-Yau 3-folds.

Let $X$ be a smooth Calabi-Yau 3-fold with Kahler form $w$, It is true that $\int c_2(TX) \wedge w \geq 0$ (for any Kahler form $w$ on $X$). Proof via algebraic geometry is rather difficult. Some ...
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### complex gradient of a function

Let $M$ be a complex $n-$dim manifold and $u : M \rightarrow \mathbb{R}$ be some smooth function. On $M$ assume that we have a Kaehler metric $h$. How is the complex gradient vectorfield defined with ...
Let $X$ be a quasi-affine scheme; that is, the natural map $$X\rightarrow \overline{X}:=Spec(\Gamma(X,\mathcal{O}_X))$$ is an inclusion. Each scheme has a quasi-coherent sheaf of Kahler ...