All Questions
Tagged with differential-forms ag.algebraic-geometry
10 questions
17
votes
3
answers
1k
views
How does one compute the space of algebraic global differential forms $\Omega^i(X)$ on an affine complex scheme $X$?
In 1963 Grothendieck introduced the algebraic de Rham cohomolog in a letter to Atiyah, later published in the Publications Mathématiques de l'IHES, N°29.
If $X$ is an algebraic scheme over $\mathbb C$...
- 47.5k
11
votes
1
answer
510
views
When is the module of Kahler volume forms torsion-free?
Let $R$ be a commutative algebra over a field $k$. Denote the $R$-module of Kahler differentials by $\Omega^1_kR$; this is the $R$-module generated by symbols of the form $da$, $a\in R$, and ...
- 13k
8
votes
1
answer
756
views
What is the geometric significance of the definition of supermanifold?
We know that a supermanifold $M$ is a locally ringed space $(M,O_M)$ which is locally isomorphic to $(U,C^\infty(U) \otimes \wedge W^\ast)$, where $U$ is an open subset of $\mathbb{R}^n$, $W$ is a ...
- 155
7
votes
0
answers
245
views
Albanese morphism induces an isomorphism on global $1$-forms
Let $X$ be a smooth projective variety over a field $k$ of characteristic zero equipped with a point $e\in X(k)$. There is Albanese morphism $a:X\to \mathrm{Alb}\,X$ which is initial among pointed ...
- 7,377
3
votes
1
answer
331
views
Degeneration of Hodge-de Rham spectral sequence, exactness of a pairing and the trace morphism
If $X\longrightarrow S$, where $S=\operatorname{Spec}(k)$, is a proper smooth morphism of dimension $p$ between $\mathbb{F}_{p}$ schemes with Frobenius $F$, then there is an exact pairing $\wedge:\...
- 1,805
2
votes
0
answers
286
views
Does the sheaf of locally exact differential forms splitting in positive characteristic
Let k be an algebraically closed field of characteristic $p>2$, $X$ a smooth projective curve of genus $g>1$ over $k$, and $F_X:X\rightarrow X$ be the absolute Frobenius morphism. Let $B^1_X$ be ...
- 85
1
vote
1
answer
147
views
existence of meromorphic differentials with non vanishing residues
Let $X$ be a smooth projective variety over a field $k$ of characteristic zero and let $D$ be a simple normal crossing divisor on $X$, with irreducible components $D_i$.
Does there exist a nonzero ...
- 11
1
vote
0
answers
127
views
Degeneration differential form nodal curve
I have a (possibly very basic) question about differential forms on nodal curves. After reading Witten's survey "Two-dimensional gravity and intersection theory on moduli space", I am ...
- 11
0
votes
1
answer
351
views
Residues and Mittag-Leffler sequence
Let $X$ be a compact Riemann surface, $\omega$ a meromorphic differential on $X$ and $f$ a meromorphic function on $X$ with poles only over the points $P_1,\dots,P_d$. The product $\;f\cdot\omega\;$ ...
- 251
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 ...
- 625