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Results tagged with complex-geometry
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user 13972
Complex geometry is the study of complex manifolds, complex algebraic varieties, complex analytic spaces, and, by extension, of almost complex structures. It is a part of differential geometry, algebraic geometry and analytic geometry.
11
votes
Accepted
Intersections in almost complex manifolds
Let me answer the easy part now and then come back later for the (slightly) more delicate singular case: If $(M,J)$ is a real-analytic almost-complex manifold and $X,Y\subset M$ are almost-complex su …
- 108k
4
votes
Accepted
Can we always solve this equation in the space of Hermitian structures on a complex vector b...
If $E$ has rank $0$ or $1$, 'yes', otherwise, 'no'. Just do a dimension count. You'll find that you have (many) more unknowns than equations and, for general $h$ and $h''$, there will be no solution …
- 108k
4
votes
Closed parallel (1,1)-forms on compact Kähler manifolds
Take a compact complex torus with a Kähler structure, say, an abelian variety, for example. Then there are lots of non-vanishing closed parallel $(1,1)$-forms, and their products will generate $(p,p) …
- 108k
3
votes
Is every positive $(n-1,n-1)$ form almost decomposable?
Not sure why you aren't asking for a $\chi$ such that $\alpha = \chi^{n-1}$, as this surely exists and doesn't require compactness. The point is that $\alpha$ is positive by construction and, for any …
- 108k
5
votes
Accepted
Pull-backs of $\sum dx_i \wedge dy_i$ under radial diffeomorphisms of $\mathbb C^n$
Yes, it's true, and it follows immediately from a simple calculation: Since $f$ is assumed to be a function of $|z|$ such that $\phi(z) = f(|z|)z$ is a smooth diffeomorphism, it follows that $\mathrm …
- 108k
3
votes
Analogue of Infinitesimal Schwarzian for holomorphic $(G,X)$-manifolds
To add to what Ben said: Typically the resolution of the sheaf mapping ${\frak{g}}\to\Gamma(TX)$ is the Spencer resolution, which is a canonical $G$-equivariant exact sequence of differential operato …
- 108k
2
votes
Accepted
pull back of a Kähler form by a smooth circle group action on a compact Kähler manifold
Now that I fully understand your question, which (apparently) is whether every circle action on a compact Kähler manifold $(M,J,\omega)$ must preserve the $J$-type of $\omega$, I can answer it. The a …
- 108k
6
votes
Accepted
Does $\partial\overline{\partial}f=0$ imply $f\equiv c$ for particular kind of $f$?
Yes, this is true. The point is that $\partial\bar\partial f = 0$ on the complement $C$ of a ball in $\mathbb{C}^n$ ($n\ge 2$) implies that $f = h_+ + \overline{h_-}$ for some functions $h_\pm$ that …
- 108k
11
votes
Accepted
How to deduce this equation for a 4-dim almost Kahler manifold?
Here is one way to do this computation:
Choose a local coframing $\eta = (\eta_i)$
in which $g = {\eta_1}^2{+}{\eta_2}^2{+}{\eta_3}^2{+}{\eta_4}^2$
and $\omega = \eta_1\wedge\eta_2{+}\eta_3\wedge\et …
- 108k
4
votes
Accepted
almost complex $\mathbb{Z}^{6}$-action
Here is a construction of a family of such examples that will work, but you will have to choose a particular map to get an explicit example.
Let's use coordinates $v_1,v_2,v_3,v_4,x,y$ (each periodic …
- 108k
4
votes
Accepted
about Kahler curvature tensor on page 77 of Besse's book "Einstein Manifolds"
Well, $B$, as defined in Besse on page 77, is not the Bochner curvature because it is not traceless when $m>1$ (cf. (2.64)). I believe that $B_0$ is some version of the Bochner curvature tensor.
I …
- 108k
9
votes
Accepted
First Chern class vanishes on a Lagrangian submanifold
Yes, if you ignore 2-torsion: After all, if you give $TM$ a complex structure compatible with $\omega$, say $J$, then you'll have that $TM$ pulls back to $L$ to become isomorphic to $TL\oplus T^*L\si …
- 108k
6
votes
Accepted
What is the definition of a Calabi-Yau metric on a non-compact manifold?
There are two slightly different definitions. The first is that it is a Kähler metric that is Ricci-flat, and the second is that it is a Kähler metric on a (usually connected) complex $n$-manifold wi …
- 108k
5
votes
Accepted
Can the $(n-1)$ power of the hermitian form on a compact complex $n$-fold be $\partial {\bar...
Yes, this can happen. For a simple example, consider $M = \mathrm{SL}(2,\mathbb{C})/\Lambda$ where $\Lambda\subset \mathrm{SL}(2,\mathbb{C})$ is a discrete, co-compact lattice. Then $M$ is a compact …
- 108k
12
votes
Accepted
Determine whether a (1,2) tensor is Nijenhuis tensor
Yes, there are pointwise algebraic conditions on a section $N$ of $T\otimes\Lambda^2(T^*)$ in order for $N$ to equal $N_J$ for some almost complex structure $J$, but there are differential conditions …
- 108k