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Results tagged with complex-geometry
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user 3377
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.
0
votes
Moduli space of complex and anti-complex tori?
For $d$ odd these two components are disconnected, because $I$ and $-I$ induce opposite orientation. For $d$ even, you have an involution, which takes a lattice to a complex conjugate lattice. This in …
- 12.9k
5
votes
Structure of Kähler cone
Explicit description of a Kahler cone for all hyperkahler manifolds is here:
https://arxiv.org/abs/1401.0479 (Rational curves on hyperkahler manifolds,
Ekaterina Amerik, Misha Verbitsky)
- 4,713
0
votes
Accepted
Holomorphic function on $\mathbb C^n$
This function has constant Jacobian by Liouville. Then it is
map of Jacobian 1 composed with a homothety or its differential is degenerate everywhere. The constant Jacobian biholomorphisms are subject …
- 9,237
7
votes
Complex manifolds whose tangent and cotangent bundles are isomorphic as complex vector bundles
There are many such examples, for instance, all complex nilmanifolds (and most complex solvmanifolds) have tangent bundle which is topologically trivial. The Hopf manifolds also have topologically tri …
- 9,237
5
votes
When Atiyah class and Chern class coincide?
I guess this is always true, if you adjust the statement appropriately.
Consider the Bott–Chern cohomology $H^*_{BC}(M):=\dfrac{\ker d\cap \ker d^c}{\operatorname{im} dd^c}$. Since the curvature of a …
- 12.9k
8
votes
Accepted
Does it make sense to define a holomorphic structure on $\mathbb{C}P^\infty$ and vector bund...
Yes, there is lots of literature on this subject.
However, Tyurin proved that all vector bundles on $CP^\infty$ are
direct sum of line bundles. There are several more recent papers by
Penkov and Tikho …
- 35.5k
1
vote
Accepted
Can deformation equivalent Kähler manifolds always be obtained by a deformation where all th...
I don't think this is known. For hyperkahler manifolds, conjecturally,
all smooth complex deformations are class C and birational to hyperkahler.
If this is true, your conjecture would follow automati …
- 9,237
1
vote
Different algebraic structures on complements to divisors
Do you know other examples of non-isomorphic algebraic
structures on complements to square-zero curves
The easiest example is
the twisted cotangent bundle to an elliptic curve.
This space can be rea …
- 9,237
3
votes
Accepted
Bott-Chern cohomology for singular complex spaces
closed (1,1)-forms and currents on X
are not necessary locally $dd^c$-exact in general
What makes it different when X is singular?
The obstruction to local $dd^c$-lemma
is $R^1\pi_*(O_{X'})$, where
…
- 9,237
8
votes
Coincide between Chern-connection and Levi-Civita connection
It is easier to prove this result for 1-forms, instead of vector fields. On (1,0)-forms, $\nabla^{0,1}=\bar\partial$ because the Levi-Civita connection is torsion-free, hence $\bigwedge(\nabla(\eta))= …
- 9,237
5
votes
automorphism group of K3 surfaces
Calabi-Yau theorem implies that any diffeomorphism of a Calabi-Yau manifold which preserves
the complex structure and the Kahler class also preserves the Calabi-Yau metric. However, the group of isom …
- 9,237
10
votes
How restrictive is having zero Chern numbers for a compact complex manifold ? Same for negat...
When a compact Kahler manifold satisfies $c_1=0$, it admits a Ricci-flat Kahler metric by Calabi-Yau, hence its tangent bundle is polystable (direct sum of stable bundles of the same slope). Then its …
- 9,237
1
vote
Curvature forms of holomorphic line bundles
Sure, any closed 2-form $\eta$ with integer cohomology
can serve as the curvature of a connection
on a line bundle. This can be seen if you take a line bundle with
the same Chern class and connection …
- 9,237
3
votes
Can a non-Kähler complex manifold be rationally connected?
As Jason said already, there are many
examples of Moishezon manifolds which
are rationally connected. Indeed, any
manifold bimeromorphic to a rational
connected manifold is again rationally
connected, …
- 9,237
2
votes
Holomorphic vector fields acting on Dolbeault cohomology
Klemyatin proved that this action is trivial if the corresponding
${\Bbb C}$-flow is compatible with some metric (hence can be extended
to a compact torus action),
https://arxiv.org/abs/1909.04075,
(N …
- 9,237