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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
4
votes
2
answers
238
views
Locality of Kähler-Ricci flow
Let $(M,I, \omega)$ be a compact Kähler manifold with $c_1(M)=0$. Denote by $\operatorname{Ric}^{1,1}(\omega)$ the Ricci (1,1)-form, that is, the curvature of the canonical bundle. It is known ("Defor …
- 6,001
4
votes
1
answer
241
views
Equivariant projective embeddings with optimal dimension
Let $X$ be a complex projective manifold, and $f\in Aut(X)$ an automorphism, which is linearizable, that is, can be extended to an ambient projective space ${\mathbb P}^m$. I am interested to find wha …
- 6,995
8
votes
2
answers
373
views
Real analytic subvariety in complex manifold which is complex outside of its singular set
Let $M$ be a complex manifold, and $Z \subset M$ a closed real analytic subvariety. Suppose that the set of smooth points in $Z$ is complex analytic in $M$. Will it follow that $Z$ is complex analytic …
- 513
3
votes
0
answers
231
views
Kawamata BPF applied to a semi-positive line bundle using Demailly's holomorphic Morse inequ...
Let $M$ be a compact complex manifold equipped with a line bundle $L$ which has curvature which is non-negative and strictly positive outside of a measure zero set $Z$. In his paper "Holomorphic Mors …
- 295
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
8
votes
2
answers
901
views
Gorenstein varieties: why the two definitions are equivalent?
There are two definitions of Gorenstein singularities
in the literature. Using Grothendieck's (or Serre's) duality, one
defines the "dualizing sheaf" an object $\hat K_M$ of derived category
of cohere …
- 42.9k
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