40
votes

Accepted

### Elementary Proof of Riemann-Roch for Compact Riemann Surfaces

RRT
There is a big difference in difficulty between the compact Riemann surface case and the projective curve case, for reasons already mentioned. Namely a projective curve comes equipped with a ...

18
votes

Accepted

### Is every smooth projective variety contained in a chain of smooth projective varieties of increasing dimension?

Suppose that $\operatorname{dim}(X)>1$ and that such a chain exists. Since $\operatorname{Pic}(\mathbf{P}^n)\simeq \mathbf{Z}$, the variety $X_{k-1}$ is an ample divisor in $\mathbf{P}^n$, and ...

16
votes

### Can the group of holomorphic automorphisms of an open subset of the complex plane be isomorphic to the additive group of real numbers?

No, we cannot.
Let $S$ be a connected noncompact Riemann surface whose group of conformal automorphisms is nondiscrete. Then (this follows from the uniformization theorem) $S$ is either conformal to ...

15
votes

### Orientable with respect to complex cobordism?

Let me expand a bit on my comments. If $E$ is a nice enough ring spectrum (e.g. an $\mathbb{E}_2$-ring spectrum; there is also a slightly modified version that works for an $\mathbb{E}_1$-ring) then ...

15
votes

Accepted

### Exotic $\mathbb{R}^4$ with a complex structure?

It is a result of Gromov that an open manifold of dimension six or less admits a complex structure if and only if it admits an almost complex structure; see the corollary on page 103 of his book ...

14
votes

### Can the group of holomorphic automorphisms of an open subset of the complex plane be isomorphic to the additive group of real numbers?

It cannot be $R$. Consider two cases:
a) Your region $D$ is hyperbolic Then $D=H/G$, where $H$ is the upper half-plane,
and $G$ a discrete group. Let $\Gamma$ be the pullback of your group of ...

13
votes

### Elementary Proof of Riemann-Roch for Compact Riemann Surfaces

Joe Harris, as recorded in his course notes here, gives the following slick proof when both $D$ and $K-D$ are effective; it has the advantage of never mentioning $H^1$. See lecture 1 for this argument,...

13
votes

Accepted

### Artin vanishing for Stein manifolds and restriction maps

The pairs (U,X) are called Runge pairs. The homology version of your statement is proved in the paper of
Andreotti and Narasimhan Annals of Math vol 76 no 3 (1962) 499-509 using Morse
Theory.The title ...

13
votes

### Proving algebraicity of compact Riemann surfaces without Chow's theorem

Given an embedding $X\to \Bbb P^n$, it's possible to construct polynomial equations for the image.
Theorem (Narasimhan 9.6.2): Let $X$ be a compact Riemann surface and $f$, a nonconstant meromorphic ...

12
votes

Accepted

### Examples of non-Kähler compact complex manifolds which satisfy the Dolbeault isomorphism

A compact complex manifold $X$ satisfies the Hodge decomposition $$H^k_{\mathrm{DR}}(X, \, \mathbb C) = \bigoplus_{p+q=k}H^{p, q}(X)$$ (possibly without Hodge symmetry) if its Frölicher spectral ...

12
votes

Accepted

### Is every surjective holomorphic self-map on a compact complex manifold finite-to-one?

Let me give a sketch of proof for Gromov's claim in the case where $X$ is Kähler. More precisely, let me prove the following ${}$
Proposition [G03, p.223]. Let $X$, $Y$ be two complex manifolds (not ...

12
votes

Accepted

### Holomorphic Urysohn Lemma

Yes, since the morphism of sheaves $\mathcal{O}_{\mathbb{C}^n} \to \mathcal{O}_{M \cup N}$ is surjective, it is surjective on global sections by Cartan's theorem B. Thus, any holomorphic function on $...

12
votes

Accepted

### Is there a non-singular cubic surface that has a point where four lines intersect?

No, this is not possible. If p is a smooth point on any surface S, and is
contained in a line l on S, then l is contained in the tangent plane at p,
call it T_p. Now if S is a cubic then it ...

12
votes

Accepted

### Do non-projective K3 surfaces have rational curves?

Some of them do, and some don't.
Indeed, by global Torelli theorem, there is a K3 surface $X$ with $\mathrm{Pic}(X) = 0$. Such $X$ has no curves, in particular no rational curves.
On the other hand, ...

11
votes

Accepted

### How does one complexify a real $n$-dimensional Riemannian manifold $(M,g)$?

I believe the following is meant:
Every smooth (real) manifold $M$ has a (unique) real-analytic structure compatible with the smooth structure. So, cover $M$ with real-analytic charts, i.e. whose ...

11
votes

Accepted

### An almost complex structure on the real $n$-sphere $S^n$

Let $M=SU_3$, the compact semisimple Lie group.
By request of the OP, for those unfamiliar with Maurer-Cartan form, let me define it. Write each point of $SU_3$ as a matrix $g$. Left translation by $g^...

10
votes

### Examples of compact complex manifolds for which the $dd^c$ lemma does not hold

A known consequence of the $dd^c$-lemma is the vanishing of Massey products, which are certain secondary cohomology operations, see Deligne, Griffiths, Morgan, Sullivan, Real homotopy theory of Kähler ...

10
votes

Accepted

### Can a class be represented by both a $(p,q)$-form and a $(p',q')$-form?

I think you want an example of a compact complex manifold $X$ and differential forms $\gamma \in \mathcal{E}^{p,q}(X)$ and $\gamma'\in \mathcal{E}^{p',q'}(X)$ with $(p',q') \neq (p, q)$ such that $[\...

10
votes

Accepted

### Is the Bott-Chern/Aeppli cohomology determined by the de Rham and Dolbeault cohomologies?

I learnt all of the following from section $3.2$ of Angella's Cohomological Aspects in Complex Non-Kähler Geometry.
Let $R$ be a commutative ring with identity. The three-dimensional Heisenberg group ...

10
votes

Accepted

### Status of a conjecture of Hirzebruch

Suppose $X$ is diffeomorphic to $S^2\times S^2$ or $\mathbb{CP}^2\#\overline{\mathbb{CP}^2}$. Then $X$ is biholomorphic to a Hirzebruch surface.
Note that $b_1(X) = 0$, so $X$ admits a Kähler metric. ...

10
votes

Accepted

### Threefolds with the same Betti numbers and the same Chern numbers

The complex parallelizable (hence, all Chern classes are trivial) Iwasawa manifold is constructed by taking the complex Lie group of matrices of the form $$\begin{pmatrix} 1 & x & z \\ 0 & ...

9
votes

### Is the Bott-Chern/Aeppli cohomology determined by the de Rham and Dolbeault cohomologies?

As for compact complex surfaces, the Bott-Chern/Aeppli cohomologies are determined by de Rham/Dolbeault cohomologies:this is contained in Lemma 2.3 in:
Teleman, A.: The pseudo-effective cone of a non-...

9
votes

### Vanishing of Aronhold S-invariant on the cubic forms on $H^2(X, \mathbb Q)$

The Aronhold invariant vanishes when the form has border rank $\le 3$, i.e., lies in the Zariski closure of sums of three cubes of linear forms. I don't know how your specific cubic looks like but if ...

9
votes

### Top integer homology of compact analytic variety

This is indeed the case. In fact in the general case (possibily non compact) an irreducible complex analytic subvariety V has a fundamental class in Borel-Moore homology (in singular homology when $V$ ...

8
votes

### Automorphism group of compact almost complex manifold

See Kobayashi, Transformation Groups, Theorem 4.1 page 16, where the theorem is proved that the group of automorphisms of a smooth compact almost complex manifold is a finite dimensional Lie group ...

8
votes

### Given a smooth hyperplane section Y of a variety X there exists a Lefschetz pencil of hyperplane sections of X containing Y

A pencil of hyperplane sections of $X$ corresponds to a line in the dual projective space $\check{\mathbb{P}}^N$. It is a Lefschetz pencil if and only if the line is transverse to the projectively ...

7
votes

### Relation between $h^1(\mathcal{O}_X)$, $h^0(\Omega_X)$, and first betti number for general complex manifold?

There is a spectral sequence $$ E^{pq}_1 = H^q(X, \Omega^p_X) \quad\Rightarrow\quad H^{p+q}(X, \Omega^\bullet_X) \cong H^{p+q}(X, \mathbb{C})$$
(which degenerates if $X$ is Kahler). In particular, $h^...

7
votes

### Elementary Proof of Riemann-Roch for Compact Riemann Surfaces

The proof given in Otto Forster, Lectures on Riemann Surfaces (Graduate Texts in Mathematics 81), chapter 16, seems very much suited to your list of prerequisites.

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