# Tag Info

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 ...
• 12.2k
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 ...
• 15.5k

### 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 ...
• 31.1k

### 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 ...
• 13.3k
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 ...
• 19.2k

### 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 ...
• 90.6k

### 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,...
• 154k
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 ...
• 4,111

### 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 ...
• 659
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 ...
• 65.8k
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 ...
• 65.8k
Accepted

• 26k

### 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 ...
• 34.9k
Accepted

• 15.5k