Articles of complex geometry

Canonical symplectic form on cotangent bundle of complex manifold

Given any smooth manifold $M$, its cotangent bundle $T^*M$ is a symplectic manifold, with the canonical symplectic form. If $M$ is a complex manifold then $T^*M$ is also a complex manifold. Thus, $T^*M$ is a complex symplectic manifold. Does it follow that the canonical symplectic form is holomorphic? If not, what condition can be placed […]

{line bundles} $\neq$ {divisor line bundles}

Let $X$ be a compact complex manifold, and with the following sheaves $\mathscr O$, the sheaf of holomorphic function, $\mathscr O^*$, the sheaf of nonvanishing holomorphic function $\mathscr K^*$ the sheaf of nonidentically zero meromorphic function. A divisor is an element in $\Gamma(X, \mathscr K^*/\mathscr O^*)$ (Cartier divisor). The short exact sequence of sheaves $$ […]

The complex structure of a complex torus

A complex torus $X$ of complex dimension $1$ is just a quotient of $\mathbb{C}$ by a lattice $\Lambda=\mathbb{Z}\oplus \tau\mathbb{Z}$ where $\tau=a+ib$ is a complex number with $b>0$. The complex structure is determined by this $\tau$. I want to undestand this complex structure as an endomorphism $J:TX \to TX$ such that $J^2=-1$ (an almost complex structure). […]

Kähler form convention

I’ve been wondering about this for a while and I have my ideas about the answer, but I would like to make sure once and for all that I’m not missing something. Let’s look at this from a purely linear algebra perspective: let $h$ be a Hermitian inner product on a complex vector space. Should […]

Checking flat- and smoothness: enough to check on closed points?

I am currently studying varieties over $\mathbb{C}$, i know some scheme theory. Let $f: X \rightarrow Y$ be a morphism of varieties. If we want to show flatness, is it enough to check the condition only at the closed points of $Y$? If yes, could you give an argument, if not, could you give an […]

A linear system of a curve on a K3 surface.

Let $S$ be a K3 surface and $C \subset S$ be a smooth curve of genus $g$ (assume it represents a primitive homology class). Is it possible to compute the dimension of the linear system $|C|$ only from these data?

Some questions about $S^n$

I have some questions about the $n$-sphere: I know that for $n=0,1,3$, $S^n$ forms a Lie group and I also know why it’s true, but why is it not the case for other $n$? I have the same question for the $n$-spheres admitting an almost complex structures $(n=2,6)$. Is there a general reason to conclude […]

The sum of the residues of a meromorphic differential form on a compact Riemann surface is zero

How can one see that the sum of the residues of a meromorphic function on a Riemann surface $ \Sigma_g$ of positive genus is always zero? This is not true for the Riemann sphere $\mathbb{CP}^1$.

Why is an analytic variety irreducible provided that the set of its smooth points is connected?

Here is a proposition from Griffith and Harris, Principles of Algebraic Geometry. Let $V$ be an analytic variety on a complex manifold $M$, let $V^*$ be its points of smooth points. If $V^*$ is connected, then $V$ is irreducible. The proof in the book is very short. It goes like this: Suppose $V=V_1\cup V_2$ is […]

Proof that the Nijenhuis tensor vanishes in a complex manifold

I’m in trouble proving that if $(M,J)$ is a complex manifold with $J$ a compatible almost complex structure then the Nijenhuis tensor of $J$ vanishes: in other words I would like to find that for any two vector fields $X,Y$ one has $$ J[X,Y]=J[X,JY]+J[JX,Y]+[X,Y] $$ I tried applying all the definitions of commutator I actually […]