If we have a complex Morse function on a complex four-manifold, $f: X\to \mathbb{C} $, can we tell from the function how the genus of inverse images $f^{-1}(z)$ (for regular values …

Good morning, I'm just curious about the following. With a compact Kahler manifold, we can associate an Albanese torus. This helps us a lot study the manifold. My question: Are …

Is there any necessary and sufficient condition for existence of $SU(3)$-structure on 6-manifolds $M$?

This is a question about pathologies. Let $X/\mathbb{C}$ be an irreducible projective variety smooth over $\mathbb{C}$. Then, the singular cohomology groups $H^i(X, \mathbb{C})$ h …

I have two questions on complex geometry. First one is that why the existence of almost complex structure on tangent bundle on real 2n-dimension manifold is a topological question …

I am trying to solve exercise in Huybrechts's book 'Complex geometry' While solving problems, one problem kept me from going forward. That is, The surface $\Sigma_n=\mathbb{P}$ …

When a Riemannian manifold is of Hessian Type (i.e., a Riemannian manifold which its metric is Hessian)

Its known ( see " The birational geometry of degenerations") that there exist a smooth one parameter family (i.e. total space is smooth) of two dimensional complex toris over unit …

Let $X$ be a projective complex manifold. Under what condition do we have the equality $Aut(X)=Bir(X)$? Here $Aut(X)$ denotes the group of holomorphic automorphisms of $X$ and $Bir …

Let $f:X\rightarrow C$ be a fibration of complex manifold over a smooth curve $C$. Let $D$ be an irreducible component of singular fibers. How can one prove that $X$ does not admit …

In her book, after proving the hodge decomposition, Voisin spends time discussing primitive cohomology $H^r(X, \mathbb{C})_{prim} = \ker L^{n-r+1} \subset H^r(X, \mathbb{C})$ (wher …

I've been interested in the possible (singular) cohomology groups of complex projective algebraic varieties, and there are lots of theorems that give various restrictions on these …

Is there new classification of Kahler manifolds of complex dimension n and new results for necessary and sufficient conditions for a manifold being Kahler? I know if redactivity of …

A compact Riemannian manifold is called geometrically formal if the wedge product of two $d$-harmonic forms is $d$-harmonic. Are there any known results for when a non-Kahler com …

Let $X/k$ be a surface nonsingular and proper over an algebraically closed field $k$. Let $C \subset X$ be a nonsingular curve. Then it is clear that the self-intersection $(C \cdo …