Do complex schemes locally deformation retract onto closed subschemes in the analytic topology?

Let X be a scheme of finite type over C and let Z↪X be a closed subscheme. Consider the associated closed inclusion Zan↪Xan between their analytifications (regarded as topological spaces). Is this a strong neighborhood deformation retract? By this I mean, can I find for every neighborhood U of a point z in Zan another … Read more

Why is the Hodge Conjecture so important?

The Hodge Conjecture states that every Hodge class of a non singular projective variety over $\mathbf{C}$ is a rational linear combination of cohomology classes of algebraic cycles: Even though I’m able to understand what it says, and at first glance I do find it a very nice assertion, I cannot grasp yet why it is … Read more

Hodge-Weil Formula for Quaternionic-Kähler manifold

Let M be a quaternionic-Kähler manifold, with fundamental form ω, and let L be the Lefschetz operator of ω. In the Kähler and, more generally, symplectic cases, there is a mysterious relation between the Lefschetz sl2-decomposition and the Hodge operator due to Weil (see Huybrechts 1.2.31) Does there exist a quaternionic-Kähler analogue? Answer AttributionSource : … Read more

Can one integrate around a branch-cut?

How meaningful is it to try to integrate around the branch-cut of a function? For example lets say I have the function log(z2+a2) for a>0 and I choose my branch-cuts to be starting at ±ia and moving up and down the y−axis respectively. Now I am trying to integrate around a small circle around such … Read more

classification of homogenous complex manifolds

Suppose X is a complex manifold (doesn’t assume it’s Kahler), and it’s holomorhpic automorphism group is transitive. My question is that is there any classification of those manifolds ? Answer AttributionSource : Link , Question Author : user42804 , Answer Author : Community

Moduli Spaces of Higher Dimensional Complex Tori

I know that the space of all complex 1-tori (elliptic curves) is modeled by SL(2,R) acting on the upper half plane. There are many explicit formulas for this action. Similarly, I have been told that in the higher dimensional cases, the symplectic group Sp(2n,R) acts on some such space to give the moduli space of … Read more

Deformations of the moduli space of ppav’s

Consider the complex algebraic moduli space X:=Ang of ppav’s of dimension g with some high enough level n structure (so that it represents the corresponding functor). Can one compute the dimension of the infinitesimal space of deformations to X (as a function of g and the level)? If V is a smooth projective variety, then … Read more

Some examples where the plurigenera are nonconstant, when the fibres have worse singularities than canonical

Let start with a definition Invariance of plurigenera: Choose $m$ large enough so that $mK_F$ has a non-zero global section for some fibre $F$. For any fibre $F$, we have $K_F = K_{X/D}~_{|F}$. So deformation invariance of plurigenera says that the function $$\dim H^0(X_p, m K_{X/D}~_{|X_p})$$ is constant on $D$ (where now $X_p$ denotes the … Read more

K-stability on Fano fibration

Motivation: Let $\pi:X\to B$ be a holomorphic fibre space. By theorem 1.3 of Kawamata, if the central fibre be of the general type then all the fibres are of the general type see , this tells you that if the central fibre $X_0$ has Kahler-Einstein metric with negative Ricci curvature, then all the fibres … Read more

Cauchy-Riemann Operators and Selberg Zeta Function

The determinant of hyperbolic Maaß-Laplacian operator on arbitrary tensors and spinors can be written in terms of Selberg zeta function. Is there a corresponding formula for the determinant of the Cauchy-Riemann operators $\det\overline{\partial}_j$ which act on the space of $j$-differentials (i.e. arbitrary tensors and spinors) in terms of Selberg zeta function? Answer AttributionSource : Link … Read more