# Tagged Questions

**6**

votes

**1**answer

242 views

### Can we normalize a complex analytic space in a covering of an open subset?

Let $X$ be a normal connected complex analytic space, $x\in X$ a point, $f$ a nonzero holomorphic function vanishing at $x$. Denote by $U\subseteq X$ the locus where $f$ is nonzero. Suppose that ...

**8**

votes

**1**answer

200 views

### Different notions of convergence of complex subvarieties

Let $X$ be a smooth complex algebraic variety (or, better, complex analytic manifold). Let $\{C_i\}$ be a sequence of compact algebraic subvarieties (resp. analytic reduced subspaces) which converges ...

**1**

vote

**1**answer

137 views

### The locus of rational/elliptic curves on a special surface in $\mathbb{P}^3$

Let $P$ and $Q$ be two general polynomials of the same degree $d>5$. Consider the surface $S: z^2=P(x)Q(y)$ in $\mathbb{P}^3$ (after homogenization by the variable $w$). One can show that these ...

**1**

vote

**0**answers

107 views

### Exactness of the relative de Rham complex restricted to subschemes

I think that the statement below about relative de Rham complex is true (am I wrong?) If it is the case, a reference would be very helpful. (I admit that the statement sounds somewhat technical and ...

**0**

votes

**1**answer

115 views

### Hilbert scheme of a closed subscheme

Let $X$ be a complex algebraic variety. Its Hilbert scheme represents the functor $G$ from schemes to sets given by $$G(S)=\{Z\subset X\times S|\, Z \mbox{ is a closed subscheme, flat and proper over ...

**1**

vote

**1**answer

117 views

### Hilbert scheme of an infinitesimal neighborhood of a subvariety

Let $X$ be a complex algebraic variety. Let $C\subset X$ be a compact (reduced) subvariety. Let $C^{(n)}$ denote the $n$th infinitesimal neighborhood of $C$ inside $X$. Let $Hilb(X)$ denote the ...

**2**

votes

**2**answers

256 views

### Basic questions on the Hilbert scheme/ Douady space

Let $X$ be a complex projective scheme (resp. complex analytic space). The Hilbert scheme (resp. Douady space) parameterizes closed subschemes (resp. complex analytic subspaces) of $X$. More ...

**3**

votes

**1**answer

143 views

### Flat family with special fiber $\mathbb{C}\mathbb{P}^1$

Let $C=Spec \mathbb{C}[t]/(t^{n+1})$. Let $X$ be an algebraic (or complex analytic) scheme, flat over $C$ with the structure morphism $f\colon X\to C$. Assume that the special fiber is isomorphic to ...

**1**

vote

**1**answer

285 views

### When flatness of a morphism implies smoothness?

EDIT: Let $f\colon X\to C$ be a flat proper morphism of complex algebraic (or analytic) varieties. Assume the special fiber over a point $p\in C$ is smooth.
Is it true that there exists a ...

**1**

vote

**1**answer

226 views

### A generalization of the Grauert direct image theorem

EDIT: Let $f\colon X\to Y$ be proper holomorphic submersive map of complex analytic manifolds. Let $\mathcal{F}$ be the sheaf of holomorphic sections of a holomorphic vector bundle over $X$. Assume ...

**1**

vote

**2**answers

648 views

### Spicing up Riemann surfaces course (revised)

I am a master's student planning to write a master's thesis on Riemann surfaces. I plan to study Forster's Lectures on Riemann surfaces. What side topics could one study to spice up the thesis? I am ...

**10**

votes

**2**answers

583 views

### Is there a holomorphic Morse-Bott lemma?

It asks for a generalization of the question in the post
Normal form for a holomorphic Morse function
Suppose $f$ is a holomorphic function on a complex manifold $M$ which has Bott type critical ...

**1**

vote

**0**answers

90 views

### Algebraic set given by sequence of polynomials

When working on some problem, I have end up with a following situation. Suppose
$P(z)=z^d+a_{d-1}z^{d-1}+\ldots+a_1z+a_0$ is a complex polynomial $d\geq2$ and that $\gamma$ and $\delta$ are non-zero ...

**2**

votes

**1**answer

299 views

### Definition of a complex space

In the definition of a complex space (in the sense of Grauert), one defines a model space (one to which we require a complex space to be locally isomorphic) to be the support of the quotient sheaf ...

**3**

votes

**1**answer

298 views

### Automorphisms of complete discrete valuation ring

Let ${\Bbb F}_2[[T]]$ be a c.d.v.r over ${\Bbb F}_2$. We consider the automorphism $\sigma$ of ${\Bbb F}_2[[T]]$ such that $\sigma \colon T \mapsto T + c_2T^2 + c_3T^3 + \cdots$ with $c_i \in {\Bbb ...

**0**

votes

**0**answers

48 views

### Finding stable ideals of F_3[[X,S]] by group action.

Let k > 1 be a positive integer and define the action σ_k on F_3[[X,S]] by
σ_k: X ---> X + S + X^k
σ_k: S ---> S + S^3.
Then,
Conjecture: There exists a principal ideal (a) other than (S) such ...

**2**

votes

**2**answers

290 views

### Topological information via cohomology of sheaves

On a projective smooth variety $X$ over complex numbers (or rather compact Kahler) we have a specific set of sheaves, namely sheaves of holomorphic forms ${\mathcal \Omega}^p$ of various degrees. The ...

**3**

votes

**1**answer

258 views

### Normal form for a holomorphic Morse function

Similarly to Morse lemma, a holomorphic Morse function can be written, near a critical point, as $W_1^2+W_2^2+...+W_n^2+C$ , for well chosen coordinates $W_1,W_2,...,W_n$. I want to cite this result ...

**3**

votes

**1**answer

151 views

### Weighted projective space with rational or real weights

The most common formulation of the weighted projective space is perhaps the global quotient
$$
(\mathbb{C}^{n+1} \setminus \{(0,\ldots,0\}) / \mathbb{C}^\ast
$$
with the $\mathbb{C}^\ast$ group action ...

**8**

votes

**1**answer

288 views

### Local polynomial form of holomorphic functions

It is well-known that a germ of a holomorphic function of $n\geq 2$ variables, with at most an isolated singularity (i.e. the singularity of the analytic variety defined by the zero locus of the ...

**4**

votes

**2**answers

410 views

### When is $\mathrm{Aut}(X)\times \mathrm{Aut}(Y)$ of finite index in $\mathrm{Aut}(X\times Y)$?

Let $\mathrm{Aut}(X)$ denote the group of biholomorphic autmorphisms of the (non-compact) complex manifold $X$. If $X$ and $Y$ are two (non-compact) complex manifolds, then $\mathrm{Aut}(X)$ and ...

**1**

vote

**0**answers

56 views

### irreducible analytic decomposition of sets invariant under a group action

Let $U$ be a complex analytic space with an action of a finitely generated group $\Gamma$. Under what assumptions
is the following true:
Every $\Gamma$-invariant closed analytic subset of $U$ ...

**2**

votes

**0**answers

122 views

### topology generated by irreducible componets of $\Gamma$-invariant closed sets

For an analytic space $U$ equipped with an action of a group $\Gamma$,
call a subset $Z\subseteq U$ $\Gamma$-closed iff
it is a closed analytic subset and each of its irreducible components
is an ...

**1**

vote

**0**answers

54 views

### Is it obvious that the defining conditions to obtain a particular singularity are well defined on the quotient space?

Let $~f:\mathbb{C}^2 \rightarrow \mathbb{C}$ be a holomorphic function
vanishing at the origin, with
the following properties:
$$ f_{00}, ~f_{10}, ~f_{01}, ~f_{20}, ~f_{11} =0,~~f_{20} \neq 0 ...

**0**

votes

**0**answers

233 views

### Local System and Gauss-Manin connection

Fix a complex manifold $X$. Then if we have a line bundle $L=\mathcal{O}(D)$ together with Gauss-Manin connection $\nabla: L \rightarrow L \otimes \Omega^{1}_X$, we get the locally constant sheaf $F$ ...

**1**

vote

**1**answer

74 views

### Finite construction of lacunary functions using algebraic and certain analytic operations

Algebraic functions have a discrete set of singularities. Lacunary functions, e.g. $f(z)=\sum_{n=0}^\infty z^{2^n}$, have a continuum of singularities at every point of the boundary of their disk of ...

**2**

votes

**2**answers

391 views

### What is the closure of space of polynomials in a dense subspace along with a marked point equal to?

EDIT
Let $\mathbb{C}^{m*}$ be the space of non zero polynomials of degree at most
$d$ in two variables. So an element of this space is essentially
$$ f:=f_{00} + f_{10} x + f_{01} y + \ldots ...

**5**

votes

**1**answer

208 views

### Zariski's main theorem in the complex analytic category

Hello,
I am looking for a reference to something like that: if $f\colon X\to Y$ is a finite (i.e., proper with finite fibers) morphism of reduced and irreducible normal (or at least smooth) complex ...

**0**

votes

**1**answer

338 views

### Dolbeault cohomology

Let $X=S^{2n-1} \times S^1$. I have to compute $H^{(1,0)}_{\bar{\partial}} (X)$ and
$H^{(0,1)}_{\bar{\partial}}(X)$ . I don't know how to do this but if we use Kunnet formula we
have that ...

**3**

votes

**2**answers

243 views

### j-invariant duplication, triplication and quintuplication formulae… how?

I am interested in finding the derivation of the duplication, triplication and quintuplication formulae for Klein’s j-invariant, which are equations (13) – (24) of the corresponding page (Klein’s ...

**1**

vote

**0**answers

289 views

### The relation between the weak Lefschetz theorem and the strong Lefschetz theorem

The Weak Lefschetz Theorem states that for a compact Kahler manifold, $Pic(X) \rightarrow H^{1,1}(X, \mathbb{Z})$ is surjective.
The Hard Lefschetz Theorem states that for a compact Kahler manifold ...

**3**

votes

**9**answers

894 views

### functions of one complex variable: geometric theory

Can someone recommend a good textbook on functions of one complex variable which have good chapters on geometric theory, in English?
When I studied complex analysis, I used two
textbooks:
An ...

**16**

votes

**2**answers

437 views

### A direct proof of the Harer-Zagier recursion enumerating the ways to paste a 2n-gon to get a genus g surface?

In a 1986 paper, Harer and Zagier proved the recursion:
$$(n+1)e(g,n)=(4n-2)e(g,n-1)+(2n-1)(n-1)(2n-3)e(g-1,n-2)$$
where e(g,n) is the number of ways of grouping sides $S_1...S_{2n}$ of a 2n-gon ...

**2**

votes

**2**answers

348 views

### Bolza curve admits no anticonformal fixedpointfree involution

The Bolza curve B double covers the Riemann sphere with branching at the vertices of a regular octahedron. An affine model is given by the locus of $y^2=x^5-x$. How does one show that B does not ...

**1**

vote

**0**answers

198 views

### bivariate polynomial

Hello,
Let $p(x,y) = \sum_{m=1}^M\sum_{n=1}^N a_{m,n}x^{m-1}y^{n-1}$ be a bivariate polynomial where $\{a_{m,n}\}$ are complex.
If $(x_k, y_k), k=1,2,\cdots, MN-1$ are roots of $p(x,y)=0$ where ...

**3**

votes

**1**answer

459 views

### Complete intersections in complex and algebraic geometry

I'm wondering why (and therefore also if) the notions of "a projective variety/submanifold of projective space is a complete intersection" as used in algebraic geometry and the theory of, say, Riemann ...

**2**

votes

**1**answer

121 views

### on the density of hypersurfaces in complex projective spaces

Good morning,
Let $H$ be a hypersurface in a complex projective space $\mathbb{CP}^N.$ Let $d$ be the distance de Fubini-Study on $\mathbb{CP}^N.$
Let $x = [x_0: \ldots :x_N]$ and ...

**5**

votes

**0**answers

205 views

### Automorphisms of Compact Riemann Surfaces

I read a statement that for a compact Riemann surface $C$ with genus $g\geq 2$, one has
for the Jacobian $J(C)$ of the curve $C$:
$$ Aut (J(C))\sim Aut C$$
when $C$ is hyperelliptic and
...

**6**

votes

**1**answer

856 views

### Poincare line bundle

I am being stuck by the proof of the existence of Poincare line bundle of complex torus in Griffiths-Harris. Here is the question:
Let $M$ be a complex torus and $M'$ be the complex torus dual to ...

**5**

votes

**0**answers

135 views

### Perturbations of zero-dimensional algebraic varieties

Let $P(z,w):\mathbb C^2\to\mathbb C$ be a certain polynomial, and consider $p(s,t)=P(e^{is},e^{it}):\mathbb T^2\to \mathbb C$ its restriction to the real torus. Assume generic situation, so by ...

**2**

votes

**0**answers

171 views

### ALE Kähler manifolds are birational to deformations of $\mathbb{C}^n/G$.

I am reading Dominic Joyce's book 'Compact Manifolds with Special Holonomy' and I am struggling to understand a remark he makes at the end of chapter 8. The assertion is (I think) the following:
...

**5**

votes

**0**answers

156 views

### Complex manifold with non-finitely generated canonical ring

P.M.H. Wilson has an example of a compact non-Kahler manifold whose canonical ring is not finitely generated; see his article and this MO question. I'm trying to understand his construction and have ...

**1**

vote

**0**answers

68 views

### Does a (NOT necessarily positive) current have a decomposition formula?

It is well-known that for any positive (1,1)-current $T$, there is a decomposition formula according to [Siu74]. That is, $T$ can be written as an infinite sum of prime divisors plus an extra part. In ...

**15**

votes

**1**answer

844 views

### Any algebraic substitute for Morse theory (and homology) in arbitrary characteristic?

As far as I know, Morse theory yields much information on the topology of smooth manifolds; in particular, it can be used to prove Artin's vanishing (that the singular cohomology of smooth complex ...

**3**

votes

**1**answer

215 views

### Smoothness of solution to a PDE

Let $X$ be a Riemann surface and let $E$ be a smooth complex vector bundle on $X$ with a connection $D$. We can write the connection $D$ as the sum $D'+D''$ where $D'$ is the (1,0) part and $D''$ is ...

**3**

votes

**3**answers

318 views

### When can Hodge filtrations (decompositions?) be described explicitly in terms of periods?

It seems that there is no chance to explain the Hodge theory (to students) in an hour or so. Yet do there exist any cases when the Hodge filtration (or the Hodge decomposition) of the cohomology of a ...

**1**

vote

**1**answer

255 views

### Automorphic and modular forms for subgroups of modular group and fuchsian groups

Is there a well-understood correspondence between subgroups G of $SL_2(\mathbb{Z})$ (not necessarily of finite index) and graded algebras of modular forms invariant under G?
Given an algebra of ...

**11**

votes

**4**answers

2k views

### Geometric invariant theory for geometers

I am trying to learn "Geometric invariant theory" like it was introduced by Mumford. But I do not have a strong background in algebraic geometry since I work in geometric topology and geometry.
So ...

**5**

votes

**0**answers

342 views

### Jet differentials and hyperbolicity: possible mistake in the literature?

I was reading this note by Jingzhou Sun http://arxiv.org/abs/1109.1329
about Demailly's approach to hyperbolicity using jet differentials. The author seems to claim that there is a mistake in one of ...

**12**

votes

**2**answers

762 views

### Determining rational functions by their critical points

Fix an integer $d > 1$ and $2d-2$ points $P_1, \ldots, P_{2d-2}$ in the Riemann sphere (not necessarily distinct). Thanks to the work of Eisenbud and Harris on limit linear series (Inventiones, ...