Incidence structures on algebraic curves

Posted on June 7, 2016 by Dima

tags: incidence structures, zariski geometries

Abstract projective geometry

Let \(k\) be a field. Consider the set of points \(P\) and lines \(L\) on the projective \(n\)-space \({\operatorname{\mathbb{P}}}^n(k)\) over \(k\). One finds that the following properties hold:

A tuple \((P,L,I)\) where \(P\) is the set of points, \(L\) is the set of lines and \(I \subset P \times L\) is the incidence relation, is an abstract projective geometry if the properties above hold. These properties in particular hold if \(P\), \(L\) are sets of points and lines on a projective plane over a (skew) field \(k\). There are examples of abstract projective geometries that do not arise this way. The reason is that all geometries arising from a (skew) field additionally satisfy Desargues axiom:

Given two triangles \(ABC\) and \(A'B'C'\) as above, the lines \(AB\) and \(A'B'\), \(BC\) and \(B'C'\), and \(AC\) and \(A'C'\) intersect in three points that lie on the same line.

There are two affine statements that follow from this axioms: called small and big Desargues axioms.

small Desargues: given \(A\) and \(A'\), \(B\) and \(B'\), \(C\) and \(C'\) lying on three parallel lines, \(BC\) and \(B'C'\) are parallel.

big Desargues: given that the lines \(AB\) and \(A'B'\), and \(AC\) and \(A'C'\) are pairwise parallel, the lines \(BC\) and \(B'C'\) are parallel.

If \(k\) is commutative, then a Pappus axiom is also satisfied:

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ddc lemma

Posted on May 4, 2016 by Dima

tags: kahler manifolds, ddc

In this note I will prove the easy local case of \(dd^c\) lemma.

Let \((M,I)\) be a complex manifold. Extend the action of the complex structure on exterior powers of the complexified tangent bundle forms by \[ \mathbf{I}: \bigwedge{}^* M \to \bigwedge{}^* M, \mathbf{I} := \sum i^{p-q} \cdot \Pi^{p,q} \] where \(\Pi^{p,q}: \bigwedge^{p+q} \to \bigwedge^{p,q}\) is the natural projection. Define a dwisted differential \(d^c = I \circ d \circ I^{-1}\). On the cotangent bundle define it to be \(\mathbf{I}(\alpha)(v_1, \ldots, v_n) := \alpha(\mathbf{I}v_1, \ldots, \mathbf{I}v_n)\).

Proposition. \(\partial = \dfrac{d + i \cdot d^c}{2}, \bar \partial = \dfrac{d - i \cdot d^c}{2}\), where \(\partial\) and \(\bar\partial\) are projections of \(d\) on \((\cdot +1, \cdot)\) and \((\cdot, \cdot+1)\) components.

Proof. Indeed \[ \partial + \bar\partial + i \mathbf{I}^{-1} (\partial + \bar\partial) \mathbf{I} = \partial + \bar\partial + i (i^{q-p-1} \partial + i^{q-p+1}\bar\partial)i^{q-p}=\\ = \partial + \bar\partial + \partial + i^2\bar\partial=2\partial \] Similarly for the second equality. \(\square\)

In particular, \(\partial\bar\partial = -\dfrac{i}{2}dd^c\) on a complex manifold.

Lemma (Poincaré lemma). If \(\alpha\) is a closed form on a polydisc then it is exact.

Lemma (Poincaré-Dolbeault-Grothendieck lemma). If \(\alpha\) is a \(\partial\)-closed, not holomorphic (i.e. \(\alpha \notin A^{n,0} M\)), form on a polydisc then it is \(\bar\partial\)-exact.

(this lemma means in particular that Dolbault resoltions of sheaves of holomorphic forms are acyclic)

Note that since \(\bar(\partial\alpha) = \bar\partial \bar\alpha\), the PDG lemma also holds for \(\partial\).

Lemma (local \(dd^c\) lemma). Let \(\eta\) be a \((1,1)\)-form on a polydisc. Assume either

Then there exists a function \(\chi\) such that \(\eta = dd^c \chi\).

Proof. First establish the equivalence of the assumptions. Recall that \(d^2=\partial^2=\bar\partial^2=0\), where \(d=\partial+\bar\partial\), and therefore \(\bar\partial\partial=-\partial\bar\partial\). If \(\eta = d\alpha\) then there exist forms \(\beta \in A^{1,0}, \gamma\in A^{0,1}\) such that \(\eta = \partial\beta + \bar\partial\gamma\) and \(\bar\partial\beta=\partial\gamma=0\). Then \(\partial(\eta) = \partial^2 \beta + \partial\bar\partial\gamma=-\bar\partial\partial\gamma=0\). Similarly for \(\bar\partial\eta\). Apply PDG and \(\partial\)-PDG to get \(\partial\) and \(\bar\partial\)-exactness of \(\eta\).

By Poincaré-Dolbeault-Grothendieck lemma, there exists a form \(\alpha \in A^{1,0}\) such tha \(\bar\partial\alpha=0\). Deduce \(\bar\partial\partial \alpha = -\partial\bar\partial \alpha=-\partial \eta = 0\).

Therefore \(\partial\alpha\) is a closed form: \((\partial + \bar\partial)\alpha = \partial^2 \alpha + \bar\partial\alpha=0\). By Poincar'e lemma, there exists a form \(\alpha'\) such that \(d\alpha' = \partial\alpha' + \bar\partial\alpha = \partial\alpha\). By grading considerations, \(\bar\partial\alpha = 0\).

Consider the form \(\beta = \alpha - \alpha' \in A^{1,0}\). Then \(\partial\beta = \partial\alpha - \partial\alpha' = 0\) and \(\bar\partial\beta = \bar\partial\alpha - \bar\partial\alpha' = \eta\). By \(\partial\)-PDG, there exists a function \(\psi\) such that \(\partial\psi = \beta\), so \(\bar\partial\partial\psi=\eta\). Put \(\chi=-i\psi\). \(\square\)

Interestingly, this statement is true globally, i.e. for global \((p,q)\)-forms on a manifold \(M\), if the manifold is Kähler.

Lemma (global \(dd^c\) lemma). Let \(\alpha\) be a form on a Kähler manifold, and assume that \(\eta\) is either \(d\)-, \(\partial\)- or \(\bar\partial\)-exact. Then \(\alpha\) is \(dd^c\)-exact (or, which is the same, \(\partial\bar\partial\)-exact).

This result is not as trivial, and requires Hodge theory.


The Picard scheme

Posted on April 24, 2016 by Dima

tags: moduli, picard

Functors \(Pic\) and \(Div\)

We will denote base change with a subscript: \(X_T = X \times T\).

If \(X\) is a scheme, the Picard group of \(X\) is defined to be the group of isomorphism classes of invertible sheaves on \(X\). The relative Picard functor of an \(S\)-scheme \(X\) is defined as \[ {\operatorname{Pic}}_{X/S}(T) := {\operatorname{Pic}}(X_T) / {\operatorname{Pic}}(T) \] where the embedding \({\operatorname{Pic}}(T) \hookrightarrow {\operatorname{Pic}}(X_T)\) is given by the pullback along the structure maps \(X_T \to T\).

Caveat: all representability results work with a sheafification of this functor in some topology, Zariski, étale, or fppf. Unless \(X \to S\) is proper and has a section, they need not be isomorphic.

An effective divisor is a closed subscheme such that its ideal is invertible. If \(f: X \to S\) is a morphism of schemes then a relative effective divisor on \(X\) is an effective divisor \(D\) such that \(D\) is flat over \(S\). For a morphism \(X \to S\) define the functor of relative divisors \[ {\operatorname{Div}}_{X/S}(T) := {\{\ \textrm{ relative effective divisors on } X_T/T \ \}} \]

We are interested in representability of this functor, so to this end we prove a little lemma.

Lemma. Let \(X \to S\) be a flat morphism. Let \(D\) be a closed subscheme of \(X\) flat over \(S\). Then \(D\) is relative effective divisor in a neighbourhood of \(x \in X\) if and only if \(D_s\) is cut out in a neighbourhood of \(x\) in \(X_s\), where \(s\) is the image of \(x\), by a single non-zero element of \({\operatorname{\mathcal{O}}}_{X_s,x}\) (which amounts to being an effective divisor in \(X_s\) but we won’t prove it).

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