Holomorphic functions on normalization quotients #
Given an equivalence relation on a complex manifold, define a sheaf on its topological quotient: a section is a holomorphic function on the inverse image of an open set whose values agree at related points. In particular, for the normalization of a nodal curve these are precisely the holomorphic functions satisfying the matching condition at every pair of node flags. No hypersurface charts or stability assumptions enter this construction.
quotientSheaf is a subsheaf of the direct image of the native holomorphic structure sheaf.
quotientStalkBranch restricts its germs to a point of the normalization. At two identified
points, quotientStalkToNode combines these restrictions into the existing branch-pair node
ring.
When the identifications are isolated, these maps are isomorphisms. At a point near which no two
points are identified, quotientStalkBranch_bijective shows that the stalk of the quotient sheaf
is the holomorphic stalk of the normalization. At two distinct points x and y identified only
with each other, near which nothing else is identified, quotientStalkToNode_bijective shows
that the stalk is the branch-pair node ring: a germ on the quotient is the same as a pair of
holomorphic germs at x and y with equal values. The node case needs the normalization to be
Hausdorff, so that the two branch germs can be glued on disjoint neighbourhoods.
The construction works for arbitrary equivalence relations: the sheaf and the comparison maps do not need finiteness of the flags, compactness, or connectedness of the curve. It makes no claim that an arbitrary such quotient is a complex analytic space.
The sheaf proof uses Mathlib's direct-image sheaf and unique-gluing API. The local ring and matching condition follow the normalization description of nodal curves in Joel W. Robbin and Dietmar A. Salamon, A construction of the Deligne–Mumford orbifold, JEMS 8 (2006).
Holomorphic functions on the normalization over U which agree at identified points.
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The characteristic matching condition for sections on the quotient.
On an open set over which no points are identified, every holomorphic function descends. This applies in particular to opens in the smooth locus of a nodal normalization quotient.
The sheaf of holomorphic functions on the normalization which have matching values on each fibre of the quotient map. For paired node flags this is the normalization construction of the structure sheaf of a nodal curve.
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Read a quotient section as its matching holomorphic function upstairs.
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Quotient sections are equal if their holomorphic functions upstairs agree pointwise.
Constant functions take equal values at identified points, so the quotient sections form a
ℂ-algebra.
The constant quotient section c is the constant function c upstairs.
The normalization pullback includes the quotient sheaf into the direct image of the holomorphic structure sheaf.
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Normalization pullback reads the underlying holomorphic function.
The quotient sheaf is a subobject of the direct image of the normalization sheaf.
Restriction in the quotient sheaf is restriction of the holomorphic section upstairs.
A quotient germ restricts to a holomorphic germ at any point over its base point.
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Restricting a quotient section's germ to a branch gives its ordinary holomorphic germ.
Evaluating a quotient germ through a branch over its base point evaluates the section.
Evaluation of a quotient germ through a branch over its base point is surjective.
The branch restrictions of a quotient germ have equal values at any two identified points.
At two identified normalization points, restriction gives a homomorphism from the quotient stalk to the branch-pair node ring.
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The first branch of the node comparison is normalization restriction at the first flag.
The second branch of the node comparison is normalization restriction at the second flag.
Stalks at smooth points and at nodes #
Away from the identified points, and at a pair of points identified only with each other, the stalks of the quotient sheaf are the expected local rings. The hypotheses are local: near the relevant points no other identifications occur.
Stalks at smooth points. At a point of the normalization near which no two points are identified, restriction to the branch is an isomorphism from the stalk of the quotient sheaf to the holomorphic stalk of the normalization.
At two points identified with each other and with no other point, near which nothing else is identified, a germ on the quotient is determined by its two branch germs.
At two distinct points of a Hausdorff normalization, identified with each other and with no other point, near which nothing else is identified, every pair of holomorphic branch germs with equal values comes from a germ on the quotient.
Stalks at nodes. Let x ≠ y be identified with each other and with no other point, and
suppose no further identifications occur near x and y. Then the branch restrictions identify
the stalk of the quotient sheaf at their common image with the branch-pair node ring.