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We will use Sing X to denote the closed subset of
singular (i.e. not simple) points. If X is normal then
Sing X has codimension at least two. If Sing X is empty
we say that X is non-singular. We will usually use the
letter M to denote a non-singular variety. Such varieties
enjoy many geometric properties. Some are stated below,
and in §3.
Theorem 3 . Suppose that M is non-singular.
(i) If X c M is irreducible and of codimension one
then on each set U of some affine open cover of M there
is a regular function f such that X = V(f). Such an f
is called a local equation for X on U. Sing X is defined
locally by the equations df = 0 and f = 0.
(ii) Suppose that N is a non-singular subvariety of
codimension n. Then on each U of some cover there are
local parameters u1,...,u<j such that u^ = ... = un = 0
are equations defining N.
The proof of this result can be found in [35, C h . II §3].
We use the notion of a transversal intersection of
subvarieties - a definition can be found in [35, C h . II §2.1]
A divisor D on a smooth variety M is said to have normal
crossings if the components of its support
Supp D are transverse. Equivalent to this is the
condition that on each open subset U of some open cover of
M there are local parameters u^,...,u^ such that u* = 0 is
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3. Poincarg residue
The Poincarg residue homomorphism is a generalization
to higher dimension of the Cauchy residue of a contour
integral in the theory of functions of a single complex
variable. In the latter theory one takes, for instance,
a function f(z) with a simple pole at the origin and
applies to it the operator dz where C is a circular
path around the origin. The conclusion is that
g(O) 2tt/=T = [ f(z) dz 'C
where g(z) = zf(z).
In defining the Poincard residue, we make a number of
changes:
the definition will be purely algebraic
we will 'integrate' a differential form w,
not a function, which is of top degree
the differential w will have a simple pole
along a subvariety X of codimension one
the residue will be a differential form of
top degree on X.
The definition will first be made in a special case
and then extended.
Theorem 4. Suppose that X c u is a non-singular sub-
variety of the smooth variety U. Suppose also that f = 0
is a local equation for X and that w is a simple differential
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Consider the equation
w = M a n . (2 )
Then (i) All solutions to ( 2 ) have the same restriction
to X.
(ii) The equation (2) has a solution.
Definition. The restriction to X of a solution q of
(2) is called the Poincaré residue Res(w) of w on X.
Theorem 4 (Cont.)
(iii) Res(w) is Kahler on X
(iv) Res(w) depends only on X and w.
Proof. (i) If q and q' are two solutions then their
difference n will satisfy the equation df a q « 0. As X
is non-singular there is, at least locally, a system
of local parameters for U such that u^ = f .
Clearly, q necessarily lies in the ideal of the Grassmann
algebra generated by d f . So the restriction of q to X is
zero.
(ii) Using the above system of local parameters
we can write u as i u df a du_ a ... a du, where to is a
f o 2 d o
function regular on U. Then the differential (-1) times
wo du2 a ... a du^ will satisfy (2). More generally,
if u^.-.-.u^ are local parameters and df * £ f ■ du^ then
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du. a ... du. ... du, . .
»„ c-i--- (-i)1’1
( 3 )
so long as f. does not vanish on X. As X is non-singular,
at none of its points do all the f^ vanish.
It is clear from the proof of (ii) that (iii) is
satisfied.
(iv) The condition on w clearly does not depend
upon the choice of local equation f for X. Any other
local equation f' is equal to vf for some regular function
v not vanishing on X.
We will use the notation of (ii) and formula ( 3) for
n . The partial derivatives f^ of f satisfy the equation
fi = V + V fA
while the co-efficient u> satisfies o
0) = 0) V
o o
and so the difference between the two expressions ( 3 ) for
n is that one has
S0/*i = % y/(vif + vfi^
in the place of u^/f^. But as f vanishes on X, the two
expressions are equal on X.
We can now conclude the definition. If X c M is a
irreducible and of codimension one, we can find an open
** *
the theorem that the choice of U is irrelevant. However, ’
the argument shows only that Res (to) is Kahler at the
non-singular points of X.
Definition. A differential to on a variety X is regular
if it is Kahler at the non-singular points of X.
The next result can be proved by use of local parameters
to express differentials.
Theorem 5. Suppose that 10 is a rational differential
of the smooth variety X. Then the set Z of points at which
a) is not KMhler is closed, and each of its components
has codimension one.
4. Comments
The behaviour of differential forms on a singular space
is delicate. Already we have the concepts of Kahler,
regular, and rational.
In Chapter II we introduce two new concepts, those of
logarithmic and of the first kind. The following inclusions
exist.
Kahler c logarithmic c first kind c regular c rational
The conditions of Chapter II respect proper birational
modifications. In other words, if X' •+ X is such then to
is logarithmic or the first kind on X just in case it
is on X ' .
The same is not true for Kahler and regular.
§3 . n o n-s i n g u l a r v a r i e t i e s
1. Rational maps
Theorem 1 . Suppose that X is a non-singular variety
and that <í>: X - P n is a rational map. Then there is a
set Z of codimension at least two such that <j> is regular
on X-Z. (The proof appears in [35,Ch.II §3.]] and depends
upon the fact that in the local rings of X factorization
is unique. That X maps to P n is not important.)
Corollary. Suppose that X is non-singular and <J>: X - ■+• Y
is a rational map and Y is complete. Then Z as above
exists.
Proof. We may suppose by Chow's lemma that Y c P n .
Let i|>:X-Z -*■ IPn be given by the theorem. Necessarily, the
image of tp lies in Y.
2. Birational morphisms
Suppose that \p:X’ -*■ X is a birational morphism. Then
there are open subsets U' and U of X' and X that are
isomorphic via <{>. The complement E in X' of the largest
such U' is called the exceptional locus of X' X.
Suppose that P ' is a point of X' and that P = <t>(P') is
the corresponding point of X. Always, the local ring of
P is contained in that of P'. Moreover, P' is in E just
in case these rings are unequal. Thus, E consists of all
points P' of X' such that the rational map <{> *:X - X'
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Theorem 2 . Suppose that <t>:X' -*• X is a birational
map and X is non-singular. Then E is a union of irreducible
subvarieties of codimension one, and <KE) has dimension
two or more. In other words, E is contracted by <p.
Proof. Suppose that P' e E and P = <f>(P'). As the
local rings of P and P' are different, we can find a
function f regular at P b u t not at P. Moreover, we can
insist that f(P')= 0. But the map is birational so
f = g/h for functions g and h regular at P. Because P
is a simple point we can choose g and h so that the
varieties V(g) and V(h) have no components in common.
Now consider the equations g = h = 0. On X they define
a subvariety of codimension two. But on X', fh = g and
so they define a subvariety of codimension one. This
concludes the proof.
We do not need to assume that X is non-singular. It
is enough that every element f of k(X) have a power f
such that fr = g/h for functions g and h regular at P for
which the subvarieties V(g) and V(h) have no components
in common. This is true iff the local class group at
every point P of X is a torsion group. Equivalently, if
every Weil divisor has a multiple which is locally principal.
The theorem is thus valid under this weaker hypothesis.
This result tells us something about small resolutions
(Chapter II §1.2) and the like.