WEIGHTED PROJECTIVE VARIETIES
by
Igor Dolgachev
Contents
0.
Introduction
i.
Weighted projective space
i.i.
Notations
1.2.
Interpretations
1.3.
The first properties
1.4.
Cohomology of
0F(n)
1.5.
Pathologies
2.
Bott's theorem
2.1.
The sheaves
~(n)
2.2.
Justifications
2.3.
~$(n)
3.
Weighted complete intersections
3.1.
Quasicones
3.2.
Complete intersections
3.3.
The dualizing sheaf
3.4.
The Poincare series
3.5.
Examples
4.
The Hodge structure on cohomology of weighted hypersurfaces.
4.1.
A resolution of
~i
X
4.2.
The Griffiths theorem
4.3.
Explicit calculation
4.4.
Examples and supplements
O.
Introduction.
In this paper I discuss the technique of weighted homogeneous coordinates
which has appeared in works of various geometers a few years ago and it seems has
been appreciated and armed by many people.
In many cases this technique allows one
to present a nonsingular algebraic variety as a hypersurface in a certain space (a
weighted projective space) and deal with it as it would be a nonsingular hypersur
face in the projective space.
A generalization of this approach is the technique
of polyhedral projective spaces for which we refer to [5, 6, 15].
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Section 1 deals with weighted projective spaces, the spectrums of graded poly
nomial rings.
Most of the results from this section Can be found in [7].
In section 2 we generalize the Bott theorem on the cohomology of twisted
sheaves of differentials to the case of weighted projective spaces.
Another proof
of the same result can be found in [23] and a similar result for torical spaces is
discussed in [5].
In section 3 we introduce the notion of a quasismooth subvariety of a weighted
projective space.
For this we define the affine quasicone over a subvariety and
require that this quasicone is smooth outside its vertex.
We show that quasismooth
weighted complete intersections have many properties of ordinary smooth complete
intersections
in a projective space.
The work of Mori [19] contains a similar re
sult but under more restrictive conditions.
Rather surprisingly not everything
goes the same as for smooth complete intersections.
For example, recent examples
of Catanese and Todorov show that the local Torelli theorem fails for some quasi
smooth weighted complete intersections
(see [4, 24]).
In section 4 we generalize to the weighted case the results concerning the
Hodge structure of a smooth projective hypersurfaces.
Our proof is an algebraic
version of one of Steenbrink
[23] and can be applied to the calculation of the De
Rham cohomology of any such hypersurface over a field of characteristic
zero.
The
present paper is partially based on my talks at a seminar on the HodgeDeligne
theory at Moscow State University in 1975/76.
It is a pleasure to thank all of its
participants for their attention and criticism.
i.
Weishted projective space.
i.i.
Notations
Q = {qo,ql,.
..,qr } ,  a finite set of positive integers;
IQI = qo +.
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 Fall '06
 IgorDolgachev
 Calculus, Manifold, Algebraic geometry, Algebraic variety, exact sequence, ince

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