Adjunction theory
In its basic form, the study of the interplay between an embedding of a projective manifold (cf. Projective algebraic set) into projective space and its canonical bundle,
, where
is the cotangent bundle of
. For simplicity, below the objects are considered over the complex numbers. The book [a1] is a general reference with full coverage of the literature for the whole theory with its history. The book [a4] is a fine reference on related material.
Classical adjunction theory.
To study a smooth projective algebraic curve , i.e., a compact Riemann surface, a major approach in the 19th century was to relate properties of the curve to properties of the canonical mapping of the curve, i.e., the mapping of the curve into projective space given by sections of
. To study a two-dimensional algebraic submanifold of projective space
, it was natural to try to reduce questions about the surface
to the hyperplane sections of
, i.e., to the curves obtained by slicing
with linear hyperplanes
. This led to the study of the adjoint bundle
, where
is the restriction to
of the hyperplane section bundle of
, i.e., the line bundle on
whose sections vanish on hyperplanes of
. The restriction of the adjoint bundle to a hyperplane section
of
is the canonical bundle of
, and, except in a few trivial cases,
is the only line bundle that has this property. Therefore, if the mapping associated to
exists, it would tie together the canonical mappings of the smooth hyperplane sections of
. In the 19th century geometers, especially G. Castelnuovo and F. Enriques, used this rational mapping to study the embedding of
into
. The general procedure was to consider the sequence
. A classical result was that adjunction terminates, i.e., there is a positive
such that
has no sections, if and only if
is birational (cf. Birational morphism) to the product of an algebraic curve and
. A key point was that if the above
was the first positive
such that
has no sections, then
is one of only a "very short list of pairs" . Classically, it was not known if
was spanned and, therefore, this procedure did not usually lead to a biregular classification. (A line bundle
on an algebraic set is said to be spanned if global sections of
surject onto the fibre of
at any point of the algebraic set.) There were also analogous classical procedures on threefolds due to G. Fano and U. Morin (see [a5], [a1]).
A.J. Sommese [a6] started the modern study of the mapping (which he called the adjunction mapping) associated to , where
is a very ample line bundle (cf. Ample vector bundle) on an
-dimensional projective manifold. The complete story with history of the adjunction mapping can be found in [a1], Chapts. 8–11. The fundamental results of this theory are that except for a few special varieties, the pair
can be replaced by a closely related pair
with
the blow-up of an
-dimensional projective manifold
at a finite set,
an ample line bundle,
, and
very ample. Moreover, if
then, except for a few more examples,
is numerically effective, or nef, i.e.,
for any effective curve
. These results allow the classical birational results alluded to above to be both considerably extended and redone biregularly.
The major open question for this part of adjunction theory is to what extent the mapping associated to is well-behaved when
is very ample,
is smooth, and
. For example, it is known when
that, except for obvious counterexamples,
, and if the Kodaira dimension of
is non-negative, then
unless
is a quintic threefold with
trivial. See [a2], [a1] for this and some further discussion of this problem.
General adjunction theory.
A more abstract approach to adjunction theory is to start with a pair with
an ample line bundle on a projective variety
having at worst mild singularities, e.g., terminal singularities. Then, assuming
is not nef, Kawamata's theorem asserts that there is a rational number
with
, called the nefvalue of the pair such that
is nef but not ample. By the Kawamata–Shokurov theorem,
is spanned for all sufficiently large positive integers
. The morphism
, from
to a normal projective variety
, associated to
, is independent of
for all sufficiently large
. This mapping
, called the nefvalue morphism of the pair, has connected fibres, and at least one positive-dimensional fibre. If
, one can express
as a very special fibration of Fano varieties, e.g., if
, then
is a contraction of an extremal ray (see [a3] and [a1], Chapt. 6). If
, then one can replace
with
and repeat the procedure. This works well for
(see [a1], Chapts. 6, 7).
Define the spectral value of the pair
as the infimum of the positive rational numbers
with
such that there is some positive integer
with
. A major conjecture in this part of the theory is that if
, then
is the nefvalue of the pair
and the nefvalue morphism has a lower-dimensional image.
References
[a1] | M.C. Beltrametti, A.J. Sommese, "The adjunction theory of complex projective varieties" , Expositions in Mathematics , 16 , De Gruyter (1995) MR1318687 Zbl 0845.14003 |
[a2] | M.C. Beltrametti, A.J. Sommese, "On the dimension of the adjoint linear system for threefolds" Ann. Scuola Norm. Sup. Pisa Cl. Sci. Ser. (4) , XXII (1995) pp. 1–24 MR1315348 Zbl 0857.14001 |
[a3] | M.C. Beltrametti, A.J. Sommese, J.A. Wiśniewski, "Results on varieties with many lines and their applications to adjunction theory (with an appendix by M.C. Beltrametti and A.J. Sommese)" K. Hulek (ed.) T. Peternell (ed.) M. Schneider (ed.) F.-O. Schreyer (ed.) , Complex Algebraic Varieties, Bayreuth 1990 , Lecture Notes in Mathematics , 1507 , Springer (1992) pp. 16–38 |
[a4] | T. Fujita, "Classification theories of polarized varieties" , London Math. Soc. Lecture Notes , 155 , Cambridge Univ. Press (1990) MR1162108 Zbl 0743.14004 |
[a5] | L. Roth, "Algebraic threefolds with special regard to problems of rationality" , Springer (1955) MR0076426 Zbl 0066.14704 |
[a6] | A.J. Sommese, "Hyperplane sections of projective surfaces, I: The adjunction mapping" Duke Math. J. , 46 (1979) pp. 377–401 MR0534057 Zbl 0415.14019 |
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