Difference between revisions of "K3-surface"
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A smooth projective [[Algebraic surface|algebraic surface]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055040/k0550402.png" /> for which the [[Canonical class|canonical class]] is trivial and for which the dimension, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055040/k0550403.png" />, of the space of one-dimensional differential forms on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055040/k0550404.png" /> is 0. The values of the following invariants for a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055040/k0550405.png" />-surface are known: the [[Geometric genus|geometric genus]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055040/k0550406.png" />, the [[Euler characteristic|Euler characteristic]] of the structure sheaf, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055040/k0550407.png" />, the étale or (over the field of complex numbers) the topological Betti numbers (cf. [[Betti number|Betti number]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055040/k0550408.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055040/k0550409.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055040/k05504010.png" />, and the Euler–Poincaré characteristic, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055040/k05504011.png" />. The Riemann–Roch formula for a one-dimensional invertible sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055040/k05504013.png" /> on a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055040/k05504014.png" />-surface takes the form | A smooth projective [[Algebraic surface|algebraic surface]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055040/k0550402.png" /> for which the [[Canonical class|canonical class]] is trivial and for which the dimension, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055040/k0550403.png" />, of the space of one-dimensional differential forms on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055040/k0550404.png" /> is 0. The values of the following invariants for a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055040/k0550405.png" />-surface are known: the [[Geometric genus|geometric genus]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055040/k0550406.png" />, the [[Euler characteristic|Euler characteristic]] of the structure sheaf, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055040/k0550407.png" />, the étale or (over the field of complex numbers) the topological Betti numbers (cf. [[Betti number|Betti number]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055040/k0550408.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055040/k0550409.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055040/k05504010.png" />, and the Euler–Poincaré characteristic, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055040/k05504011.png" />. The Riemann–Roch formula for a one-dimensional invertible sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055040/k05504013.png" /> on a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055040/k05504014.png" />-surface takes the form | ||
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====References==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> D. Burns, M. Rappoport, "On the Torelli theorem for Kählerian <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055040/k05504079.png" /> surfaces" ''Ann. Sci. Ec. Norm. Sup.'' , '''8''' (1975) pp. 235–274</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> A., et al. (ed.) Beauville, "Géométrie des surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055040/k05504080.png" />: modules et périodes. Sem. Palaiseau" ''Astérisque'' , '''126''' (1985)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> A. Ogus, "Periods of integrals in characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055040/k05504081.png" />" , ''Internat. Congress Mathematicians (Warszawa, 1982)'' , PWN & North-Holland (1983) pp. 753–762 {{MR|0804731}} {{ZBL|0585.14015}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> Y. Siu, "Every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055040/k05504082.png" /> surface is Kähler" ''Invent. Math.'' , '''73''' (1983) pp. 139–150 {{MR|}} {{ZBL|0557.32004}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> J. Giraud (ed.) et al. (ed.) , ''Surfaces algébrique'' , ''Lect. notes in math.'' , '''868''' , Springer (1981) {{MR|0638596}} {{ZBL|0452.00012}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> A. Beauville, "Surfaces algébriques complexes" ''Astérisque'' , '''54''' (1978) {{MR|0485887}} {{ZBL|0394.14014}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> A. van de Ven, "Compact complex surfaces" , Springer (1984) {{MR|}} {{ZBL|0718.14023}} </TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> D. Burns, M. Rappoport, "On the Torelli theorem for Kählerian <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055040/k05504079.png" /> surfaces" ''Ann. Sci. Ec. Norm. Sup.'' , '''8''' (1975) pp. 235–274</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> A., et al. (ed.) Beauville, "Géométrie des surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055040/k05504080.png" />: modules et périodes. Sem. Palaiseau" ''Astérisque'' , '''126''' (1985)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> A. Ogus, "Periods of integrals in characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055040/k05504081.png" />" , ''Internat. Congress Mathematicians (Warszawa, 1982)'' , PWN & North-Holland (1983) pp. 753–762 {{MR|0804731}} {{ZBL|0585.14015}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> Y. Siu, "Every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055040/k05504082.png" /> surface is Kähler" ''Invent. Math.'' , '''73''' (1983) pp. 139–150 {{MR|}} {{ZBL|0557.32004}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> J. Giraud (ed.) et al. (ed.) , ''Surfaces algébrique'' , ''Lect. notes in math.'' , '''868''' , Springer (1981) {{MR|0638596}} {{ZBL|0452.00012}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> A. Beauville, "Surfaces algébriques complexes" ''Astérisque'' , '''54''' (1978) {{MR|0485887}} {{ZBL|0394.14014}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> A. van de Ven, "Compact complex surfaces" , Springer (1984) {{MR|}} {{ZBL|0718.14023}} </TD></TR></table> | ||
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Revision as of 21:19, 20 December 2014
2020 Mathematics Subject Classification: Primary: 14J28 [MSN][ZBL]
A smooth projective algebraic surface for which the canonical class is trivial and for which the dimension,
, of the space of one-dimensional differential forms on
is 0. The values of the following invariants for a
-surface are known: the geometric genus,
, the Euler characteristic of the structure sheaf,
, the étale or (over the field of complex numbers) the topological Betti numbers (cf. Betti number)
,
and
, and the Euler–Poincaré characteristic,
. The Riemann–Roch formula for a one-dimensional invertible sheaf
on a
-surface takes the form
![]() |
where is the index of self-intersection for the class of divisors corresponding to
(see Riemann–Roch theorem). If
corresponds to an effective irreducible divisor, then
.
The formula for the arithmetic genus of an irreducible curve on
also has a simple form:
![]() |
As a result one obtains , with
only for smooth rational curves. It also follows that
is an even number for any divisor
. Let
be the Néron–Severi group of the surface
, i.e. the group of divisor classes on
with respect to algebraic equivalence. Then
is a free Abelian group of rank
, where
if the characteristic of the ground field
is 0, and
or
if
. The intersection index defines an integer-valued bilinear form on
for which the square of any element is even. Surfaces with
(for
) are called singular, while those with
(for
) are called supersingular.
Another numerical invariant for a surface is the minimal possible index
of self-intersection of an effective very ample divisor on
, i.e. the minimum possible degree of a polarization on
. If
, the surface
can be imbedded in
-dimensional projective space and cannot be imbedded in a projective space of lower dimension.
An important tool in studying -surfaces is to represent them as families (pencils) of elliptic curves. A surface
is represented as a family of elliptic curves if one is given a regular mapping
all fibres of which, apart from a finite number, are non-singular elliptic curves. A surface
can be represented in that form if and only if the group
contains a non-zero element with self-intersection index 0, where all such representations correspond to the classes of effective divisors with self-intersection index 0. If a surface represented as a family of elliptic curves is a
-surface, it does not have multiple fibres. The Jacobi elliptic family constructed from such a family is again a
-surface.
An important class of -surfaces is provided by the Kummer surfaces. A Kummer surface is a non-singular model for the quotient of a two-dimensional Abelian variety
with respect to the automorphism subgroup generated by the sign-change mapping. In particular, the surface given by the equation
in
is a Kummer surface. Any smooth surface of order four in
is a
-surface. A smooth surface obtained as the intersection of three hypersurfaces of order two (quadrics) in
and a double covering of
ramified in a smooth curve of order six are
-surfaces.
All -surfaces over the field of complex numbers are diffeomorphic; their moduli variety is connected and has dimension 19. One examines the structure of this variety and of
-surface automorphisms by means of the period mapping. The period mapping for a
-surface over the field of complex numbers is bijective (a theorem of Torelli type) [2].
If one is given a one-dimensional family of -surfaces (over
) with one singular fibre, then the surface can be rearranged after covering the base, without alterations outside the singular fibre, so that that fibre either becomes non-singular or else is one of two types: a) the components of the singular fibre and the intersection curves are rational, and the dual polyhedron for the singular fibre has the topological type of a two-dimensional sphere; or b) the components of the singular fibre constitute a chain, and non-empty intersections occur only for adjacent surfaces, while the two extreme surfaces are rational surfaces, the middle ones are elliptic ruled surfaces and the intersection curves are elliptic. Types a) or b) arise when the monodromy of the family is non-trivial [2].
-surfaces over an algebraically closed field of positive characteristic allow of a lifting into characteristic zero, their crystalline cohomology spaces do not have torsion, and their ranks coincide with the corresponding Betti numbers. For supersingular surfaces, an analogue of the period mapping has been constructed and a theorem of Torelli type has been proved. In this case the period variety is irreducible, complete, has dimension 9, and is unirational. All possible intersection forms on
have been described for supersingular surfaces, and there are nine of them for each value of the characteristic of the ground field [4].
References
[1] | I.R. Shafarevich, "Algebraic surfaces" Proc. Steklov Inst. Math. , 75 (1967) Trudy Mat. Inst. Steklov. , 75 (1965) MR1392959 MR1060325 Zbl 0830.00008 Zbl 0733.14015 Zbl 0832.14026 Zbl 0509.14036 Zbl 0492.14024 Zbl 0379.14006 Zbl 0253.14006 Zbl 0154.21001 |
[2] | V.S. Kulikov, "Degenerations of ![]() |
[3] | A.N. Rudakov, I.R. Shafarevich, "On the degeneration of ![]() |
[4] | A.N. Rudakov, I.R. Shafarevich, "Surfaces of type ![]() |
[5] | D. Husemoller, "Classification and embeddings of surfaces" R. Hartshorne (ed.) , Algebraic geometry (Arcata, 1974) , Proc. Symp. Pure Math. , 29 , Amer. Math. Soc. (1975) pp. 329–420 MR0506292 Zbl 0326.14009 |
Comments
Proofs of some of the results mentioned about -surfaces over fields of positive characteristic can be found in [a3] and [a5]. The notion of a
-surface applies also to (not necessarily algebraic) complex surfaces. Each such a surface has a structure of a Kählerian variety (cf. Kähler manifold) [a4]. There is an analogue of Torelli's theorem (cf. Torelli theorems) for complex
-surfaces [a1].
References
[a1] | D. Burns, M. Rappoport, "On the Torelli theorem for Kählerian ![]() |
[a2] | A., et al. (ed.) Beauville, "Géométrie des surfaces ![]() |
[a3] | A. Ogus, "Periods of integrals in characteristic ![]() |
[a4] | Y. Siu, "Every ![]() |
[a5] | J. Giraud (ed.) et al. (ed.) , Surfaces algébrique , Lect. notes in math. , 868 , Springer (1981) MR0638596 Zbl 0452.00012 |
[a6] | A. Beauville, "Surfaces algébriques complexes" Astérisque , 54 (1978) MR0485887 Zbl 0394.14014 |
[a7] | A. van de Ven, "Compact complex surfaces" , Springer (1984) Zbl 0718.14023 |
K3-surface. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=K3-surface&oldid=35762