Vector bundles on the cubic threefold

Arnaud Beauville

Description

Vector bundles on the cubic threefold Arnaud BEAUVILLE To Herb Introduction Let X be a smooth cubic hypersurface in P4 . In their seminal paper [C-G], Clemens and Griffiths showed that the intermediate Jacobian J(X) , an abelian vari- ety defined analytically through Hodge theory, is a fundamental tool to understand the geometry of X . They studied the Fano surface F of lines contained in X , proving that the Abel-Jacobi map embeds F into J(X) and induces an isomor- phism Alb(F) ??? J(X) . They were able to deduce from this the Torelli theorem and the non-rationality of X (a problem which had resisted the efforts of the Italian geometers)1. Mumford noticed that one can express J(X) as a Prym variety and thus give an alternate proof for the non-rationality of X ([C-G], Appendix C); the other results of [C-G] can also be obtained via this approach [B2]. Later Clemens observed that one could use the twisted cubics as well, giving an elegant parametrization of the theta divisor (see (4.2) below). At this point the cubic threefold could be considered as well understood, and the emphasis shifted to other Fano threefolds.

• semi-stable sheaves


• abel-jacobi map embeds


• smooth


• using chern


• map ?


• vector bundle


• p5 -bundle over


• curve ? ?


• stable rank

Sujets

• Beauville
• Smooth
• Vector bundle

Informations

• Publié par: pefav
• Nombre de lectures: 20
• Langue: English

Extrait

POLYNOMIAL INVARIANTS OF LINKS SATISFYING CUBIC SKEIN RELATIONS
PAOLO BELLINGERI AND LOUIS FUNAR
Abstract. The aim of this paper is to define two link invariants satisfying cubic skein relations. In the
hierarchyofpolynomialinvariantsdeterminedbyexplicitskeinrelationstheyarethenextlevelofcomplexity
afterJones,HOMFLY,KauffmanandKuperberg’sG quantuminvariants. Ourmethodconsistsinthestudy2
of Markov traces on a suitable tower of quotients of cubic Hecke algebras extending Jones approach.
1. Introduction
1.1. Preliminaries. J.Conwayshowed that the Alexander polynomial of a knot, when suitably normalized,
satisfies the following skein relation:
  ! !
−1/2 1/2  ∇ −∇ = (t −t )∇
Given a knot diagram one can always change some of the crossings such that the modified diagram
represents the unknot. Therefore one can use the skein relation for a recursive computation of∇, although
this algorithm is rather time consuming, since it is exponential.
In the mid eighties V.Jones discovered another invariant verifying a different but quite similar skein
relation, namely:
  ! !
−1 −1/2 1/2  t V −tV = (t −t )V ,
1/2 −1/2which was further generalized to a 2-variable invariant by replacing the factor (t −t ) with a new
variablex. The latter one was shown to specialize to both Alexander and Jonespolynomials. The Kauffman
polynomial is another extension of Jones polynomial which satisfies a skein relation, but this time in the
realm of unoriented diagrams. Specifically, the formulas:
! ! ! !!
Λ +Λ =z Λ +Λ

Λ =aΛ( )
define a regular isotopy invariant of links, which can be renormalized, by using the writhe of the oriented
diagram, in order to become a link invariant. Remark that some elementary manipulations show that Λ
verifies a cubical skein relation:
 
     
  1 z 1       Λ = +z Λ − +1 Λ + Λ 
  a a a
1991 Mathematics Subject Classification. 16S15, 57M27, 81R15.
Key words and phrases. skein relation, cubic Hecke algebras, Markov trace.
Partially supported by a Canon grant.
12 P.BELLINGERI AND L.FUNAR
It has been recently proved ([12], and Problem 1.59 [18]) that this relation alone is not sufficient for a
recursive computation of Λ. Whenever the skein relations and the value of the invariant for the unknot are
sufficienttodetermine itsvaluesforalllinksthe systemofskeinrelationswillbe saidtobe complete. Several
results concerning the incompleteness of higher degree unoriented skein relations and their skein modules
have been obtained by J.Przytycki and his students (see e.g. [12, 28, 29]).
These invariants were generalized to quantum invariants associated to general Lie algebras and super-
algebrasand their representations. V.Turaev([33]) identified the HOMFLY and Kauffman polynomialswith
the invariants obtained from the series A and B ,C ,D respectively. G.Kuperberg ([19]) defined then n n n
G -quantum invariant of knots by means of skein relations, by making use of trivalent graphs diagrams and2
exploited further these ideas for spiders of rank 2 Lie algebras. The skein relations satisfied by the quantum
invariants coming from simple Lie algebras were approached also via weight systems and the Kontsevich
integral in ([22, 23]), for the classical series, and in ([1, 2]) for the case of the Lie algebra of G .22
Notice that any link invariant coming from some R-matrix R verifies a skein relation of the type:

*  +n X
a j twists = 0j
j=0 
which can be derived from the polynomial equation satisfied by the R-matrix R.
Let us mention that the skein relations are somewhat related to the representation theory of the Hopf
algebra associated to the R-matrix R. In particular, there are no other known invariants given by means
of a complete skein relations, but those from above. Moreover, one expects that the quantum invariants
associated to other Lie (super) algebras or by cabling the previous ones satisfy skein relations of degree at
least 4, as already the G invariant does.2
This makes the search for an explicit set of complete skein relations, in which at least one relation is
cubical, particularly difficult and interesting. This problem was first considered in [13] and solved in a
particular case. The aim of this paper is to complete the result of [13] by constructing a deformation of the
previously constructed quotients of the cubic Hecke algebras and of the Markov traces supported by these
(z,δ)algebras. We obtain in this way two link invariants, denoted by I and I , which are recursively(α,β)
computable and uniquely determined by two skein relations. Explicit computations show thatI detects(α,β)
the chirality of the knots with number crossing at most 10 where HOMFLY, Kauffman and their 2-cablings
fail. On the other hand, as HOMFLY, Kauffman and their 2-cablings ([24, 27]), it seems that our invariants
do not distinguish between mutant knots. We recall that the some mutant knots can be distinguished by
the 3-cablings of the HOMFLY polynomial (see [25]).
Acknowledgements. Part of this work was done during the second author’s visit to the Tokyo Institute of
Technology, whose support and hospitality are gratefully acknowledged. The authors are thankful to Chris-
tian Blanchet, Emmanuel Ferrand, Thomas Fiedler, Louis Kauffman, Teruaki Kitano, Sofia Lambropoulou,
Ivan Marin, Jean Michel, Hugh R. Morton, Luis Paris and Vlad Sergiescu for useful discussions, remarks
and suggestions.
1.2. The main result. The aim of this paper is to define two link invariants by means of a complete set of
skein relations. More precisely we will prove the following Theorem (see section 5):
(z,δ)Theorem 1.1. There exist two link invariants I and I which are uniquely determined by the two(α,β)
skein relations shown in (1) and (2) and their value for the unknot, which is traditionally 1. These invariants
take values in:
2 ±/2 2 ±/2Z[α, β, (α −2β) , (β +2α) ]
,
(H )(α,β)
and respectively:
±/2 ±/2Z[z ,δ ]
,
(z,δ)(P )
gPOLYNOMIAL INVARIANTS OF LINKS 3
where −1∈{0,1} is the number of components modulo 2 and:
6 5 2 4 4 4 3 3 3 2 5 2 2H := 8α −8α β +2α β +36α β−34α β +17α +8α β +32α β −(α,β)
4 6 3
−36αβ +38αβ+8β −17β +8,
and respectively
(z,δ) 23 18 16 2 14 3 9 4 7 5 6 5 7P :=z +z δ−2z δ −z δ −2z δ +2z δ +δ z +δ .
Here (Q) denotes the ideal generated by the element Q in the algebra under consideration.
2 3(1) = w + β w + wα
-2 -1 -1 -1
A w +B w +B w +C w+ +D
+E +E +F +Gw w +H w+F +G
(2)
2 2 2 2+H w +I w +L w w +M w w+L +M
43 3w w +P w = 0+N +O
The values of the polynomials A,B,C,...,P corresponding to I are given in the table below. In order(α,β)
(z,δ) 4 1/2 7 2 4to obtain those corresponding to I it suffices to set w = (−z /(δz)) and replace α =−(z +δ )/(z δ)
2 3and β = (δ−z )/z in the other entries of table 1.
2 2 1/2 2w =((α +2β)/(2α−β )) A =(β −α)
2 2 2 2B = (α −αβ −β) C =(α −αβ )
2 2 3 2 2 3D = (1+2αβ +α β −α ) E = (1+αβ +α β −α )
3 3 2F = (1+2αβ−β ) G =(αβ −2α−2α β)
3 2 2 4 3 2 2H =(αβ −2α−2α β +β ) I = (α −α β −2α β−3α)
3 2 2 3 2 4 2 2L =(2α β+3α −α β −αβ ) M =(β −2β−3αβ +α )
2 2 3 4 3 2 2 3 4N =(1+4αβ+3α β −α −αβ −β ) O = (1+3αβ+3α β −α −αβ )
2 5 2 3P = (3β −β −2α−3α β +4αβ )
Table 1
1.3. Properties of the invariants. The following summarize the main features of these invariants (see
section 6):
(1) theydistinguishallknotswithnumbercrossingatmost10thathavethesameHOMFLYpolynomial,
and thus they are independent from HOMFLY. However, like HOMFLY and Kauffman polynomials,
they seem to not distinguish among mutants knots: in particular they don’t separate the Kinoshita-
Terasaka knot from the Conway knot, which are the simplest non-equivalent mutant knots.
(2) I = I for amphicheiral knots, and I detects the chirality of all those knots with(α,β) (−β,−α) (α,β)
number crossing at most 10, whose HOMFLY, Kauffman polynomials as well as the 2-cabling of
HOMFLY fail to dete