on kobayashi hyperbolicity - université paris-saclaymerker/...2013/05/22 · "geometry and...
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On Kobayashi hyperbolicityof Zariski-generic projective algebraic hypersurfaces
JOËL MERKER
Département de Mathématiques d’Orsay
www.math.u-psud.fr/∼merker/
I. Picard theorem and genus formula
II. Kobayashi Hyperbolicity Conjectures
III. Review of existing results
IV. Coordinate Construction of Jet Differentials
"Geometry and Analysis on Manifolds"Memorial Symposium for Professor Shoshichi Kobayashi
Tokyo, Wednesday 22 May – Saturday 25 May, 2013
Graduate School of Mathematical Sciences, Tokyo UniversityTakushiro Ochiai, Keizo Hasegawa, Toshiki Mabuchi, Yoshiaki Maeda,
Junjiro Noguchi, Yoshihiko Suyama, Takashi Tsuboi
2
I – Picard theorem and genus formula
• Complex numbers :C.
• Riemann sphere :P1C = C ∪ {∞}.
• Picard theorem : Every holomorphic map :
f : C −→ P1C∖{
trois distinct points}
is necessarily constant.
C
D
projection
P1C
f
lifting f
holomorphicRiemann sphere
unit disc
Proof :
� The universal cover P1C∖{
three points}
is the unitdisc D.
� Because C is simply connected, the map f can belifted as a holomorphic map f : C→ D.
� The Liouville theorem forces f , hence f , both tobe constant. �
3
• Complex projective space of dimension 2 :
P2C := C3\{0} modulo dilations centered at the origin.
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• Curve Picard Theorem : Every holomorphic map :
f : C −→ C
valued in a smooth projective algebraic curve :
C ⊂ P2Cof genus > 2 is necessarily constant.
C
D
projection
f
P2C
C
lifting f
holomorphic
unit disc
� Same proof, because the universal cover of thecurve C is again the unit disc D. �
4
• Riemann-Poincaré-Koebe theorem : Every ab-stract compact Riemann surface which is of genus> 2 has the unit disc D as its universal cover.
• Genus formula : Every irreducible complex projec-tive algebraic curve with normal crossings :
C ⊂ P2
possesses, after desingularization as a 2-dimen-sional orientable real surface, a genus :
genus(C) =
(degree(C)− 1
)(degree(C)− 2
)2
−# crossings.
C2
with the line at infinitytransverse intersection
w
z C1∞,z
∞w
Curve C
Philipp A. Griffiths :« The genus formula is of great importance, be-cause it exposes the relationship between the ‘in-trinsic’ topological invariant g of the curve C andthe ‘extrinsic’ quantity d».
• Corollary : Every irreducible smooth curve :C ⊂ P2
which is of degree > 4 is uniformised by D.
5
• Major consequence :
Most smooth algebraic curves C ⊂ P2
contain no transcendental copy of C.
• Synthetic, visual geometry of P2(C) :
z
C2
w
0z
C2
w
0
C1∞,z
∞w
∼=
P1∞
P1∞
∼=
C2
w
0z
C1∞,w
∞z
• Principle for the normalisation (desingularization)of a plane curve having only simple normal crossings :
charts 1 charts 2
6
C1∞,z
C2z,w
∞w
C
z
w
C1∞,z
∞w
C2z,w
C
C
w
z
p1
E
E
p3 q2
p2
q1
{R = 0
}
{Rx = 0
}
C2x,y
{Ry = 0
}
C1x,∞
7
• Normalization map :
C1∞,z
C2z,w
∞w
C
σ C
z
w
• Application of Riemann-Hurwitz formula :
C1∞,z
C2z,w
∞w
σ
πzΦ
P1z
C
C
z
w
8
II – Kobayashi Hyperbolicity Conjectures
• Question : What happens in higher dimension ? Forinstance :
f : C −→ X2 = smooth algebraic surface ⊂ P3
B B B There does not exist yet a satisfactory, completeRiemann-Poincaré-Koebe uniformization theorem in di-mension > 2 ! !
Search for general Picard theorem in higher dimension
• Kobayashi 1970 conjecture : Most complex algebraic(smooth) hypersurfaces :
Xn ⊂ Pn+1
of high degree should contain no nonconstant entireholomorphic curves :
f : C −→ Xn .
• First, original 1970 formulation : Complements ofgeneric hypersurfaces :
Pn∖Xn−1
of high degree should contain no nonconstant entireholomorphic curves :
f : C→ Pn∖Xn−1.
n = ambiant dimension of the target space
9
• Picard-Cartan-Green-Griffiths-Fujimoto :
Pn∖2n + 1 hyperplanes in general position
contains no nonconstant entire holomorphic curveand is hyperbolically embedded in Pn.
• Kobayashi pseudometric :• X a connected complex manifold ;• p, q ∈ X two arbitrary points ;• chains of (small) holomorphic discs joining p to q ;
D
D
D
D
copies of theunit disc
curved holomorphic discs in the complex manifold
Kobayashi-pseudodistance(p, q
):=
:= infall chainsof discs
∑Poincaré distances in unit discs
• Kobayashi-Royden infinitesimal metric :• X a connected complex manifold ;
10
• p ∈ X a point ;• vp ∈ TpX a tangent vector ;• Look at largest holomorphic discs through (p, vp
):
Royden-lengthp(vp):=
:= inf
{ε > 0 : there exists a holomorphic disc f : D→ X
with f (0) = p and tangential f∗
(∂
∂x
)=
vpε
}.
D∂∂x
f
p
X
f∗(
∂∂x
)
• Definition-Proposition : The manifold X is saidKobayashi-hyperbolic if the integrated pseudodistance :
infγ : p→q
∫ 1
0
Roydenγ(t)(γ′(t)
)dt =: Kobayashi-pseudodistance
(p, q
)(which satisfies the triangle inequality) is a true distance :Kobayashi-pseudodistance
(p, q
)> 0 whenever p 6= q .
• Brody theorem (1978) : A compact complexmanifold X, e.g. a projective Xn ⊂ Pn+c(C), isKobayashi-hyperbolic if and only if every entireholomorphic curve f : C→ X is constant.
11
III – Review of existing results
• Remind : Picard-Cartan-Green-Griffiths-Fujimoto :
Pn∖2n + 1 hyperplanes in general position
is Kobayashi-hyperbolic and is hyperbolically embeddedin Pn.
=⇒ main justification to believe in :
• Optimal Kobayashi conjecture : Complements :Pn
∖Xn−1
of Zariski-generic hypersurfaces of degree :degX > 2n + 1
should be hyperbolic and hyperbolically embedded.
Theorem. [Tiba 2011] The complement :
Pn∖H1 ∪ · · · ∪Hn ∪Hn+1 ∪ Z
of (n+1) hyperplanes union a generic hypersurface(all in general position) of degree :
degZ > n
is also Kobayashi-hyperbolic and hyperbolicallyembedded in Pn.
• Observation : The sum of the degrees :1 + · · · + 1 + 1 + n = 2n + 1
is again optimal !
12
• Optimal Picard-Brody-Kobayashi conjectures : Nononconstant entire holomorphic curves should exist :� in Zariski-generic hypersurfaces Xn ⊂ Pn+1 when :
degX > 2n + 1,
� in Zariski-generic complements Pn∖Xn−1 when :
degX > 2n + 1.
n = ambiant dimension of the target space
• Major obstacle in the field : Results are much lessaccessible when the hypersurface :
Xn−1 ⊂ Pn
is irreducible and smooth, which is the generic case in theparameter space P(
n−1+dd )−1 of degree d hypersurfaces.
• Masuda-Noguchi 1996 : Very first examples ofsmooth Kobayashi-hyperbolic hypersurfaces Xn ⊂Pn+1 with Kobayashi-hyperbolic complements in arbi-trary dimension n.� In P4, the hypersurface :
0 = z1921 + · · · + z1925 − z1281 z642 − z1282 z643 + z1283 z644 + z1284 z641 .
� In P5, the hypersurface :0 = z11251 + · · · + z11256 − z2251 z8002 − · · · − z2255 z8001 +
+ z4501 z6753 + · · · + z4505 z6752 .
� Algorithms for examples in arbitrary dimension n.
13
• Zaıdenberg 1989 : Example of an irreducible quinticX1 ⊂ P2C whose complement contains no nonconstantentire holomorphic curve.
Theorem. [Siu-Yeung, Inventiones 1996] Comple-ments P2
∖X1 of generic curves X1 are Kobayashi-
hyperbolic as soon as :degX > 1013.
14
III.1 – Link with the Green-Griffiths-Lang conjecture
• Abstract compact n-dimensional complex mani-fold :
Xn .
• Canonical line bundle :KX := ΛnT ∗X .
• General type : When m→∞, require :dimH0(X, K⊗mX
)> c ·mdimX ,
for a certain constant c > 0.
• Equivalent characterization for hypersurfacesXn ⊂ Pn+1(C) :
degX > dimX + 3 .
• Green-Griffiths-Lang conjecture in the case of hy-persurfaces : If the projective algebraic hypersurfaceXn ⊂ Pn+1(C) is generic of degree d > n + 3, thenthere should exist a proper algebraic subvariety Y ⊂ Xsuch that every nonconstant entire holomorphic curvef : C → X is necessarily completely contained insideY , namely : f (C) ⊂ Y .
Y
f (C)
Xf
C
15
Theorem. [Diverio-Merker-Rousseau, Inventiones 2010] IfXn is a Zariski-generic hypersurface of Pn+1, thenthere exists a proper algebraic subvariety Y $ Xsuch that f (C) ⊂ Y for every nonconstant entireholomorphic curve as soon as :
degX > 2n5.
• Add a one-half page trick : Can even deduce from theproof that :
codimX(Y ) > 2.
Corollary. [Almost implicit in Rousseau 2007 ; note pub-lished by Diverio-Trapani] Generic hypersurfaces X3 ⊂P4 of degree :
degX > 593
are Kobayashi-hyperbolic.
• Still open : No completely rigorously convincingproof of Kobayashi hyperbolicity exists in dimension :
n > 4.
• Main obstacle towards Kobayashi hyperbolicity :The equations of the subvariety Y ⊂ X are not known.
16
Theorem. [Darondeau 2013, Ph.D.] If Xn−1 ⊂ Pn is ageneric hypersurface of degree :
degX > 5nnn,
then there is a proper algebraic subvariety Y ⊂ Pnof codimension :
codimX(Y ) > 2,
inside which all nonconstant entire maps :f : C −→ Pn
∖X
must necessarily land :f (C) ⊂ Y.
• Same degree bound in the "compact case" :
f : C −→ Xn ⊂ Pn+1.
• Recent progress on degree bounds :
� Diverio-Merker-Rousseau 2010 : [Compact case]
degX > 2n5.
� Berczi 2011 : [Compact case]
degX > n8n.
� Darondeau 2012 : [Complement case]
degX > n3n.
17
� Demailly 2012 : [Compact case]
degX > n4
3
(n log
(n(log(24n)
))n∼
(logn
)nnn.
� Darondeau 2013 : [Compact and complement cases]
degX > 5nnn.
• Comparative synthesis : When n→∞ :2n
5� nn2︸︷︷︸MERKER 2010
unpublished
� n8n︸︷︷︸BERCZI 2011
published
� n3n︸︷︷︸DARONDEAU 2012
in progress
� (n log n)n︸ ︷︷ ︸DEMAILLY 2012
Hanoi
� (5n)n︸ ︷︷ ︸DARONDEAU 2013completely checked
.
• New tools :
� As Dethloff-Lu and as Rousseau, Darondeau uses theconcept of logarithmic jet differentials, first introducedand developed by Noguchi in 1981.
� Berczi and Darondeau use equivariant cohomologycomputations and multidimensional residue formulas.
• Some reasonable hope exists to reach :degX > constantn.
� The choice of a ratio r > 3 being free, the difficulty isto minorate by :
(n2)!
(n!)nrn(n−1)
2
18
the constant term in the following product of multidi-mensional Laurent series :
Constant term in(A · E · F
),
where :A :=
∑06k26n
06k36n+k2·····················06kn6n+kn−1
(n2)!
(n + kn)! (n + kn−1 − kn)! · · · (n + k2 − k3)! (n− k2)!
rn(n−1)
2 −k2−···−kn 1
(w2)k2 · · · (wn)kn,
where :E =
1− w2
1− 2w2
1− w2w3
1− 2w2w3· · · · · · · · · 1− w2w3w4 · · ·wn
1− 2w2w3w4 · · ·wn,
and where :F :=
1− w3
1− 2w3 + w2w3
1− w3w4
1− 2w3w4 + w2w3w4· · · · · · 1− w3 · · ·wn
1− 2w3w4 · · ·wn + w2 · · ·wn
1− w4
1− 2w4 + w3w4· · · · · · · · · 1− w4 · · ·wn
1− 2w4 · · ·wn + w3w4 · · ·wn
· · · · · · · · · · · · · · · · · · 1− wn
1− 2wn + wn−1wn.
• Computational difficulty : Simultaneous presence ofpositive contributions and of negative contributions.
19
III.2 – Jet bundles strategy
• Represent the entire map in a local chart :C −→ Cn
ζ 7−→ f (ζ) =(f1(ζ), . . . , fn(ζ)
).
• Algebraic differential operator of order κ :
P(f ′, f ′′, . . . , f (κ)
)=
∑α1,α2,...,ακ∈Nn
pα1α2...ακ(f )·
· (f ′)α1(f ′′)α2 . . . (f (λ))αλ . . . (f (κ))ακ,the sum being finite, where the pα1α2...ακ are holomor-phic functions, and where :
(f (λ))αλ = (f(λ)1 )αλ,1 . . . (f
(λ)n )αλ,n.
• Green-Griffiths jet bundle : Denote by :
Gκ,m
the bundle — introduced by Green-Griffiths in 1979 —whose sections are differential operators of order κ thatare homogeneous of fixed weight :
|α1| + 2|α2| + · · · + κ|ακ| =: m
m = total number of primes�� 1.
20
Theorem. [Bloch, Ahlfors, Noguchi, Green-Griffiths, De-mailly, Siu] Let X ⊂ Pn+1 be a complex projectivealgebraic hypersurface, let A be an ample line bun-dle on X — take e.g. simply A = OX(1) — and let :
P ∈ H0(X,Gκ,m⊗A−1)
be a nonzero global section. Then every noncon-stant entire holomorphic curve f : C → X satisfiesthe corresponding differential equation :
P(f ′, . . . , f (κ)
)≡ 0.
• Current main strategy in Kobayashi hyperbolicity :� Step I : Find many jet polynomials :
P(zj, z
′j1, z′′j2, . . . , z
(κ)jκ
)that provide a global algebraic differential equation :
0 ≡ P(fj(ζ), f
′j1(ζ), f ′′j2(ζ), . . . , f
(κ)jκ
(ζ))
satisfied by every nonconstant entire holomorphic curvef : C→ X .
� Step II : Find so many such differential equations
that one can eliminate all the f ′j1, f′′j2
, . . ., f (κ)jκ.
• State of the art : This second step is understoodthanks to Siu’s slanted vector fields [Merker 2009]. �So one just needs to construct sections of jet bundles
21
III.3 – Asymptotic positivity
• Diverio-Merker-Rousseau 2010 : Introduce a certainpartially controlled jet subbundle :
Jm ⊂ Gκ,mof the Green-Griffiths jet bundle with jet order :
κ = n
equal to the dimension of X .
• Denote the degree of the hypersurface by :d := degX.
• Use asymptotic Morse inequalities to show :
dimH0(X, Jm)> polynomialn(d) ·mmax+O
(mmax−1),
where :polynomialn(d) = a0(n)︸ ︷︷ ︸
positive
dn+1 + a1(n) dn + · · · .
Lemma. If the degree d of the hypersurface is :d� n,
and if number of primes m is :m� d� n,
then there exist nonzero jet differentials on X. �• Number of primes :
m � d � 2n5.
22
III.4 – Existence of jet differentials on general type hypersurfaces
Theorem. [Green-Griffiths 1980] The Euler character-istic of the bundle of Green-Griffiths jets is asymp-totically equal to :
m(κ+1)n−1
(κ!)n((κ + 1)n− 1
)! n!
[[d(d− n− 2)n (log κ)n + O
((log κ)n−1
)]+ O
(m(κ+1)n−2),
where d = degX is the degree of the hypersurface.• General type : Hence provided only that :
d > n + 3,
Euler characteristic tends to∞ when both κ,m −→∞.
Theorem. [M. 2010] Existence of algebraic differen-tial equations in optimal degree :
degX > n + 3.
More precisely :# algebraic differential equations minorated by
> m(κ+1)n−1
(κ!)n ((κ+1)n−1)!
{(log κ)n
n! d (d− n− 2)n − inferior remainder}.
• Observations :� Jet order κ→∞ is necessary.
� Yet the number of primes m� κ tends to∞.
� The approach is intrinsic, not really effective. �
23
• Young diagram :
YD(`1,...,`d1)
`2
`1
`d1d1
2
1 `1
`d1d1
2
1
λ1d1
λ12
λ11 λ2
1 λ 1
1
`2
YD(`1,...,`d1)(λj
i )
• Semi-standard tableau :
λ11 λ1
nλ15λ1
4λ13λ1
2
Weakly increasingλb1
λa3λa
2λa1
2 4 5 711 2 413
Strictly increasingExample :
• Hasse diagram :
24
1
2
3
4
5
12
45
35
25
15
34
24
23
13
14
• Determine the leading terms of a canonical Gröb-ner basis :
incomparable pairs
2
3
4
5
1
12
45
35
15 24
23
13
14
25 34
25
• Paths :
κ
12
µν1κ
12
3
λ
κ− 2κ− 1
κ, κ− 1
23
2434
1413
15
path κ 7→ 12
Bifurcation point
• Dickson lemma :
x10
x2
• Quotient by an ideal :
26
x10
x2
j
1
2
n−1n
i
d1
`n
`n−1 dj
`i
`2
`1
d`1
1 32 `1
weak increase
increasestrict
λ12
λ1i
λ21λ1
1 λj1
λji
λ`11
λ`22
λjdj
λ`ii
λ1d1 λ
`d1d1
d`1`2
`1
`i
d1
`d1
`d1−1
dj
27
j
1
2
i
`2
`1d`1
1 32 `1
`d1d1
d1−1d1
`d1−1
dj
dj
`i
dj
j
d`i
`i
`i+1
zoom
i+1
i
i− 1
28
III.5 – Proof of algebraic degeneracy
Theorem. [D.-M.-R. 2010] If the projective algebraichypersurface Xn ⊂ Pn+1(C) is generic of degreed > 2n
5, then there exists a proper algebraic sub-
variety Y ⊂ X such that every nonconstant entireholomorphic curve f : C → X is necessarily com-pletely contained inside Y , namely : f (C) ⊂ Y .
Y
f (C)
Xf
C
• Apply Morse inequalities to get at least one nonzerojet differential :
0 6= P ∈ H0(X, Jm).
• Universal hypersurface : Define, in a system of ho-mogeneous coordinates :
[Z] = [Z0 : Z1 : · · · : Zn : Zn+1] ∈ Pn+1
[A] =[(Aα)α∈Nn+2, |α|=d
]∈ P
(n+1+d)!(n+1)! d!
−1,
the universal hypersurface of degree d :{0 =
∑|α|=d
Aα0,...,αn+1 Zα00 · · ·Z
αn+1n+1
}⊂ Pn+1 × P
(n+1+d)!(n+1)! d!
−1.
29
• Extend abstractly the jet differential generically :
P(z, A
)=
∑|α1|+···+n|αn|=m
pα(z, A
) (z′)α1 · · · (z(n))αn.
• Initial (generic) hypersurface :
X = X0←→ parameter A0.
• Siu’s slanted vector fields [M. 2009] : Construct first-order differential operators on the universal hypersurfacein order to eliminate the true derivatives z′, z′′, . . . , z(n).
• Maximal pôle order of these slanted vector fields [M.2009] :
PO∞(slanted vector fields
)= n2 + 2n.
V
f(C)
f(t0)
fiber
JnVf(t0)
JnVf(t0)
X?
Y
differentialLV P
another jet
{P =0}
{LV P =0}
constructing
jnf(t0)
jnf(t0)
jnf(t0)
• Eliminating true derivatives consists in applying :[derivations
(∂
∂z′, . . . ,
∂
∂z(n)
)+ slant
∂
∂A
](P(z, A)
).
• Insure easily that :
30
P(z, A) vanishes to sufficiently high order on Pn∞.
Proposition. [Siu 2004, DMR 2010] Nonconstant en-tire holomorphic curves algebraically degenerateinside :
Y :={z ∈ X : pα
(z, A0
)= 0, ∀ |α|n = m︸ ︷︷ ︸
all coefficients, very numerous
}.
31
III.6 – Comments, speculation, transition
• Tantalizing paradox :
� One receives extremely many algebraic equations :
pα(z, A0
)= 0.
� Their total number is quite impressive :
≈ mn�(2n
5)n.
� Naturally, one suspects that the common zero-setof all these pα
(z, A0
)is automatically empty, because
just (n + 1) independent equations would suffice.
� Hence Kobayashi’s conjecture — not in optimaldegree — seems well to be almost established !
� However, all intrinsic techniques :
• decomposition of jet bundles in Schur bundles,
• asymptotic Morse inequalities,
• probabilistic curvature estimates,
are intimately unable to provide a partial explicitexpression of even a single algebraic coefficientpα(z, A0
)!
32
Observation. [Siu, M.] Existing techniques and re-sults in hyperbolicity problems took an elaborate as-ymptotic cohomology detour which, quite probably,is in fact not adequate to solve Kobayashi’s hyper-bolicity conjecture.
Thesis. [M.] Something more explicit and systematiclies behind.
One must construct explicit sections of jet bundles
33
IV – Coordinate Construction of Jet Differentials
• Affine coordinates :(z1, z2, . . . , zn
)∈ Cn ⊂ Pn.
• Smooth algebraic hypersurface :Xn−1 ⊂ Pn(C).
• Polynomial defining equation :
0 = R(z1, . . . , zn
)=
∑α1+···+αn6d
Rα1,...,αn︸ ︷︷ ︸coefficients
(z1)α1 · · · (zn)αn︸ ︷︷ ︸monomials
.
� Its degree will be very high :d := degR.
� Its coefficients will be Zarizki-generic :(Rα1,...,αn
)∈ P(
n+dn )−1.
• 1970 Shoshichi Kobayashi conjecture : The goal isto prove that all entire holomorphic curves in the com-plement :
f : C −→ Pn∖X
are constant.
• Siu-Yeung 1996 : True for n = 2.
34
• Associated d-sheeted cover : Introduce one morevariable w ∈ C and consider the algebraic hypersurfaceof Y n ⊂ Cn+1 defined by :
0 = wd −R(z1, . . . , zn) .
Lemma. If the hypersurface Xn−1 ⊂ Pn is smooth,the associated hypersurface Y n ⊂ Pn+1 is alsosmooth and it projects as a d-sheeted holomorphiccover which ramifies exactly along X. �
Pn
Y Y Y
X X X
Pn+1
Consequence. Any nonconstant entire holomorphiccurve :
f : C −→ Pn∖X
can be lifted up as a nonconstant entire curve :g : C −→ Y n.
Pn
Y
X
Cg
f
35
Theorem. [April 2012] If the hypersurface Xn−1 ⊂ Pnis Zariski-generic and of degree :
d > 2n3,
then there exists a proper algebraic subset :Z ⊂ Pn
of dimension :dimZ 6 n− 2,
inside which all nonconstant entire curves :f : C −→ Pn
∖X
must in fact necessarily lie :f (C) ⊂ Z,
and this Z depends explicitly on the defining equa-tion {R = 0} of X.
But main goal is : Force entire curves to be constant !
Theorem. [25 July 2011] Moreover, under some —presumably removable — assumption about thebehavior of a huge linear system, one has :
Z = ∅,hence an answer to Kobayashi’s original conjecturein arbitrary dimension n > 1 (with non-optimal de-gree bound).
36
• Description of the technical assumption :Z ⊂
{A∗1
(Rz1
)m= 0
}· · · · · · · · · · · · · · · · · · · · ·Z ⊂
{A∗n
(Rzn
)m= 0
}.
• Very huge number of unknowns :A• = Aj11,...,j
n1 ,p1············
j1n,...,jnn,pn
which are plain complex numbers.
• Specific, selected unknowns :A∗1, . . . . . . , A
∗n which are some of the A•.
• Less huge linear system satisfied by all the A• :0 = coeff A∗ + coeff A• + · · · + coeff A•,· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·0 = coeff A∗ + coeff A• + · · · + coeff A•,
relatively
small
number
Lemma. Number equations� Number unknowns A•.• From general linear algebra, three cases can occur :
� A∗ is a free variable.
� A∗ is dependent and is not identically zero.
� A∗ is dependent, but is ≡ 0.
Proposition. [25 July 2011] Brody hyperbolicity ofgeneric Pn\Xn−1 of degree > 2n
3is completely
settled in the first two cases. �
37
IV.1 – Effective Cech cohomology
Main goal towards Kobayashi conjecture :Completely rebuild a more effective theoryof holomorphic sections of jet bundles,of exact sequences of vector bundles,and of their Cech cohomology groups.
• Open cover of a general abstract complex mani-fold :
X =⋃α∈A
Uα.
• Family of charts :xα : Uα −→ Cn
• Changes of charts :Uα ∩ Uβ
xαvvmmmmmmmmmmmmmm xβ
((QQQQQQQQQQQQQ
xα(Uα ∩ Uβ
) xβ◦x−1α//xβ
(Uβ ∩ Uα
).
• Coordinates in the source space and in the targetspace :
xα =(xα1, . . . , xαn
)∈ Cn,
xβ =(xβ1, . . . , xβn
)∈ Cn,
38
• Notation for the change-of-chart maps :xα 7−→ ϕβα(xα) =: xβ
• General rank r > 1 holomorphic vector bundle :Eπ
��
X,
• Local trivialization maps :Tα : π−1(Uα)
∼ //
π��
Uα × Cr
p1uukkkkkkkkkkkkkkkkkkkk
Uα
• Change of trivialization :(Uα ∩ Uβ
)× Cr
(Uβ ∩ Uα
)× Cr
//Tβ◦T−1α
π−1(Uα ∩ Uβ
)Tα
jjTTTTTTTTTTTTTTTTT Tβ
44jjjjjjjjjjjjjjj
π��
Uα ∩ Uβ,
• Linearity in the fibers :Tβ ◦ T−1α :
(x, v1, . . . , vr︸ ︷︷ ︸
∈Cr
)7−→
7−→(x,
r∑k=1
G1kβα(x) vk, . . . . . . ,
r∑k=1
Grkβα(x) vk
),
39
for a certain matrix-valued invertible map :
Uα ∩ Uβ 3 x 7−→(Gjkβα(x)
)16k6r
16j6r∈ GLr(C).
• Holomorphic section : Consists in a collection ofholomorphic maps :
x 7−→ vα(x)
defined in Uα, valued in Cr, which satisfy on each doubleintersection Uα ∩ Uβ :
vβ(x) = Gβα(x) vα(x) .
• Simultaneous change of chart and of trivilization :xβ = ϕβα(xα),
vβ = Gβα(xα) vα.
Work with a finite natural coveringU =
(Ui)06i6n
of an n-dimensional complex manifold :
X =⋃
06i6n
Ui
and build a cohomology theory with this covering
• Concrete, accessible example :The projective space Pn(C).
40
• Associated locally free sheaf :EE := sheaf of sections of E over U0, . . . ,Un.
• Cochains in Cech cohomology for a finite, fixed cov-ering :
Cq(U,EE) :=∏
06i0<i1<···<iq6n
Γ(EE, Ui0 ∩ Ui1 ∩ · · · ∩ Uiq
).
• Coboundary operator :coboundaryq : Cq−1(U,EE) −→ Cq(U,EE)
defined by :coboundaryq
((fi0,i1,...,iq−1
)06i0<···<iq−16n
):=
:=(fi1,...,iq − · · · + (−1)q fi0,...,iq−1
)06i0<i1<···<iq−1<iq6n
:=
( q∑r=0
(−1)r fi0,...,ir,...,iq
)06i0<i1<···<iq−1<iq6n
.
• Cocycles :Zq(U, EE
):= Ker
(coboundaryq
).
• Coboundaries :Bq(U, EE
):= Im
(coboundaryq−1
).
• Finite Cech cohomology :
Hq(U, EE):= Zq(U, EE
)/Bq(U, EE
).
41
Observation. One need to express a more rigor-ously and more effectively coboundaries in coordi-nates, for instance :
σj − σi ≡ Transfertij(σj)− σi.
• Collection of sections :σj : Cn 3 xj 7−→ vj(xj) ∈ Cr,
• Transition maps :
Tj : π−1(Uj)∼ //
π��
Uj × Cr
p1vvlllllllllllllllllll
Uj.
Tji :(Ui ∩ Uj
)× Cr −→
(Uj ∩ Ui
)× Cr(
xi, vi)7−→
(ϕji(xi), Gji(xi) · vi
)=:
(xj, vj
),
• Composition diagram for the transfer :
σi
σj
σi
σj
ϕji
Tij
Tranferij(σj
)
• Complete geometric diagram for the transfer :
42
Cn
Ui
π−1(Uj)π−1(Ui)
xi(Ui)
Cn xi
xi 7−→ ϕji(xi) =: xj
xj
X
Uj
π
(xi, vi) 7−→(ϕji(xi), Gji · vi
)=: (xj, vj)
xi 7→ vi(xi)section σi
xj(Uj)
section σj
xj 7→ vj(xj)π
X
(Ui ∩ Uj)× Cr (Uj ∩ Ui)× Cr
change of trivialisation
change of chart
• Expression of the first coboundary operator :coboundary1
((σi)06i6n
):=
(Tij ◦ σj ◦ ϕji − σi
)06i<j6n
.
• Expression of the second coboundary operator :coboundary2
((σi,j)06i<j6n
):=
:=(Tij ◦ σj,k ◦ ϕki − σi,k + σi,j
)06i<j<k6n
.
Main goal is :develop a finite cohomology theory :
43
IV.2 – Illustration : Cech cohomology of OP2(t)
• Projective 2-dimensional space :P2 3 [T : X : Y ].
• Three systems of affine coordinates :
x0 :=X
T, y0 :=
Y
T,
x1 :=T
X, y1 :=
Y
X,
x2 :=T
Y, y2 :=
X
Y.
• Two changes of affine charts :
� (1/x0)-change :(x0, y0) 7−→
( 1x0, y0x0
)=: (x1, y1);
� (1/y0)-change :(x0, y0) 7−→
(x0y0, 1y0
)=: (x2, y2).
• Canonical line bundles for t ∈ Z :OP2(t)
• Three chart-trivializations :((x0, y0), `0
),(
(x1, y1), `1),(
(x2, y2), `2),
44
• Identifications : `1 =1
(x0)t`0,
`2 =1
(y0)t`0,
`0 =1
(x1)t`1,
`2 =1
(y1)t`1,
`0 =1
(y2)t`2,
`1 =1
(x2)t`2.
• Two intersections :U0 ∩ U1 =
{[T6=0
: X6=0
: Y ]}={[1 : x06=0
: y0]}=
= C∗ × C in the space of (x0, y0),U0 ∩ U2 =
{[T6=0
: X : Y6=0]}={[1 : x0 : y0
6=0]}=
= C× C∗ in the space of (x0, y0),U1 ∩ U2 =
{[T : X6=0
: Y6=0]}={[x1 : 1 : x0
6=0]}=
= C× C∗ in the space (x1, y1),
• Three intersections :U0 ∩ U1 ∩ U2 =
{[T6=0
: X6=0
: Y6=0]}={[1 : x06=0
: y06=0]}=
= C∗ × C∗ in the space (x0, y0).
• Re-prove elementary result :
• h0(P2, OP2(t)
)=
0 for t 6 −1,(t + 1)(t + 2)
1 · 2for t > −1;
• 0 = h1(P2, OP2(t)
), always ;
45
• h2(P2, OP2(t)
)=
(t + 1)(t + 2)
1 · 2for t 6 −2,
0 for t > −1.• Connecting homomorphism in finite cohomology :On a complex manifold X of dimension n > 1 equippedwith an open cover U =
(Ui)06i6n, let a sequence of
sheaf homomorphisms be given :
0 −→ Fα−→ G
β−→ H −→ 0,
which satisfies, for i = 0, 1, . . . , n, finite exactness :
0 −→ F (Ui)αUi−→ G (Ui)
βUi−→ H (Ui) −→ 0.
Then there is a connection homomorphism :connection1 : H0(U,H )
−→ H1(U,F),
satisfying :
0 −→ H0(U,F
) α0−→ H0(U,G
) β0−→ H0(U,H
) connection1−→connexion1−→ H1
(U,F
) α1−→ H1(U,G
) β1−→ H1(U,H
).
and one has the exactness :0 = Kerα0,
Imα0 = Kerβ0,Im β0 = Ker
(connection1
),
Im(connection1
)= Kerα1,
Imα1 = Kerβ1.
46
IV.3 – Cech cohomology of vector bundles on hypersurfaces
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U1
U2 V2V1
ϕ1
ϕ2
M
E
M
E
X1P2
X1P2
∞y
∞x
• Proposition (Choice of adapted coordinates for asmooth curve X1 ⊂ P2). There always exist affine coor-dinates :
(x, y) ∈ C2 ⊂ P2
satisfying :(i) ∞x 6∈ X1,∞y 6∈ X1 ;
47
(ii) the intersection P1∞ ∩X1 =: X0∞ is transversal and
consists in exactly d distinct points located on the line atinfinity P1∞.In such a system of coordinates, the affine cuve X1 ∩ C2
is the zero-locus of a certain degree d polynomial :0 = R(x, y),
and the two open sets :
UC2
x := {Ry 6= 0} ∩X1,
UC2
y := {Rx 6= 0} ∩X1
prolong as two global open sets in P2 :UP2x :=
{p ∈ X1 such that P1(p,∞y) is transversal to TpX
1},
UP2y :=
{p ∈ X1 such that P1(p,∞x) is transversal to TpX
1},
which cover the curve :X1 ⊂ UP
2
x ∪ UP2
y .
Moreover, one single of these two open sets suffices tocover the part of the curve at infinity :
X1 ⊂ UC2
x ∪(UP
2
x ∩ P1∞)︸ ︷︷ ︸
partie à l’infini︸ ︷︷ ︸=UP2x
⋃UC
2
y︸︷︷︸affine
.
More precisely, if the 1x-change of chart :
(x1, y1) =(1x,
yx
)
48
transforms the equation of the curve in :
0 = (x1)dR
(1x,
yx
)=: R1(x1, y1),
then the part at infinity of the first open set :
UP2
x ∩ P1∞becomes : {
R1,y1 6= 0}.
P2
∞y
∞x
p
X1
p ∈ UP2
y
tangency : p 6∈ UP2
x
Proposition (Choice of coordinates adapted to a sur-face X2 ⊂ P3). Given an arbitrary smooth complex al-gebraic surface :
X2 ⊂ P3(C)of any degree d > 1, it is always possible to find a systemof affine coordinates :
(x, y, z) ∈ C3 ⊂ P3
satisfying the following geometric situation condution :(i) ∞x 6∈ X2,∞y 6∈ X2,∞z 6∈ X2 ;(ii) the intersection P2∞ ∩X2 =: X1
∞ is transversal andit defines a smooth degree d curve in the plane at infinityP2∞ ;
49
(iii) the three lines P1(∞x,∞y), P1(∞x,∞z),P1(∞y,∞z) are transversal in P2∞ to this curve X1
∞at infinity.
In such a system of coordinates, in which the affineequation of the surface X2∩C3 is the zero-set of a certaindegree d polynomial :
0 = R(x, y, z),
the three affine open sets :
UC3
xy :={Rz 6= 0
}∩X2,
UC3
xz :={Ry 6= 0
}∩X2,
UC3
yz :={Rx 6= 0
}∩X2,
prolong as three global open sets in the projective space :
UP3
xy :={p ∈ X2 tels que P1(p,∞z) transverse à TpX
2},UP
3
xz :={p ∈ X2 tels que P1(p,∞y) transverse à TpX
2},UP
3
yz :={p ∈ X2 tels que P1(p,∞x) transverse à TpX
2}which cover the surface :
X2 ⊂ UP3
xy ∪ UP3
xz ∪ UP3
yz.
Moreover, two among these three open sets suffice tocover the part of the surface which lies at infinity :
X2 ⊂ UC3
xy ∪(UP3xy ∩ P2
∞)︸ ︷︷ ︸
part at infinity︸ ︷︷ ︸=UP3
xy
⋃UC3
xz ∪(UP3xz ∩ P2
∞)︸ ︷︷ ︸
part at infinity︸ ︷︷ ︸=UP3
xz
⋃UC3
yz︸︷︷︸affine
.
50
Three fundamental exact sequences to determine thecohomology of the cotangent bundle T ∗X to a curveX1 ⊂ P2 :• First exact sequence : (Euler)
0 −→ T ∗P2 −→ OP2(−1)⊕3 −→ OP2(0) −→ 0.
• Second exact sequence : (Normal exact sequencetensored by T ∗P2)0 −→ T ∗P2 ⊗ OP2(−d) −→ T ∗P2 ⊗ OP2(0) −→ T ∗P2 ⊗ OX1(0) −→ 0.
• Third exact sequence :0 −→ OX1(−d) −→ T ∗P2 ⊗ OX1(0) −→ T ∗
X1 −→ 0.
Theorem [Brückmann 1971]. In Pn+1, with n > 1, theEuler-Poincaré characteristic of the twisted Schur bun-dle :
S (`1,...,`n,0)T ∗Pn+1 ⊗ OPn+1(t)with `1 > · · · > `n > 0 arbitrary and with t ∈ Z, isgiven, if one sets :
r := `1 + · · · + `n,
and if one introduces the decreasing sequence of inte-gers :
t1 := r + `1 − 1,
· · · · · · · · · · · · · · ·tn := r + `n − n,
tn+1 := r − n− 1,
51
by the explicit formula :
χ(Pn+1, S (`1,...,`n,0)T ∗Pn+1 ⊗ OPn+1(t)
)=
1
1! · · · n! (n + 1)!
∏16i<j6n+1
(ti − tj
) ∏16i6n+1
(t− ti
).
Moreover, in the expression of this characteristic interms of individual cohomology groups :
χ =
n+1∑q=0
dimHq(Pn+1, S (`1,...,`n,0)T ∗Pn+1⊗OPn+1(t)
),
in reality, for every fixed integer t ∈ Z, at most one ofthese cohomological dimensions is nonzero, as is visuallyexpressed by the following diagram :
Z tn+1 tn t2 t1 Z
t1 6 t
tn+1 6 t 6 tn
t2 6 t 6 t1
t 6 tn+1,
h1 6= 0
h0 6= 0
hn 6= 0
hn+1 6= 0
that is to say more precisely, for every fixed t satisfyingtq+1 6 t 6 tq, one has :
dimHq(Pn+1, S (`1,...,`n,0)T ∗Pn+1 ⊗ OPn+1(t)
)=
= (−1)q χ(Pn+1, S (`1,...,`n,0)T ∗Pn+1 ⊗ OPn+1(t)
),
while all the other cohomological dimensions vanish.