• Aucun résultat trouvé

# Show that the following properties are equivalent: (a)u isd-exact (i.e

N/A
N/A
Protected

Partager "Show that the following properties are equivalent: (a)u isd-exact (i.e"

Copied!
2
0
0

Texte intégral

(1)

M2R Université Grenoble Alpes 2019-2020 Introduction to analytic geometry (course by Jean-Pierre Demailly)

Sheet number 9, 06/12/2019 1. (The∂∂-lemma).

Let (X, ω) be a compact Kähler manifold. Let u be a smooth d-closed (p, q)-form (i.e. ∂u = ∂u = 0.

Show that the following properties are equivalent:

(a)u isd-exact (i.e. there is a smooth form v of degreek−1, wherek=p+q, such thatu=dv).

(b)u is∂-exact (i.e. there is a smooth form v of bidegree(p−1, q) such thatu=∂v).

(c)u is∂-exact (i.e. there is a smooth form v of bidegree(p, q−1)such thatu=∂v).

(d) u is Aeppli exact (i.e. there are smooth forms v, v0 of bidegrees (p−1, q) and (p, q−1) such that u=∂v+∂v0).

(e)u is∂∂-exact (i.e. there is a smooth forms of bidegree(p−1, q−1)such thatu=∂∂s).

(f)u⊥Hp,q(X,C)(orthogonality to harmonic (p, q)-forms).

Hint. It is easy to show that (e) implies (a), (b), (c) and (d). Also, each of the four properties (a), (b), (c), (d) implies (f), so it is enough to show that (f) implies (e). For this, assuming (f), show thatu can be decomposed as ∂v+∂w and observe that ∂w must be zero. Then use a three term decomposition ofv with respect to ∂ and ∂ and conclude.

2. (Bott-Chern cohomology).

If X is any complex manifold, one defines HBCp,q(X,C) to be the quotient of the space of (p, q)-forms u such that∂u=∂u= 0 by ∂∂-exact (p, q)-forms ∂∂v.

(a) Show that there are always natural morphisms HBCp,q(X,C)→Hp,q

(X,C), M

p+q=k

HBCp,q(X,C)→HDRk (X,C).

(b) When (X, ω) is compact Kähler, show that the above natural morphisms are isomorphisms and correspond to the Hodge decomposition in terms ofω-harmonic forms. Derive from there that the Hodge decomposition is independent of the choice of the Kähler metric ω.

3. (First Chern class and Lelong-Poincaré equation).

(a) Let L → X be a holomorphic line bundle on any complex manifold X, and h a smooth Hermitian metric onL. One defines the first Chern class ofL, denotedc1(L), to be the cohomology class{i ΘL,h} associated with the curvature tensor ΘL,h of the Chern connection. Show that this defines a real coho- mology class in HBC1,1(X,C), and also a De Rham cohomology class in HDR2 (X,R), independent of the metrich.

Hint. If h0 is another metric, there exists a global weight ψ ∈ C(X,R) such that h0 = he−ψ. Then relateΘL,h0 and ΘL,h.

(b) IfX is compact Kähler of dimensionn, one defines the degree of Lwith respect to ω to be degω(L) :=

Z

X

c1(L)∧ {ω}n−1= Z

X

i

2πΘL,h∧ωn−1.

This degree is independent ofh. Show that this is true more generally ifω satisfies ∂∂(ωn−1) = 0(such a metric is called a Gauduchon metric; Paul Gauduchon has shown in 1977 that such a metric exists on any compact complex manifold).

(c) Let s be a meromorphic section of L that does not vanish identically on any connected component of X (this means that locally s = f e where e is a non vanishing holomorphic section and f = g/h a meromorphic section). Let D = P

mjDj, mj ∈ Z, be the divisor of zeroes and poles of s. Prove the Lelong-Poincaré formula, asserting that

i

2π∂∂log|s|2h= [D]− 1 2πΘL,h

1

(2)

in the sense of currents. Infer that c1(L) coincides with the cohomology class of [D]in HBC1,1(X,C). In particular, in dimensionn= 1,

deg(L) :=

Z

X

c1(X) = degD:=X

mj ∈Z. 4. (Projective space and the O(m) line bundles)

On complex projective spacePn=Cn+1r{0},n≥1, one defines the “tautological line bundle”O(−1)to be the subbundle of the trivial bundleCn+1 such the fiber at a point x= [z] = [z0:z1:. . .:zn]consists of the lineCz, i.e. O(1)[z]=Cz. Its “tautological metric” is the one induced by the standard Hermitian metric of Cn+1, |z|2 =P

|zj|2. Finally, one defines O(1) to be the dual O(−1) =O(−1)−1 and O(m), m∈Z, to be them-th tensor product ofO(1)(or the (−m)-th tensor product ofO(−1)).

(a) IfP(z) =P

|α|=maαzα is a homogeneous polynomial of degreem on Cn+1, i.e. P(z) =a·zm where zm ∈ Sm(Cn+1) and a = P

aα(e)α ∈ Sm(Cn+1) with respect to the canonical basis (ej) of Cn+1, thenP can be seen as a section of O(m). In particular, every linear form in (Cn+1) defines a section of O(1) =O(−1).

(b) Conversely, ifσ is a holomorphic section ofO(m) overPn, then f(z) =σ([z])·zm (which should be interpreted as a division σ([z])/zm for m <0) defines a holomorphic function on Cn+1r{0} such that f(λz) = λmf(z). Then must be a homogeneous polynomial of degree m for m ≥ 0, and must vanish identically ifm <0.

(c) Show that the curvature ofO(1) with respect to its standard Hermitian metric yields i ΘO(1),std =

i

∂∂log|z|2FS, i.e. the Fubini-Study metric. In particular O(1)has positive curvature.

(d) Ifσ is a holomorphic section ofO(m)and Y its zero divisor (in particular, ifY is a smooth degreem hypersurface), then

Z

Y

ωFSn−1= Z

Pn

[Y]∧ωn−1FS =m Z

Pn

ωnFS. Hint. Use the Lelong-Poincaré formula.

(e) Use (d) and induction onnto infer that R

PnωFSn = 1 by takingY to be the hyperplane zn = 0.

5. (Riemann-Roch formula for compact Riemann surfaces, i.e. compact complex curves) On a curve, a divisor D= P

mj[pj]is a just a sum of points pj ∈X with multiplicities mj ∈ Z. The line bundle (or rather, invertible sheaf)OX(D) consists of the sheaf of germs of meromorphic functions f such thatdiv(f) +D≥0 (orf = 0).

(a) One defines the “tautological meromorphic section” sD of OX(D) to be given by the meromorphic function1. Show that it is holomorphic as a section ofOX(D) if and only ifD≥0and, in general, that its zero and pole divisor, as a section ofOX(D) (and not as a meromorphic function!), is equal toD.

For a line bundleL→X on a compact complex manifold, one defines its Euler characteristic to be χ(X, L) =

n

X

q=0

(−1)qhq(X, L)−h1(X, L), hq(X, L) = dimHq(X, L) = dimH0,q

(X, L),

thus, ifdimX= 1 (which we assume from now on),χ(X, L) =h0(X, L)−h1(X, L), and by Serre duality, h1(X, L) =h0(X,Ω1X⊗L−1).

(b) Using the long exact sequence of cohomology associated with the short exact sequence 0→L⊗OX(−[p])→L→Qp→0

whereQp is the skyscraper sheaf of fiber'Catp, show thatχ(X, L⊗O(−[p])) =χ(X, L)−1. Similarly, show thatχ(X, L⊗O([p])) =χ(X, L) + 1by using the exact sequence0→L→L⊗OX([p])→Qp→0.

(c) Infer from (b) the fundamentalRiemann-Roch formula

χ(X,OX(D)) = deg(D) + 1−g whereg=h1(X,OX) =h0(X,Ω1X) is the genus of X.

(d) Write down the long cohomology exact sequence associated withO→L→L⊗OX(m[p])→Qm,p →0 (where the stalk of Qm,p at p is ' Cm), and prove that H0(X, L⊗OX(m[p])) 6= 0 for m > h1(X, L).

If D0 ≥0 is the zero divisor of σ ∈H0(X, L⊗OX(m[p])), show thatL 'OX(D) with D=D0−m[p].

Infer that one has χ(X, L) = deg(L) + 1−g, wheredeg(L) = deg(D) = deg(D0)−m.

2

Références

Documents relatifs

Write down the transition matrix between the bases

We prove that arbitrary actions of compact groups on finite dimensional C ∗ -algebras are equivariantly semiprojec- tive, that quasifree actions of compact groups on the Cuntz

For conservative dieomorphisms and expanding maps, the study of the rates of injectivity will be the opportunity to understand the local behaviour of the dis- cretizations: we

joli (jolie), pretty mauvais (mauvaise), bad nouveau (nouvelle), new petit (petite), little vieux (vieille), old.. ordinal

While a CA is typically advised against issuing a certificate with a validity period that spans a greater period of time than the validity period of the CA’s certificate that

We show that when two closed λ-terms M, N are equal in H ∗ , but different in H + , their B¨ ohm trees are similar but there exists a (possibly virtual) position σ where they

Definition 2.1 ([Gro93, 0.5.E]) Two countable groups Γ1 and Γ2 are Measure Equivalent (ME) if there exist commuting, measure preserving, free actions of Γ1 and Γ2 on some

3Deportment of Soil Science of Temperate Ecosystems, lJniversity of Gottingen, Gdttingen, Germony 4Chair of Soil Physics, lJniversity of Bayreuth, Bayreuth,

Prove that the gamma distribution is actually a continuous probability distributions that is it is non-negative over x &gt; 0 and its integral over R + is equal to 1.. Exercise 8

[r]

Recalling the impossibility of doubting his existence and looking back to his “sum”-intuition, the Cartesian

(15) (*) A topological space is quasi-compact if one can extract from any open covering a finite open covering.. Show that a Noetherian topological space

Last year at this time the government brought down a budget which forecast an Ordinary Account surplus of \$32.4 million, an increase in the net debt of \$68.6 million and

 Godement R., Th´ eorie des faisceaux, Hermann, 1973  Iversen B., Cohomology of sheaves, Springer 1986.  Kashiwara M., Schapira P., Sheaves on Manifolds, Grundlehren

We define a partition of the set of integers k in the range [1, m−1] prime to m into two or three subsets, where one subset consists of those integers k which are &lt; m/2,

• Show that the Bose symmetry of the photon imposes a constraint on the wave function. • Find the condition on the wave function in order to have a

Delano¨e  proved that the second boundary value problem for the Monge-Amp`ere equation has a unique smooth solution, provided that both domains are uniformly convex.. This result

Central questions in the geometry of finite dimen- sional Banach spaces are those of finding the right notion of Minkowski area and, more generally, of invariant area (see Section 2

(ii) For every n ≥ 5 there exists a strictly negative curved polyhedron of dimension n whose fundamental group G is hyperbolic, which is homeomorphic to a closed aspherical

First introduced by Faddeev and Kashaev [7, 9], the quantum dilogarithm G b (x) and its variants S b (x) and g b (x) play a crucial role in the study of positive representations

A second scheme is associated with a decentered shock-capturing–type space discretization: the II scheme for the viscous linearized Euler–Poisson (LVEP) system (see section 3.3)..

Formally prove that this equation is mass conser- vative and satisfies the (weak) maximum principle.. 4) Dynamic estimate on the entropy and the first

Consider an infinite sequence of equal mass m indexed by n in Z (each mass representing an atom)... Conclude that E(t) ≤ Ce −γt E(0) for any solution y(x, t) of the damped