Holomorphic Morse inequalities and

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Holomorphic Morse inequalities and entire curves on projective varieties Jean-Pierre Demailly Universite de Grenoble I, Institut Fourier March 2012 Abstract Plurisubharmonic functions and positive currents are an essential tool of modern complex analysis. Since their inception by Oka and Lelong in the mid 1940's, major applications to algebraic and analytic geometry have been developed in many directions. One of them is the Bochner-Kodaira technique, providing very strong existence theorems for sections of holomorphic vector bundles with positive curvature via L2 estimates; one can mention here the foundational work achieved by Bochner, Kodaira, Nakano, Morrey, Kohn, Andreotti-Vesentini, Grauert, Hormander, Bombieri, Skoda and Ohsawa- Takegoshi in the course of more than 4 decades. Another development is the theory of holomorphic Morse inequalities (1985), which relate certain curvature integrals with the asymptotic cohomology of large tensor powers of line or vector bundles, and bring a useful complement to the Riemann-Roch formula. We describe here the main techniques involved in the proof of holomorphic Morse inequalities (chapter I) and their link with Monge-Ampere operators and intersection theory (chapter II). The last two chapters III, IV provide applications to the study of asymptotic cohomology functionals and the Green-Griffiths-Lang conjecture. The latter conjecture asserts that every entire curve drawn on a projective variety of general type should satisfy a global algebraic equation; via a probabilistic curvature calculation, holomorphic Morse inequalities imply that entire curves must at least satisfy a global algebraic differential equation.

  • plurisubharmonic functions via bergman kernels

  • rham cohomology

  • vector bundles

  • compact complex

  • morse inequalities

  • atiyah-bott-patodi proof

  • laplace-beltrami operators


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Holomorphic Morse inequalities and
entire curves on projective varieties
Jean-Pierre Demailly
Universit´e de Grenoble I, Institut Fourier
March 2012
Abstract
Plurisubharmonic functions and positive currents are an essential tool of modern
complex analysis. Since their inception by Oka and Lelong in the mid 1940’s, major
applications to algebraic and analytic geometry have been developed in many directions.
One ofthemistheBochner-Kodairatechnique, providingverystrongexistencetheorems
2for sections of holomorphic vector bundles with positive curvature via L estimates;
one can mention here the foundational work achieved by Bochner, Kodaira, Nakano,
Morrey, Kohn, Andreotti-Vesentini, Grauert, H¨ormander, Bombieri, Skoda and Ohsawa-
Takegoshi in the course of more than 4 decades. Another development is the theory
of holomorphic Morse inequalities (1985), which relate certain curvature integrals with
the asymptotic cohomology of large tensor powers of line or vector bundles, and bring a
useful complement to the Riemann-Roch formula.
We describe here the main techniques involved in the proof of holomorphic Morse
inequalities (chapter I) and their link with Monge-Amp`ere operators and intersection
theory (chapter II). The last two chapters III, IV provide applications to the study
of asymptotic cohomology functionals and the Green-Griffiths-Lang conjecture. The
latter conjecture asserts that every entire curve drawn on a projective variety of general
type should satisfy a global algebraic equation; via a probabilistic curvature calculation,
holomorphic Morse inequalities imply that entire curves must at least satisfy a global
algebraic differential equation.
The author expresses his warm thanks to the GAGC organizers for their invitation
andtheopportunitytodeliverasubstantialpartoftheselecturestoanaudienceofyoung
researchers. These notes are also an expansion of a course given at the CIME School
in Pluripotential Theory held in Cetraro in July 2011, organized by Filippo Bracci and
JohnErik Fornæss.2 J.-P. Demailly Holomorphic Morse inequalities and entire curves
Contents
Chapter I. Holomorphic Morse inequalities..........................................3
0. Introduction.....................................................................................3
1. Holomorphic Morse inequalities..................................................................9
2. Applications to algebraic geometry..............................................................21
3. Morse inequalities on q-convex varieties.........................................................27
4. Holomorphic Morse inequalities for vector bundles...............................................30
Chapter II. Approximation of currents and intersection theory..............35
0. Introduction....................................................................................35
1. Pseudo-effective line bundles and singular hermitian metrics.....................................35
2. Hermitian metrics with minimal singularities and analytic Zariski decomposition.................37
3. Description of the positive cones (K¨ahler and projective cases)...................................39
4. Approximation of plurisubharmonic functions via Bergman kernels.............................. 43
5. Global approximation of closed (1,1)-currents on a compact complex manifold...................46
6. Zariski decomposition and mobile intersections..................................................53
7. The orthogonality estimate..................................................................... 61
8. Dual of the pseudo-effective cone................................................................63
Chapter III. Asymptotic cohomology functionals
and Monge-Amp`ere operators..........................................................67
0. Introduction and main definitions...............................................................67
1. Extension of the functionals to real cohomology classes..........................................68
2. Transcendental asymptotic cohomology functions................................................71
3. Invariance by modification......................................................................75
4. Proof of the infimum formula for the volume....................................................76
5. Estimate of the first cohomology group on a projective surface...................................78
6. Singular holomorphic Morse inequalities.........................................................82
7. Cohomology estimates for effective divisors......................................................83
Chapter IV. Morse inequalities and
the Green-Griffiths-Lang conjecture..................................................87
0. Introduction....................................................................................87
1. Hermitian geometry of weighted projective spaces...............................................93
2. Probabilistic estimate of the curvature of k-jet bundles..........................................97
3. On the base locus of sections of k-jet bundles.................................................. 113
References..................................................................................119Chapter I
Holomorphic Morse inequalities
Holomorphic Morse inequalities provide asymptotic bounds for the cohomology of
tensor bundles of holomorphic line bundles. They are a very useful complement to the
Riemann-Roch formula in many circumstances. They were first introduced in [Dem85],
and were largely motivated by Siu’s solution [Siu84, Siu85] of the Grauert-Riemen-
schneiderconjecture,whichwereprovehereasaspecialcaseofastrongerstatement. The
basic tool is a spectral theorem which describes the eigenvalue distribution of complex
Laplace-Beltrami operators. The original proof of [Dem85] was based partly on Siu’s
techniques and partly on an extension of Witten’s analytic proof of standard Morse in-
equalities [Wit82]. Somewhat later Bismut [Bis87] and Getzler [Get89] gave new proofs,
bothrelyingonananalysisoftheheatkernelinthespiritoftheAtiyah-Bott-Patodiproof
of the Atiyah-Singer index theorem [ABP73]. Although the basic idea is simple, Bismut
used deep results arising from probability theory (the Malliavin calculus), while Getz-
ler relied on his supersymmetric symbolic calculus for spin pseudodifferential operators
[Get83].
We present here a slightly more elementary and self-contained proof which was sug-
gested to us by Mohan Ramachadran on the occasion of a visit to Chicago in 1989.
The reader is referred to [Dem85, Dem91] for more details.
0. Introduction
0.A. Real Morse inequalities
∞Let M be a compact C manifold, dim M = m, and h a Morse function, i.e. aR
function such that all critical points are non degenerate. The standard (real) Morse
qinequalities relate the Betti numbers b =dimH (M,R) and the numbersq DR
s =# critical points of index q ,q
where the index of a critical point is the number of negative eigenvalues of the Hessian
2form (∂ h/∂x ∂x ). Specifically, the following “strong Morse inequalities” hold :i j
q q(0.1) b −b +···+(−1) b 6s −s +···+(−1) sq q−1 0 q q−1 0
for each integer q > 0. As a consequence, one recovers the “weak Morse inequalities”
b 6s and the expression of the Euler-Poincar´e characteristicq q
m m(0.2) χ(M)=b −b +···+(−1) b =s −s +···+(−1) s .0 1 m 0 1 m4 J.-P. Demailly Holomorphic Morse inequalities and entire curves
These resultas are purely topological. They are obtained by showing that M can be
reconstructed from the structure of the Morse function by attaching cells according to
theindexofthecriticalpoints; realMorseinequalitiesarethenobtainedasaconsequence
of the Mayer-Vietoris exact sequence (see [Mil63]).
0.B. Dolbeault cohomology
Instead of looking at De Rham cohomology, we want to investigate here Dolbeault
cohomology, i.e. cohomology of the ∂-complex. Let X be a compact complex manifold,
n = dim X and E be a holomorphic vector bundle over X with rankE = r. Let usC
recall that there is a canonical ∂-operator
∞ p,q ∗ ∞ p,q+1 ∗(0.3) ∂ :C (X,Λ T ⊗E)−→C (X,Λ T ⊗E)X X
actingonspacesof(p,q)-formswithvaluesinE. BytheDolbeaultisomorphismtheorem,
there is an isomorphism
p,q q p∞ p,• ∗ q(0.4) H (X,E):=H (C (X,Λ T ⊗E))≃H (X,Ω ⊗ (E))X X∂ ∂
from the cohomology of the∂-complex onto the cohomology of the sheaf of holomorphic
p-forms with values in E. In particular, we have
0,q q(0.5) H (X,E)≃H (X, (E)),

q qand we will denote as usual h (X,E)=dimH (X, (E)).
0.C. Connections and curvature
∞Leut us consider first a C complex vector bundle E → M on a real differential
manifoldM (without necessarily any holomorphic structure at this point). A connection
D on E is a linear differential operator
∞ q ∗ ∞ q+1 ∗(0.6) D :C (M,Λ T ⊗E)→C (M,Λ T ⊗E)M M
satisfying the Leibniz rule
deg f(0.7) D(f∧s) =df∧s+(−1) f∧Ds
∞ p ∗ ∞ q ∗for all forms f ∈C (X,Λ T ), s∈C (X,Λ T ⊗E). On an open set U ⊂M whereM M
rE is trivial, E ≃ U ×C , the Leibniz rule shows that a connection D can be written|U
in a unique way
(0.8) Ds≃ds+Γ∧s
∞ 1 ∗ r rwhere Γ ∈ C (U,Λ T ⊗Hom(C ,C )) is an arbitrary r×r matrix of 1-forms and dM
acts componentwise. It is then easy to check that
2(0.9) D s≃(dΓ+Γ∧Γ)∧s on U.
2 ∞ 2 ∗ThereforeD s=θ ∧s for some global 2-formθ ∈C (M,Λ T ⊗Hom(E,E)), givenD D M
by θ ≃dΓ +Γ ∧Γ on any trivializing open set U with a connection matrix Γ .D U U U U
O
O
OChapter I, Holomorphic Morse inequalities 5
i(0.10)Definition. The (normalized) curvature tensor ofD is defined to be Θ = θ ,D D2π
in other words
i 2D s= Θ ∧sD

∞ q ∗for any section s∈C (M,Λ T ⊗E).M
iThe main reason for the introduction of the factor is the well known formula for2π
the expression of the Chern classes in the ring of differential forms of even degree: one
has
2 rdet(Id+λΘ )=1+λγ (D)+λ γ (D)+...+λ γ (D),D 1 2 r
where γ (D) is a d-closed differential form of degree 2j. Moreover, γ (D) has integralj j
2jperiods, i.e. the De Rham cohomology class {γ (D)} ∈ H (M,R) is the image of anj
2jintegral class, namely the j-th Chern class c (E)∈H (M,Z).j
0.D. Hermitian connections
∞Assume now that the fibers of E are endowed with a C hermitian metric h, and
r ∞that the isomorphism E ≃ U ×C is given by a C frame (e ). Then we have a|U λ
canonical sesquilinear pairing
∞ p ∗ ∞ q ∗ ∞ p+q ∗C (M,Λ T ⊗E)×C (M,Λ T ⊗E) −→ C (M,Λ T )M M M
(u,v) −→ u{,v}h
given by
X X X
{u,v} = u ∧v he ,e i for u= u ⊗e , v = v ⊗e .h λ µ λ µ h λ λ µ µ
λ,µ
The connectionD is said to be hermitian (or compatible with the hermitian metrich) if
it satisfies the additional property
deg u(0.11) d{u,v} ={Du,v} +(−1) {u,Dv} .h h h
Assuming that (e ) is h-orthonormal, one easily checks that D is hermitian if and onlyλ
∗if the associated connection matrix Γ is skew-symmetric, i.e. Γ = −Γ. In this case
∗θ =dΓ+Γ∧Γ also satisfies θ =−θ , thusD DD
i ∞ 2 ∗(0.12) Θ = θ ∈C (M,Λ T ⊗Herm(E,E)).D D M

(0.13) Special case. For a bundle E of rank r = 1, the connection matrix Γ of a
hermitian connection D can be more conveniently written Γ = −iA where A is a real
1-form. Then we have
i 1
Θ = dΓ= dA.D
2π 2π
∞ 2 ∗Frequently, especially in physics, the real 2-form B = dA = 2πΘ ∈ C (M,Λ T )D M
is referred to as the magnetic field, and the 1-form A as its potential. A phase change
iα(x)s˜(x) =s(x)e in the isomorphism E ≃U ×C replaces A with the new connection|U
˜form A=A+dα.6 J.-P. Demailly Holomorphic Morse inequalities and entire curves
0.E. Connections on a hermitian holomorphic vector bundle
IfM =X is a complex manifold, every connectionD can be split in a unique way as
′ ′′ ′ ′′the sum D =D +D of a (1,0)-connection D and a (0,1)-connection D :
′ ∞ p,q ∗ ∞ p+1,q ∗D :C (M,Λ T ⊗E)−→C (M,Λ T ⊗E),X X
′′ ∞ p,q ∗ ∞ p,q+1 ∗D :C (M,Λ T ⊗E)−→C (M,Λ T ⊗E).X X
∞In a local trivialization given by a C frame, one can write
′ ′ ′Du=du+Γ ∧u ,
′′ ′′ ′′D u=d u+Γ ∧u ,
′ ′′ ′ ′′ ∞with Γ = Γ + Γ and d = ∂, d = ∂. If (E,h) is a C hermitian structure, the
′ ′′ ∗connection is hermitian if and only if Γ = −(Γ ) in any h-orthonormal frame. Thus
′′there exists a unique hermitian connection corresponding to a prescribed (0,1) partD .
Assume now that the hermitian bundle (E,h) has a holomorphic structure. The
′′uniquehermitianconnectionDforwhichD =∂ iscalledtheChern connectionof(E,h).
Inalocalholomorphicframe(e )ofE , themetrichisgivenbysomehermitianmatrixλ |U
H = (h ) where h = he ,e i . Standard computations yield the expression of theλµ λµ λ µ h
Chern connection :
 −1′ Ds =∂s+H ∂H∧s,
′′D s =∂s,
 −12 ′ ′′ ′′ ′θ ∧s =D s=(DD +D D )s=−∂(H ∂H)∧s.D
(0.14) Definition. The Chern curvature tensor of (E,h) is the curvature tensor of its
Chern connection, denoted
−1′ ′′ ′′ ′θ =DD +D D =−∂(H ∂H).E,h
In the special case of a rank 1 bundle E, the matrix H is simply a positive function,
−ϕ ∞and it is convenient to introduce its weightϕ such thatH =(e ) where ϕ∈C (U,R)
depends on the given trivializationE ≃U×C. We have in this case|U
i i
(0.15) Θ = θ = ∂∂ϕ on U,E,h E,h
2π 2π
and therefore Θ is a closed real (1,1)-form.E,h
0.F. Fundamental facts of Hodge theory
P
AssumeherethatM isaRiemannianmanifoldwithmetricg = g dx ⊗dx . Givenij i j
2q-forms u,v onM with values inE , we consider the globalL norm and inner product
Z Z
2 2(0.16) kuk = |u(x)| dσ(x), hhu,vii= hu(x),v(x)idσ(x),
M MChapter I, Holomorphic Morse inequalities 7
where |u| is the pointwise hermitian norm and dσ the Riemannian volume form. The
Laplace Beltrami operator associated with the connection D is
∗ ∗Δ=DD +D D,
∞ q ∗acting on any of the spaces C (M,Λ T ⊗E); hereM
∗ ∞ q ∗ ∞ q−1 ∗(0.17) D :C (M,Λ T ⊗E)−→C (M,Λ T ⊗E)M M
2 ′ ′ ′∗ ′∗ ′is the (formal) L adjoint of D. The complex Laplace operators Δ = DD +D D
′′ ′′ ′′∗ ′′∗ ′′and Δ =D D +D D aredefined similarlywhenM =X isacomplex manifold. In
∗degree 0 we simply have Δ = D D. A well-known calculation shows that the principal
2 1 2symbol of Δ is σ (x,ξ) = −|ξ| Id (while σ ′(x,ξ) = σ ′′(x,ξ) = − |ξ| Id). As aΔ Δ Δ 2
′ ′′consequence Δ, Δ, Δ are always elliptic operators.
∞ q ∗When M is compact, the operator Δ acting on any of the spaces C (M,Λ T ⊗E)M
has a discrete spectrum
λ 6λ 6···6λ 6···1 2 j
∞ q ∗and corresponding eigenfunctions ψ ∈C (M,Λ T ⊗E), depending of course on q.j M
Our main goal is to obtain asymptotic formulas for the eigenvalues. For this, we will
−tΔmake an essential use of the heat operatore . In the above setting, the heat operator
is the bounded hermitian operator associated to the heat kernel
+∞X
−λ t ∗ν(0.18) K (x,y)= e ψ (x)⊗ψ (y),t ν ν
ν=1
i.e. Z
−tΔhhu,e vii= hu(x),K (x,y)·v(y)idσ(x)dσ(y).t
M×M
Standard results of the theory of elliptic operators show that
∞K ∈C (]0,+∞[×M×M,Hom(E,E))t
and that K (x,y) is the solution of the differential equationt

(0.19) K (x,y)=−Δ K (x,y), lim K (x,y)=δ (x) (Dirac at y),t x t t y
t→0∂t +
∂ −tΔ −tΔ −0Δasfollowsformallyfromthefactthat e =−Δe ande =Id. Theasymptotic∂t
distribution of eigenvalues can be recovered from the straightforward formula
Z+∞X
−λ tν(0.20) e = tr K (x,x)dσ(x).E t
Mν=1
In the sequel, we are especially interested in the 0-eigenspace:
(0.21) Definition. The space of Δ-harmonic forms is defined to be
q ∞ q ∗(M,E)=KerΔ= u∈C (M,Λ T ⊗E); Δu=0 .MΔ
H8 J.-P. Demailly Holomorphic Morse inequalities and entire curves
When M is compact, an integration by part shows that
2 ∗ 2hhΔu,uii=kDuk +kD uk ,
∗hence u is Δ-harmonic if and only if Du = D u = 0. Moreover, as Δ is a self-ajoint
operator, standard elliptic theory implies that
q∞ q ∗(0.22) C (M,Λ T ⊗E)=KerΔ⊕ImΔ= (M,E)⊕ImΔ,M Δ
q 2and KerΔ = (M,E), ImΔ are orthogonal with respect to the L inner product.Δ
∗ ∗ClearlyImΔ⊂ImD+ImD , and bothimages ImD, ImD areorthogonal tothe space
of harmonic forms by what we have just seen. As a consequence, we have
∗(0.23) ImΔ= ImD+ImD .
(0.24) Hodge isomorphism theorem. Assume that M is compact and that D is an
2integrable connection, i.e. D = 0 (or θ = 0). Then D defines on spaces of sectionsD
∞ q ∗C (M,Λ T ⊗E) a differential complex which can be seen as a generalization of theM
2 ∗De Rham complex. The condition D = 0 immediately implies that ImD⊥ ImD and
we conclude from the above discussion that there is an orthogonal direct sum
q∞ q ∗ ∗(0.25) C (M,Λ T ⊗E)= (M,E)⊕ImD⊕ImD .M Δ
∗ ∗If we put u =h+Dv+D w according to this decomposition, then Du =DD w = 0 if
∗ ∗and only if kD wk=hhDD w,wii=0, thus
q
KerD = (M,E)⊕ImD.Δ
This implies the Hodge isomorphism theorem
q q(0.26) H (M,E):=KerD/ImD≃ (M,E).DR Δ
In case M = X is a compact complex manifold, (E,h) a hermitian holomorphic vector
′ ′′ ′′2 2bundleandD =D +D theChernconnection, theintegrabilityconditionD =∂ =0
is always satisfied. Thus we get an analogous isomorphism
0,q 0,qq(0.27) H (X, (E))≃H (X,E)≃ (M,E),′′0,q Δ∂
and more generally
p p,q p,qq(0.27) H (X,Ω ⊗ (E))≃H (X,E)≃ (M,E),p,q ′′X Δ∂
p,q ′′where (M,E) is the space of Δ -harmonic forms of type (p,q) with values in E.′′Δ
(0.28)Corollary (Hodgedecompositiontheorem). If (X,ω) is a compact Ka¨hler mani-
2fold and (E,h) is a flat hermitian vector bundle over X (i.e.D =0), then there is anE,h
isomorphism M
p,qkH (M,E)≃ H (X,E).DR ∂
p+q=k
In fact, under the condition thatω is Ka¨hler, i.e.dω =0, well-known identities of K¨ahler
′ ′′ 1geometry imply Δ =Δ = Δ, and as a consequence
2
M
p,qk (M,E)= (X,E).′′Δ Δ
p+q=k
H
H
H
H
O
H
H
H
H
H
H
OChapter I, Holomorphic Morse inequalities 9
1. Holomorphic Morse inequalities
1.A. Main statements
LetX beacompactcomplexmanifold,L→X aholomorphiclinebundleandE →X
a holomorphic vector bundle of rank r = rankE. We assume that L is equipped with a
smoothhermitianmetrichand denote accordinglyΘ itscurvature form; by definitionL,h
2this is a closed real (1,1)-form and its cohomology class c (L) ={Θ }∈H (X,R)1 R L,h DR
is the first Chern class of L.
(1.1) q-index sets. We define the q-index sets and {6q}-index sets of (L,h) to be

q negative eigenvalues
X(L,h,q)= x∈X; Θ (x) hasL,h
n−q positive eigenvalues
[
X(L,h,6q) = X(L,h,j) .
16j6q
Clearly X(L,h,q) and X(L,h,6 q) are open subsets of X, and we have a partitionS
into “chambers” X = S∪ X(L,h,q) where S = {x ∈ X; Θ (x) = 0} is theL,h06q6n
degeneration set. The following theorem was first proved in [Dem85].
k(1.2) Main Theorem. The cohomology groups of tensor powers E ⊗L satisfy the
following asymptotic estimates as k→+∞ :
(1.2) Weak Morse inequalities:WM
Znkq k q n nh (X,E⊗L )6r (−1) Θ +o(k ) .L,hn! X(L,h,q)
(1.2) Strong Morse inequalities:SM
ZnX kq−j j k q n n(−1) h (X,E⊗L )6r (−1) Θ +o(k ) .L,h
n! X(L,h,6q)06j6q
(1.2) Asymptotic Riemann-Roch formula:RR
ZnX kk j j k n nχ(X,E⊗L ) := (−1) h (X,E⊗L ) =r Θ +o(k ) .L,h
n! X06j6n
The weak Morse form (1.2) follows from strong Morse (1.2) by adding conse-WM SM
q−j q−1−jcutive inequalities for the indices q−1 and q, since the signs (−1) and (−1)
are opposite. Also, (1.2) is just a weaker formulation of the existence of the HilbertRR
polynomial, and as such, is a consequence of the Hirzebruch-Riemann-Roch formula;
n+1it follows formally from (1.2) withq =n andq =n+1, sinceh = 0identically andSM
the signs are reversed. Now, by adding (1.2) for the indices of opposite parity q +1SM
and q−2, we find
Znkq+1 k q q−1 q+1 n nh (X,E⊗L )−h (...)+h (...)6r (−1) Θ +o(k ),L,h
n! X(L,h,{q−1,q,q+1})10 J.-P. Demailly Holomorphic Morse inequalities and entire curves
whereX(L,h,{q−1,q,q+1})ismeant fortheunionofchambers ofindicesq−1,q,q+1.
As a consequence, we get lower bounds for the cohomology groups:
Znkq k q q+1 q−1 q n n(1.3) h (X,E⊗L )>h −h −h >r (−1) Θ −o(k ).L,hn! X(L,h,{q−1,q,q+1})
Another important special case is (1.2) for q = 1, which yields the lower boundSM
Znk0 k 0 1 n n(1.4) h (X,E⊗L )>h −h >r Θ −o(k ).L,hn! X(L,h,61)
As we will see later in the applications, this lower bound provides a very useful criterion
to prove the existence of sections of large tensor powers of a line bundle.
1.B. Heat kernel and eigenvalue distribution
We introduce here a basic heat equation technique, from which all asymptotic eigen-
value estimates can be derived via an explicit formula, known as Mehler’s formula.
We start withacompact Riemannian manifold (M,g)withdim M =m, and denoteR
by dσ its Riemannian volume form. Let (L,h ) (resp. (E,h )) be a hermitian complexL E
line (resp. vector bundle) on M, equipped with a hermitian connection D (resp. D ).L E
k ∗We denotebyD =D k the associatedconnection onE⊗L , andby Δ =D Dk k kE⊗L k
ktheLaplace-Beltrami operatoractingon sectionsofE⊗L (i.e.forms ofdegree 0). Asin
(0.13), we introduce the (local) connection form Γ =−iA of L and the correspondingL
∞ 2 ∗(global)curvature2-formB =dA∈C (M,Λ T ),i.e.the“magneticfield”(Γ andtheEM
corresponding curvature tensor Θ of D will not play a significant role here). Finally,E E
∞weassumethatanadditionalsectionV ∈C (M,Herm(E,E))isgiven(“electricfield”);
kfor simplicity of notation, we still denote by V the operatorV ⊗Id k acting on E⊗L .L
If Ω⊂M is a smoothly bounded open subset of M, we consider for u in the Sobolev
1 kspace W (Ω,E⊗L ) the quadratic form0
Z
1 2(1.5) Q (u) = |D u| −hVu,ui.k,Ω k

1 kHereW (Ω,E⊗L ) is the closure of the space of smooth sections with compact support0
1 k 2inΩ, takeninthe HilbertspaceW (M,E⊗L ) ofsectionsthat haveL coefficients asloc loc
well as their first derivatives. In other words, we consider the densily defined self adjoint
operator
1 ∗(1.6) = D D −Vk kkk
1 kactingintheHilbertspaceW (Ω,E⊗L ),i.e.withDirichletboundaryconditions. Again,0
1 k acting on W (Ω,E⊗L ) has a discrete spectrum whenever Ω is relatively compactk 0
(and also sometimes when Ω is unbounded, according to the behavior of B and V at
infinity; except otherwise stated, we willassume that we are in this case later on). Then,
there is an associated “localized” heat kernel
+∞X
−λ t ∗ν,k,Ω(1.7) K (x,y)= e ψ (x)⊗ψ (y)t,k,Ω ν,k,Ω ν,k,Ω
ν=1