##
**Holomorphic Morse inequalities and Bergman kernels.**
*(English)*
Zbl 1135.32001

Progress in Mathematics 254. Basel: Birkhäuser (ISBN 978-3-7643-8096-0/hbk). xiii, 422 p. (2007).

This book presents in detail various results and techniques relative to the holomorphic Morse inequalities and the asymptotic expansion of the Bergman kernel. Although this book is intended for specialists, it is self-contained and will interest certainly graduate students. The heart of the book is intended to explain analytic localization techniques developed by Bismut-Lebeau and to explain their interaction with complex, Kähler and symplectic geometry. A large number of applications are given in detail: several proofs of the fundamental Kodaira’s embedding theorem, a solution of the Grauert-Riemenschneider and Shiffman conjectures, a compactification of complete Kähler manifolds of pinched negative curvature, asymptotics of the Ray-Singer analytic torsion, the Berezin quantization, weak Lefschetz theorems, etc.

Let’s describe now precisely the structure of the book.

The first chapter is a general introduction on connections of the tangent bundle (Lévi-Civita, Chern, Bismut connections), Dirac operators, Lichnerowicz formula, and gives in details the Bochner-Kodaira-Nakano formula with boundary term. The key point is the so-called spectral gap property for higher tensor powers of line bundles, which asserts that on a projective manifold \(M\), with \(L\) an ample line bundle on \(M\) and \(p \in {\mathbb N}\), the spectrum of the Kodaira Laplacian \(\square_p\) on \(L^p\) satisfies \[ \text{Spec}(\square_p) \subset \{0\} \cup ]2\pi p- C_L, +\infty[ \]

for some constant \(C_L >0\). Note that a version of this result holds for Dirac operators. The end of this dense chapter is dedicated to explain Demailly’s holomorphic (strong and weak) inequalities. The proof is based here on an asymptotic of the heat kernel, i.e that the authors prove the local formula

\[ \exp\left(-\frac{u}{p}\square_p\right)(x,x)= \frac{1}{(2\pi)^n}\frac{\det([R_L]) e^{2u \omega_d}}{\det(1-e^{-2u[R_L]})}p^n + o(p^n) \]

where \(\dim_{\mathbb{C}}M=n\), \(u>0\), \(x\) is a point of \(M\), \([R_L]\in \text{End}(T^{(1,0)})\) is the hermitian matrix associated to the curvature \(R_L\) of \(L\), \(R_L(X,Y)=\langle [R_L]X,Y\rangle\). Moreover, we denote \(\omega_d\in \text{End}(\Lambda(T^{*(0,1)}M))\) to be the trace \(\omega_d=-\sum_{l,m} R_L(w_l,w_m) \bar{w}^m \wedge i_{\bar{w}_l}\) for \(w_j\) a local orthonormal frame of \(T^{(1,0)}\).

After recalling some basic facts on complex geometry, the second chapter gives some fundamental characterizations of Moishezon manifolds. A compact connected complex manifold \(M\) is Moishezon if it has \(\dim(M)\) algebraically independant meromorphic functions. The authors give the proof of Siu-Demailly criterion which answers the Grauert-Riemenschneider conjecture (a manifold is Moishezon if and only if it has a quasi-positive sheaf). The rest of the chapter is about some recent results related to that question. For instance Moishezon manifolds are characterized in terms of integral Kähler currents (Shiffman’s conjecture) and a singular version of holomorphic Morse inequalities is given.

Chapter 3 presents a \(L^2\)-Hodge theory on non-compact hermitian manifolds which leads to holomorphic Morse inequalities for the \(L^2\)-cohomology in that context. This leads to an extension of the Siu-Demailly criterion for compact complex spaces with isolated singularities in order to detect Moishezon spaces. Finally, the authors give a version of the Morse inequalities for \(q\)-convex manifolds and covering manifolds.

In Chapter 4, the asymptotic expansion of the Bergman kernel is given in detail. For \(x,x'\in M\), the Bergman kernel \(P_p(x,x')\) is defined as the kernel of the orthogonal \(L^2\)-projection on the space of the holomorphic sections \(H^0(L^p)\), i.e.,

\[ P_p(x,x')=\sum_{i=1}^{\dim H^0(M,L^p)} S_i(x) \otimes S_i(x')^{*_{h^p}} \]

where \(h\) is a metric on \(L\), \((S_i)_{i=1,\dots,h^0(M,L^p)}\) is an orthonormal basis of \(H^0(M,L^p)\) with respect to \(\int_M h^p(.,.)\,dV_M\). If the holomorphic line bundle \((L,h)\) polarizes the manifold \(M\), the main result of this section shows that

\[ P_p(x,x) = p^n + \frac{1}{2} \text{scal}(c_1(h))(x) p^{n-1} + O(p^{n-2}) \]

i.e., that one has a kind of pointwise Riemann-Roch formula (here \(\text{scal}(c_1(h))\) means the scalar curvature of the Kähler metric associated to \(h\)). Thanks to the spectral gap property of the Laplacian, the authors use the finite propagation speed of solutions of hyperbolic equations in order to localize the problem on \({\mathbb R}^{2n}.\) Hence their proof of the existence of this asymptotic and the computation of the coefficients is purely local and come from some functional analysis resolvent techniques on \({\mathbb R}^{2n}.\) Another fact is that outside the diagonal, \(P_p\) converges exponentially fast to \(0.\) This asymptotic result is very natural and has been studied by many authors; it has a key role in many problems in Kähler geometry. The interested reader can find other proofs more geometric in the literature [for instance by B. Berndtsson, Contemp. Math. 332, 1–17 (2003; Zbl 1038.32003), and R. Berman, B. Berndtsson and J. Sjoestrand, “Asymptotics of Bergman kernel”, arXiv:math/0506367]. On the other hand, the techniques developed in this chapter give uniform estimates under very weak assumptions for both heat kernels and Bergman kernels (the Bergman kernel is the limit when \(u \rightarrow +\infty\) of the heat kernel).

Chapter 5 gives some applications of the previous work. The authors study the metric aspect of the Kodaira map following the work of T.Bouche and obtain the famous Kodaira embedding theorem. They give a short but clear explanation of the use of the asymptotic of the Bergman kernel in Donaldson’s theory for extremal metrics. This is related to the quantization procedure of Kähler metrics with constant scalar curvature by balanced metrics. They give the analog in the case of vector bundles. Then, they study the distributions of random sections briefly. A section will interest particularly the specialists. The Bergman kernel is studied on complex orbifolds and an asymptotic formula is given together with Baily’s extension of the Kodaira theorem. The last part of this long chapter is dedicated to the asymptotic of the Ray-Singer analytic torsion when the power of the line bundle tends to infinity. This allows to understand the variation of the Quillen metrics at first order.

In Chapter 6, the authors show how to derive an asymptotic of the Bergman kernel for certain non compact manifolds and various consequences of this fact. In particular, they study the compactification of manifolds with pinched negative curvature. This comes from the relationship through the Morse inequalities between, from one hand, the growth of the space of holomorphic sections of the pluricanonical line bundle and from another hand the volume of the manifold.

Chapter 7 describe the properties of Toeplitz operators and the Berezin-Toeplitz quantization. The Toeplitz operators on \(H^0(L^k)\) are defined for smooth functions \(f\) on the manifold with real values by

\[ T_k(f)= P_p f P_f \]

The key result is here that the set of Toeplitz operators form an algebra (the set of Toeplitz operators is closed under composition of operators).

In Chapter 8 is given an asymptotic expansion of the Bergman kernel associated to the modified Dirac operator and the renormalized Bochner Laplacian.

Some appendices describe very useful tools or techniques (for instance the relation between heat kernel and the finite propagation speed of solutions of hyperbolic equations). Note that one can find on the first author’s webpage a list of errata (there are very few of them). This book covers a very wide range of techniques from modern analysis to geometry through the study of Bergman kernels and Morse inequalities. Especially, the authors have done the effort to explain carefully how the study of heat and Bergman kernels enters in various fundamental problems of complex and symplectic geometry. One could regret sometimes the high level of technicity of some sections but this is the price to pay in order to get very general results. We have no doubt that this book will become soon a reference on the subject.

Let’s describe now precisely the structure of the book.

The first chapter is a general introduction on connections of the tangent bundle (Lévi-Civita, Chern, Bismut connections), Dirac operators, Lichnerowicz formula, and gives in details the Bochner-Kodaira-Nakano formula with boundary term. The key point is the so-called spectral gap property for higher tensor powers of line bundles, which asserts that on a projective manifold \(M\), with \(L\) an ample line bundle on \(M\) and \(p \in {\mathbb N}\), the spectrum of the Kodaira Laplacian \(\square_p\) on \(L^p\) satisfies \[ \text{Spec}(\square_p) \subset \{0\} \cup ]2\pi p- C_L, +\infty[ \]

for some constant \(C_L >0\). Note that a version of this result holds for Dirac operators. The end of this dense chapter is dedicated to explain Demailly’s holomorphic (strong and weak) inequalities. The proof is based here on an asymptotic of the heat kernel, i.e that the authors prove the local formula

\[ \exp\left(-\frac{u}{p}\square_p\right)(x,x)= \frac{1}{(2\pi)^n}\frac{\det([R_L]) e^{2u \omega_d}}{\det(1-e^{-2u[R_L]})}p^n + o(p^n) \]

where \(\dim_{\mathbb{C}}M=n\), \(u>0\), \(x\) is a point of \(M\), \([R_L]\in \text{End}(T^{(1,0)})\) is the hermitian matrix associated to the curvature \(R_L\) of \(L\), \(R_L(X,Y)=\langle [R_L]X,Y\rangle\). Moreover, we denote \(\omega_d\in \text{End}(\Lambda(T^{*(0,1)}M))\) to be the trace \(\omega_d=-\sum_{l,m} R_L(w_l,w_m) \bar{w}^m \wedge i_{\bar{w}_l}\) for \(w_j\) a local orthonormal frame of \(T^{(1,0)}\).

After recalling some basic facts on complex geometry, the second chapter gives some fundamental characterizations of Moishezon manifolds. A compact connected complex manifold \(M\) is Moishezon if it has \(\dim(M)\) algebraically independant meromorphic functions. The authors give the proof of Siu-Demailly criterion which answers the Grauert-Riemenschneider conjecture (a manifold is Moishezon if and only if it has a quasi-positive sheaf). The rest of the chapter is about some recent results related to that question. For instance Moishezon manifolds are characterized in terms of integral Kähler currents (Shiffman’s conjecture) and a singular version of holomorphic Morse inequalities is given.

Chapter 3 presents a \(L^2\)-Hodge theory on non-compact hermitian manifolds which leads to holomorphic Morse inequalities for the \(L^2\)-cohomology in that context. This leads to an extension of the Siu-Demailly criterion for compact complex spaces with isolated singularities in order to detect Moishezon spaces. Finally, the authors give a version of the Morse inequalities for \(q\)-convex manifolds and covering manifolds.

In Chapter 4, the asymptotic expansion of the Bergman kernel is given in detail. For \(x,x'\in M\), the Bergman kernel \(P_p(x,x')\) is defined as the kernel of the orthogonal \(L^2\)-projection on the space of the holomorphic sections \(H^0(L^p)\), i.e.,

\[ P_p(x,x')=\sum_{i=1}^{\dim H^0(M,L^p)} S_i(x) \otimes S_i(x')^{*_{h^p}} \]

where \(h\) is a metric on \(L\), \((S_i)_{i=1,\dots,h^0(M,L^p)}\) is an orthonormal basis of \(H^0(M,L^p)\) with respect to \(\int_M h^p(.,.)\,dV_M\). If the holomorphic line bundle \((L,h)\) polarizes the manifold \(M\), the main result of this section shows that

\[ P_p(x,x) = p^n + \frac{1}{2} \text{scal}(c_1(h))(x) p^{n-1} + O(p^{n-2}) \]

i.e., that one has a kind of pointwise Riemann-Roch formula (here \(\text{scal}(c_1(h))\) means the scalar curvature of the Kähler metric associated to \(h\)). Thanks to the spectral gap property of the Laplacian, the authors use the finite propagation speed of solutions of hyperbolic equations in order to localize the problem on \({\mathbb R}^{2n}.\) Hence their proof of the existence of this asymptotic and the computation of the coefficients is purely local and come from some functional analysis resolvent techniques on \({\mathbb R}^{2n}.\) Another fact is that outside the diagonal, \(P_p\) converges exponentially fast to \(0.\) This asymptotic result is very natural and has been studied by many authors; it has a key role in many problems in Kähler geometry. The interested reader can find other proofs more geometric in the literature [for instance by B. Berndtsson, Contemp. Math. 332, 1–17 (2003; Zbl 1038.32003), and R. Berman, B. Berndtsson and J. Sjoestrand, “Asymptotics of Bergman kernel”, arXiv:math/0506367]. On the other hand, the techniques developed in this chapter give uniform estimates under very weak assumptions for both heat kernels and Bergman kernels (the Bergman kernel is the limit when \(u \rightarrow +\infty\) of the heat kernel).

Chapter 5 gives some applications of the previous work. The authors study the metric aspect of the Kodaira map following the work of T.Bouche and obtain the famous Kodaira embedding theorem. They give a short but clear explanation of the use of the asymptotic of the Bergman kernel in Donaldson’s theory for extremal metrics. This is related to the quantization procedure of Kähler metrics with constant scalar curvature by balanced metrics. They give the analog in the case of vector bundles. Then, they study the distributions of random sections briefly. A section will interest particularly the specialists. The Bergman kernel is studied on complex orbifolds and an asymptotic formula is given together with Baily’s extension of the Kodaira theorem. The last part of this long chapter is dedicated to the asymptotic of the Ray-Singer analytic torsion when the power of the line bundle tends to infinity. This allows to understand the variation of the Quillen metrics at first order.

In Chapter 6, the authors show how to derive an asymptotic of the Bergman kernel for certain non compact manifolds and various consequences of this fact. In particular, they study the compactification of manifolds with pinched negative curvature. This comes from the relationship through the Morse inequalities between, from one hand, the growth of the space of holomorphic sections of the pluricanonical line bundle and from another hand the volume of the manifold.

Chapter 7 describe the properties of Toeplitz operators and the Berezin-Toeplitz quantization. The Toeplitz operators on \(H^0(L^k)\) are defined for smooth functions \(f\) on the manifold with real values by

\[ T_k(f)= P_p f P_f \]

The key result is here that the set of Toeplitz operators form an algebra (the set of Toeplitz operators is closed under composition of operators).

In Chapter 8 is given an asymptotic expansion of the Bergman kernel associated to the modified Dirac operator and the renormalized Bochner Laplacian.

Some appendices describe very useful tools or techniques (for instance the relation between heat kernel and the finite propagation speed of solutions of hyperbolic equations). Note that one can find on the first author’s webpage a list of errata (there are very few of them). This book covers a very wide range of techniques from modern analysis to geometry through the study of Bergman kernels and Morse inequalities. Especially, the authors have done the effort to explain carefully how the study of heat and Bergman kernels enters in various fundamental problems of complex and symplectic geometry. One could regret sometimes the high level of technicity of some sections but this is the price to pay in order to get very general results. We have no doubt that this book will become soon a reference on the subject.

Reviewer: Julien Keller (Marseille)

### MSC:

32-02 | Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces |

53-02 | Research exposition (monographs, survey articles) pertaining to differential geometry |

58-02 | Research exposition (monographs, survey articles) pertaining to global analysis |

32A25 | Integral representations; canonical kernels (Szegő, Bergman, etc.) |

58J35 | Heat and other parabolic equation methods for PDEs on manifolds |

58J50 | Spectral problems; spectral geometry; scattering theory on manifolds |

58J52 | Determinants and determinant bundles, analytic torsion |

53D50 | Geometric quantization |