Cycle spaces of flag domains. A complex geometric viewpoint.

*(English)*Zbl 1084.22011
Progress in Mathematics 245. Boston, MA: Birkhäuser (ISBN 0-8176-4391-5/hbk; 0-8176-4479-2/ebook). xx, 339 p. (2006).

Cycle space theory is a basic chapter in complex analysis. Since the 1960s its importance has been underlined by its role in the geometry of flag domains, and by applications in the representation theory of semisimple Lie groups. This theory developed very slowly until a few years ago when methods of complex geometry, in particular those involving Schubert slices, Schubert domains, Iwasawa domains and supporting hypersurfaces, were introduced. In the late 1990s, and continuing through the early 2002, the authors developed those methods and used them to give a precise description of cycle spaces for flag domains. This effectively enabled them to use double fibration transforms in all flag domain situations. That has very interesting consequences for the geometric construction of representations of semisimple Lie groups, especially for the construction of singular representations. It also has many potential interesting consequences for automorphic cohomology, other aspects of the variation of the Hodge structure, and the moduli of compact complex manifolds. In this book the authors prove these recent results, filling in the background as necessary, and present new results that complete the picture. The authors begin with a real linear semisimple group \(G_0\) which for all practical purposes can be assumed to be simple. It is embedded in its complexification \(G\) and acts naturally on every \(G\)-homogeneous manifold \(Z = G/\mathcal Q\). Here the authors restrict themselves to the case of flag manifolds, in other words \(Z\) is assumed to be compact Kähler, or equivalently \(Z\) is \(G\)-equivariantly projective algebraic. Also equivalently, \(\mathcal Q\) is a parabolic subgroup.

Part I, Introduction to Flag Domain Theory, is primarily devoted to preliminary and foundational material, and also to older results concerning flag manifolds and flag domains.

Chapter 1, Structure of Complex Flag Manifolds, starts with a review of the structure and finite-dimensional representation theory for semisimple Lie groups and algebras, introduces the structure theory for parabolic subalgebras and parabolic subgroups and ends with a discussion of homogeneous vector bundles and the Bott-Borel-Weil Theorem.

Chapter 2, Real Group Orbits, contains the first combinatorial results for the \(G_0\)-action on \(Z\). In particular, it is shown that the set \(\text{Orb}_Z(G_0)\) of \(G_0\)-orbits in \(Z\) is finite. Hence, by dimension at least one \(G_0\)-orbit is open in \(Z\). This research monograph is devoted to the study of the cycle spaces of \(G_0\)-orbits on \(Z\).

In Chapter 3, Orbit Structure for Hermitian Symmetric Spaces, the authors give a complete description of the \(G_0\)-orbit structure for the case where \(G_0\) is the group of a bounded symmetric domain and \(Z\) is the dual compact Hermitian symmetric space, for example where \(G_0\) corresponds to the open unit ball in \(\mathbb C^n\) and \(Z =\mathbb C\mathbb P^n\).

The authors then turn the attention, in Chapter 4, Open Orbits, to a discussion of the first results for the open \(G_0\)-orbits \(D\) in \(Z\). They discuss their structure, compact subvarieties, and holomorphic functions. The compact subvarieties are of fundamental importance; they are the base cycles in the treatment of cycle spaces.

Certain foundational results for the Hermitian case are proved in Chapter 5, The Cycle Space of a Flag Domain.

Part II, Cycle Spaces as Universal Domains, is a systematic presentation of the authors’ recent work which describes in a precise way the cycle spaces associated to the \(G_0\)-orbits in \(Z = G/\mathcal Q\). Let Orb\(_Z(G_0)\) denote the set of all \(G_0\)-orbits on \(Z\). While much of the work is devoted to the case of an open orbit \(D\), any \(\gamma\in \text{Orb}_Z(G_0)\) has a naturally associated cycle space \(\mathcal C(\gamma)\) that can be realized as an open set in a \(G\)-orbit in the appropriate cycle space \(\mathcal C_q(Z)\). If \(\gamma\) is an open orbit \(D\), it follows by definition that \(\mathcal C(\gamma)=\mathcal M_D\) – a possibly smaller space. The main result for the cycle domains \(\mathcal C(\gamma)\) can be stated as follows: For every real form \(G_0\) there exists a precisely computable universal domain \(\mathcal U\) so that, with a few well-understood exceptions which only occur in the Hermitian cases, \(\mathcal C(\gamma) =\mathcal U\) for all \(\gamma\in \text{Orb}_Z(G_0)\).

In Chapter 6, Universal Domains, the authors begin to describe results of D. Burns, S. Halverscheid and R. Hind which show, in particular, that \(\mathcal U\) can be naturally identified with the maximal domain of existence \(\Omega_{\text{adpt}}\) of the adapted complex structure in the tangent bundle \(T\Omega_0\). This underlines the complex differential geometric importance of \(\mathcal U\).

In Chapter 7, \(B\)-Invariant Hypersurfaces in \(\mathcal M_Z\), the authors introduce basic Schubert incidence geometry which is used for the description of the space \(\mathcal M_D\), or more generally for \(\mathcal C(\gamma)\) for any \(\gamma\in\text{Orb}_Z(G_0)\).

The symplectic geometric approach to orbit duality is explained in Chapter 8, Orbit Duality via Momentum Geometry. This is based on a fundamental idea of Uzawa, and in this context was first carried out in detail for the case of \(G/B\) by I. Mirković, T. Uzawa and K. Vilonen. It was extended to the general case of \(Z = G/\mathcal Q\) by R. Bremigan and J. Lorch, and in Chapter 8, the authors present their proof of the orbit duality theorem.

With the basic results on duality, it is possible to prove the existence of Schubert Slices in the Context of Duality, Chapter 9. Having shown that \(\mathcal U\subset\mathcal M_D\), in Chapter 10, Analysis of the Boundary of \(\mathcal U\), the authors give a precise description of \(\mathcal M_D\). The description of the cycle spaces associated to open \(G_0\)-orbits in \(Z\) is completed in Chapter 11, Invariant Kobayashi-Hyperbolic Stein Domains.

In Chapter 12, Cycle Spaces of Lower-Dimensional Orbits, the authors turn to the cycle space \(\mathcal C(\gamma)\) of an arbitrary \(G_0\)-orbit \(\gamma\in\text{Orb}_Z(G_0)\). The final chapter in Part II, Chapter 13, Examples, is devoted to three types of examples. First, the authors discuss very roughly the first results for \(SL(n;\mathbb R)\) which are proved by the Schubert slice method. Second, using Grassmann geometry, a differential geometric characterization of \(\mathcal U\) is given, and the characterization of \(\mathcal M_D\) which is discussed above. Third, the authors look at the simplest Hermitian examples and compare the slice methods presented here with those coming from the classical theory.

Part III, Analytic and Geometric Consequences, is where the authors apply the results of Parts I and II to the mechanism of the double fibration transform, as well as to certain other matters. The authors start in Chapter 14 with a general discussion of The Double Fibration Transform. In Chapter 15, Variation of Hodge Structure, the authors present a very brief exposition of Griffiths’ period map. The period domains in this case are closely related to Hermitian symmetric spaces of noncompact type, i.e., to bounded symmetric domains. In Chapter 16, Cycles in the K3 Period Domain, the authors outline moduli space results for K3 surface which are marked with a basis for their integral homology.

Part IV, The Full Cycle Space, is devoted to considerations of the full cycle space \(\mathcal C\) of an open \(G_0\) orbit \(D\), i.e., the irreducible component containing \(C_0\) in \(\mathcal C_q(D)\). This contains \(\mathcal M_D\) as a closed submanifold. The results proved in Part IV give a local description of \(\mathcal C\) at the base cycle \(C_0\) in an arbitrary open \(G_0\)-orbit \(D\) in an arbitrary flag manifold \(Z = G/\mathcal Q\). The cases where \(\mathcal M_D=\mathcal C\) are precisely described.

In Chapter 17, Combinatorics of Normal Bundles of Base Cycles, the authors compute the normal bundle \(\mathbb N_Z(C_0)\) of a closed \(K\)-orbit \(C\) in \(Z\) in abstract terms, as a holomorphic \(K\)-homogeneous vector bundle. The goal is to prove the vanishing of \(H^1(C_0;\mathcal O(\mathbb N_Z(C_0)))\), i.e., the smoothness of \(\mathcal C_q(Z)\) at \(C_0\), and to compute the tangent space \(H^0(C_0;\mathcal O(\mathbb N_Z(C_0)))\) as a \(K\)-module. The tangent space in that case has a natural decomposition \(H^0(C;\mathcal O(\mathbb N_Z(C))) =\mathfrak s\oplus H^1(C;\mathcal O(\mathbb E((\mathfrak q + \theta(\mathfrak q))_{\mathfrak q}))),\) where \(\mathfrak g =\mathfrak k +\mathfrak s\) is the decomposition of the Lie algebra \(\mathfrak g\) of \(G\) under the Cartan involution \(\theta\) of the real group \(G_0\), and where \((\mathfrak q + \theta(\mathfrak q))_{\mathfrak q}\) is the \(\mathfrak s\)-component of \(\mathfrak q +\theta(\mathfrak q)\), and where \(\mathbb E((\mathfrak q +\theta(\mathfrak q))_{\mathfrak q})\) is the \(K\)-homogeneous holomorphic vector bundle \(K\times_{K\cap \mathcal Q}(\mathfrak q +\theta(\mathfrak q))_{\mathfrak q}\) defined by the (isotropy) representation of \(K\cap\mathcal Q\) on \((\mathfrak q +\theta(\mathfrak q))_{\mathfrak q}\). One is naturally led to apply the Bott-Borel-Weil Theorem to compute \(H^1(C;\mathcal O(\mathbb E((\mathfrak q + \theta(\mathfrak q))_{\mathfrak q})))\) as a \(K\)-module. Since the isotropy representation of \(K\cap\mathcal Q\) on \(\mathfrak q_{\mathfrak s}\) is usually reducible, one must first compute the cohomology groups of the quotient bundles which arise from a natural filtration of \(\mathbb E((\mathfrak q +\theta(\mathfrak q))_{\mathfrak q})\). The filtration is constructed so that the Bott-Borel-Weil Theorem can indeed be applied to these bundles. One must compute the weights \(\lambda\) with \(\lambda + \rho\) regular and of index 1 and 2. A starting point for this computation is the description of the highest weights of the representation of \(\mathfrak k\) on \(\mathfrak s\).

These highest weights are known, and in Chapter 18, Methods for Computing \(H^1(C;\mathcal O(\mathbb E((\mathfrak q+\theta\mathfrak q)_{\mathfrak s})))\), the authors indicate the connection with weighted affine Dynkin diagrams. The knowledge of these weights for a fixed Cartan subgroup (maximal complex torus) in the isotropy subgroup of \(K\) leads to the knowledge of all weights of that torus on \(\mathfrak s\). Then in turn one can calculate the weights \(\lambda\) on \((\mathfrak q_{\mathfrak s} + \theta(\mathfrak q_{\mathfrak s}))_{\mathfrak s}\) with \(\lambda +\rho\) regular and of index 1 and 2. This work is carried out in Chapter 18 and in Chapter 19, Classification for Simple \(\mathfrak g_0\) with \(\text{rank}\,\,\mathfrak k < \text{rank}\,\,\mathfrak g\). It gives an explicit method for computing the cohomology groups of the quotient bundles. The essential result is the String Lemma. The Cohomology Lemma then gives the method of computing the original cohomology groups \(H^\ast(C;\mathcal O(\mathbb E((\mathfrak q+\theta(\mathfrak q_{\mathfrak s}))_{\mathfrak s})))\) from those of the quotient bundles. The String and Cohomology lemmas are quite explicit, and it is clear that within finite time one should be able to compute the complementary space \(H^1(C;\mathcal O(\mathbb E((\mathfrak q_{\mathfrak s}+\theta(\mathfrak q_{\mathfrak s}))_{\mathfrak s})))\) to the tangent space \(T_C(G.C)\) of the \(G\)-orbit \(G.C\) in \(T_C(\mathcal C_q(Z))\).

Chapter 20, Classification for \(\text{rank}\,\,\mathfrak k =\text{rank}\,\,\mathfrak g\), is devoted to these calculations. It is necessary to consider five infinite series and three exceptional cases. In most cases, depending on the weights which are determined by the base cycle at hand, both vanishing and nonvanishing occur.

Part I, Introduction to Flag Domain Theory, is primarily devoted to preliminary and foundational material, and also to older results concerning flag manifolds and flag domains.

Chapter 1, Structure of Complex Flag Manifolds, starts with a review of the structure and finite-dimensional representation theory for semisimple Lie groups and algebras, introduces the structure theory for parabolic subalgebras and parabolic subgroups and ends with a discussion of homogeneous vector bundles and the Bott-Borel-Weil Theorem.

Chapter 2, Real Group Orbits, contains the first combinatorial results for the \(G_0\)-action on \(Z\). In particular, it is shown that the set \(\text{Orb}_Z(G_0)\) of \(G_0\)-orbits in \(Z\) is finite. Hence, by dimension at least one \(G_0\)-orbit is open in \(Z\). This research monograph is devoted to the study of the cycle spaces of \(G_0\)-orbits on \(Z\).

In Chapter 3, Orbit Structure for Hermitian Symmetric Spaces, the authors give a complete description of the \(G_0\)-orbit structure for the case where \(G_0\) is the group of a bounded symmetric domain and \(Z\) is the dual compact Hermitian symmetric space, for example where \(G_0\) corresponds to the open unit ball in \(\mathbb C^n\) and \(Z =\mathbb C\mathbb P^n\).

The authors then turn the attention, in Chapter 4, Open Orbits, to a discussion of the first results for the open \(G_0\)-orbits \(D\) in \(Z\). They discuss their structure, compact subvarieties, and holomorphic functions. The compact subvarieties are of fundamental importance; they are the base cycles in the treatment of cycle spaces.

Certain foundational results for the Hermitian case are proved in Chapter 5, The Cycle Space of a Flag Domain.

Part II, Cycle Spaces as Universal Domains, is a systematic presentation of the authors’ recent work which describes in a precise way the cycle spaces associated to the \(G_0\)-orbits in \(Z = G/\mathcal Q\). Let Orb\(_Z(G_0)\) denote the set of all \(G_0\)-orbits on \(Z\). While much of the work is devoted to the case of an open orbit \(D\), any \(\gamma\in \text{Orb}_Z(G_0)\) has a naturally associated cycle space \(\mathcal C(\gamma)\) that can be realized as an open set in a \(G\)-orbit in the appropriate cycle space \(\mathcal C_q(Z)\). If \(\gamma\) is an open orbit \(D\), it follows by definition that \(\mathcal C(\gamma)=\mathcal M_D\) – a possibly smaller space. The main result for the cycle domains \(\mathcal C(\gamma)\) can be stated as follows: For every real form \(G_0\) there exists a precisely computable universal domain \(\mathcal U\) so that, with a few well-understood exceptions which only occur in the Hermitian cases, \(\mathcal C(\gamma) =\mathcal U\) for all \(\gamma\in \text{Orb}_Z(G_0)\).

In Chapter 6, Universal Domains, the authors begin to describe results of D. Burns, S. Halverscheid and R. Hind which show, in particular, that \(\mathcal U\) can be naturally identified with the maximal domain of existence \(\Omega_{\text{adpt}}\) of the adapted complex structure in the tangent bundle \(T\Omega_0\). This underlines the complex differential geometric importance of \(\mathcal U\).

In Chapter 7, \(B\)-Invariant Hypersurfaces in \(\mathcal M_Z\), the authors introduce basic Schubert incidence geometry which is used for the description of the space \(\mathcal M_D\), or more generally for \(\mathcal C(\gamma)\) for any \(\gamma\in\text{Orb}_Z(G_0)\).

The symplectic geometric approach to orbit duality is explained in Chapter 8, Orbit Duality via Momentum Geometry. This is based on a fundamental idea of Uzawa, and in this context was first carried out in detail for the case of \(G/B\) by I. Mirković, T. Uzawa and K. Vilonen. It was extended to the general case of \(Z = G/\mathcal Q\) by R. Bremigan and J. Lorch, and in Chapter 8, the authors present their proof of the orbit duality theorem.

With the basic results on duality, it is possible to prove the existence of Schubert Slices in the Context of Duality, Chapter 9. Having shown that \(\mathcal U\subset\mathcal M_D\), in Chapter 10, Analysis of the Boundary of \(\mathcal U\), the authors give a precise description of \(\mathcal M_D\). The description of the cycle spaces associated to open \(G_0\)-orbits in \(Z\) is completed in Chapter 11, Invariant Kobayashi-Hyperbolic Stein Domains.

In Chapter 12, Cycle Spaces of Lower-Dimensional Orbits, the authors turn to the cycle space \(\mathcal C(\gamma)\) of an arbitrary \(G_0\)-orbit \(\gamma\in\text{Orb}_Z(G_0)\). The final chapter in Part II, Chapter 13, Examples, is devoted to three types of examples. First, the authors discuss very roughly the first results for \(SL(n;\mathbb R)\) which are proved by the Schubert slice method. Second, using Grassmann geometry, a differential geometric characterization of \(\mathcal U\) is given, and the characterization of \(\mathcal M_D\) which is discussed above. Third, the authors look at the simplest Hermitian examples and compare the slice methods presented here with those coming from the classical theory.

Part III, Analytic and Geometric Consequences, is where the authors apply the results of Parts I and II to the mechanism of the double fibration transform, as well as to certain other matters. The authors start in Chapter 14 with a general discussion of The Double Fibration Transform. In Chapter 15, Variation of Hodge Structure, the authors present a very brief exposition of Griffiths’ period map. The period domains in this case are closely related to Hermitian symmetric spaces of noncompact type, i.e., to bounded symmetric domains. In Chapter 16, Cycles in the K3 Period Domain, the authors outline moduli space results for K3 surface which are marked with a basis for their integral homology.

Part IV, The Full Cycle Space, is devoted to considerations of the full cycle space \(\mathcal C\) of an open \(G_0\) orbit \(D\), i.e., the irreducible component containing \(C_0\) in \(\mathcal C_q(D)\). This contains \(\mathcal M_D\) as a closed submanifold. The results proved in Part IV give a local description of \(\mathcal C\) at the base cycle \(C_0\) in an arbitrary open \(G_0\)-orbit \(D\) in an arbitrary flag manifold \(Z = G/\mathcal Q\). The cases where \(\mathcal M_D=\mathcal C\) are precisely described.

In Chapter 17, Combinatorics of Normal Bundles of Base Cycles, the authors compute the normal bundle \(\mathbb N_Z(C_0)\) of a closed \(K\)-orbit \(C\) in \(Z\) in abstract terms, as a holomorphic \(K\)-homogeneous vector bundle. The goal is to prove the vanishing of \(H^1(C_0;\mathcal O(\mathbb N_Z(C_0)))\), i.e., the smoothness of \(\mathcal C_q(Z)\) at \(C_0\), and to compute the tangent space \(H^0(C_0;\mathcal O(\mathbb N_Z(C_0)))\) as a \(K\)-module. The tangent space in that case has a natural decomposition \(H^0(C;\mathcal O(\mathbb N_Z(C))) =\mathfrak s\oplus H^1(C;\mathcal O(\mathbb E((\mathfrak q + \theta(\mathfrak q))_{\mathfrak q}))),\) where \(\mathfrak g =\mathfrak k +\mathfrak s\) is the decomposition of the Lie algebra \(\mathfrak g\) of \(G\) under the Cartan involution \(\theta\) of the real group \(G_0\), and where \((\mathfrak q + \theta(\mathfrak q))_{\mathfrak q}\) is the \(\mathfrak s\)-component of \(\mathfrak q +\theta(\mathfrak q)\), and where \(\mathbb E((\mathfrak q +\theta(\mathfrak q))_{\mathfrak q})\) is the \(K\)-homogeneous holomorphic vector bundle \(K\times_{K\cap \mathcal Q}(\mathfrak q +\theta(\mathfrak q))_{\mathfrak q}\) defined by the (isotropy) representation of \(K\cap\mathcal Q\) on \((\mathfrak q +\theta(\mathfrak q))_{\mathfrak q}\). One is naturally led to apply the Bott-Borel-Weil Theorem to compute \(H^1(C;\mathcal O(\mathbb E((\mathfrak q + \theta(\mathfrak q))_{\mathfrak q})))\) as a \(K\)-module. Since the isotropy representation of \(K\cap\mathcal Q\) on \(\mathfrak q_{\mathfrak s}\) is usually reducible, one must first compute the cohomology groups of the quotient bundles which arise from a natural filtration of \(\mathbb E((\mathfrak q +\theta(\mathfrak q))_{\mathfrak q})\). The filtration is constructed so that the Bott-Borel-Weil Theorem can indeed be applied to these bundles. One must compute the weights \(\lambda\) with \(\lambda + \rho\) regular and of index 1 and 2. A starting point for this computation is the description of the highest weights of the representation of \(\mathfrak k\) on \(\mathfrak s\).

These highest weights are known, and in Chapter 18, Methods for Computing \(H^1(C;\mathcal O(\mathbb E((\mathfrak q+\theta\mathfrak q)_{\mathfrak s})))\), the authors indicate the connection with weighted affine Dynkin diagrams. The knowledge of these weights for a fixed Cartan subgroup (maximal complex torus) in the isotropy subgroup of \(K\) leads to the knowledge of all weights of that torus on \(\mathfrak s\). Then in turn one can calculate the weights \(\lambda\) on \((\mathfrak q_{\mathfrak s} + \theta(\mathfrak q_{\mathfrak s}))_{\mathfrak s}\) with \(\lambda +\rho\) regular and of index 1 and 2. This work is carried out in Chapter 18 and in Chapter 19, Classification for Simple \(\mathfrak g_0\) with \(\text{rank}\,\,\mathfrak k < \text{rank}\,\,\mathfrak g\). It gives an explicit method for computing the cohomology groups of the quotient bundles. The essential result is the String Lemma. The Cohomology Lemma then gives the method of computing the original cohomology groups \(H^\ast(C;\mathcal O(\mathbb E((\mathfrak q+\theta(\mathfrak q_{\mathfrak s}))_{\mathfrak s})))\) from those of the quotient bundles. The String and Cohomology lemmas are quite explicit, and it is clear that within finite time one should be able to compute the complementary space \(H^1(C;\mathcal O(\mathbb E((\mathfrak q_{\mathfrak s}+\theta(\mathfrak q_{\mathfrak s}))_{\mathfrak s})))\) to the tangent space \(T_C(G.C)\) of the \(G\)-orbit \(G.C\) in \(T_C(\mathcal C_q(Z))\).

Chapter 20, Classification for \(\text{rank}\,\,\mathfrak k =\text{rank}\,\,\mathfrak g\), is devoted to these calculations. It is necessary to consider five infinite series and three exceptional cases. In most cases, depending on the weights which are determined by the base cycle at hand, both vanishing and nonvanishing occur.

Reviewer: Vasily A. Chernecky (Odessa)

##### MSC:

22E46 | Semisimple Lie groups and their representations |

22-02 | Research exposition (monographs, survey articles) pertaining to topological groups |

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

32L25 | Twistor theory, double fibrations (complex-analytic aspects) |

32N10 | Automorphic forms in several complex variables |

32Q28 | Stein manifolds |

53C30 | Differential geometry of homogeneous manifolds |

22F30 | Homogeneous spaces |

32F10 | \(q\)-convexity, \(q\)-concavity |

32M10 | Homogeneous complex manifolds |

81R25 | Spinor and twistor methods applied to problems in quantum theory |

81S10 | Geometry and quantization, symplectic methods |