Holomorphic Action Research Articles

Conic reductions for Hamiltonian actions of U(2) and its maximal torus

Suppose given a Hamiltonian and holomorphic action of G=U(2) on a compact Kähler manifold M, with nowhere vanishing moment map. Given an integral coadjoint orbit mathcal {O} for G, under transversality assumptions we shall consider two naturally associated ‘conic’ reductions. One, which will be denoted overline{M}^G_{mathcal {O}}, is taken with respect to the action of G and the cone over mathcal {O}; another, which will be denoted overline{M}^T_{varvec{nu }}, is taken with respect to the action of the standard maximal torus Tleqslant G and the ray mathbb {R}_+,imath varvec{nu } along which the cone over mathcal {O} intersects the positive Weyl chamber. These two reductions share a common ‘divisor’, which may be viewed heuristically as bridging between their structures. This point of view motivates studying the (rather different) ways in which the two reductions relate to the the latter divisor. In this paper we provide some indications in this direction. Furthermore, we give explicit transversality criteria for a large class of such actions in the projective setting, as well as a description of corresponding reductions as weighted projective varieties, depending on combinatorial data associated to the action and the orbit.

Complexified Hermitian Symmetric Spaces, Hyperkähler Structures, and Real Group Actions

There is a known hyperkähler structure on any complexified Hermitian symmetric space G/K, whose construction relies on identifying G/K with both a (co)adjoint orbit and the cotangent bundle to the compact Hermitian symmetric space Gu/K0. Via a family of explicit diffeomorphisms, we show that almost all of the complex structures are equivalent to the one on G/K; via a family of related diffeomorphisms, we show that almost all of the symplectic structures are equivalent to the one on \(T^{*}\left (G_{u}/K_{0}\right )\). We highlight the intermediate Kähler structures, which share a holomorphic action of G related to the one on G/K, but moment geometry related to that of \(T^{*}\left (G_{u}/K_{0}\right )\). As an application, for the real form G0 ⊂ G corresponding to G0/K0, the Hermitian symmetric space of noncompact type, we give a strategy for study of the action on G/K using the moment-critical subsets for the intermediate structures. We give explicit computations for SL(2).

Equivariant fixed point formulae and Toeplitz operators under Hamiltonian torus actions and remarks on equivariant asymptotic expansions

Suppose given a holomorphic and Hamiltonian action of a compact torus on a polarized Hodge manifold. Assume that the action lifts to the quantizing line bundle, so that there is an induced unitary representation of the torus on the associated Hardy space which decomposes into isotypes. The main result of this paper is the description of asymptotics along rays in weight space of traces of equivariant Toeplitz operators composed with quantomorphisms for the torus action. The main ingredient in the proof is the micro-local analysis of the equivariant Szegő kernels. As a particular case we obtain a simple approach for asymptotics of the Lefschetz fixed point formula and traces of Toeplitz operators in the setting of ladder representations. We also consider equivariant asymptotic when the decomposition given by the standard circle action is taken into account, in this case one can recall previous results of X. Ma and W. Zhang or of R. Paoletti. We address some explicit computations for the action of the special unitary group of dimension two.

A Splitting Result for Real Submanifolds of a Kähler Manifold

Let (Z,omega ) be a connected Kähler manifold with an holomorphic action of the complex reductive Lie group U^mathbb {C}, where U is a compact connected Lie group acting in a hamiltonian fashion. Let G be a closed compatible Lie group of U^mathbb {C} and let M be a G-invariant connected submanifold of Z. Let xin M. If G is a real form of U^mathbb {C}, we investigate conditions such that Gcdot x compact implies U^mathbb {C}cdot x is compact as well. The vice-versa is also investigated. We also characterize G-invariant real submanifolds such that the norm-square of the gradient map is constant. As an application, we prove a splitting result for real connected submanifolds of (Z,omega ) generalizing a result proved in Gori and Podestà (Ann Global Anal Geom 26: 315–318, 2004), see also Bedulli and Gori (Results Math 47: 194–198, 2005), Biliotti (Bull Belg Math Soc Simon Stevin 16: 107–116 2009).

Canonical structure of minimal varying Λ theories

Minimal varying Λ theories are defined by an action built from the Einstein–Cartan–Holst first order action for gravity with the cosmological constant Λ as an independent scalar field, and supplemented by the Euler and Pontryagin densities multiplied by 1/Λ. We identify the canonical structure of these theories which turn out to represent an example of irregular systems. We find five degrees of freedom on generic backgrounds and for generic values of parameters, whereas if the parameters satisfy a certain condition (which includes the most commonly considered Euler case) only three degrees of freedom remain. On de Sitter-like backgrounds the canonical structure changes, and due to an emergent conformal symmetry one degree of freedom drops from the spectrum. We also analyze the self-dual case with an holomorphic action depending only on the self-dual part of the connection. In this case we find two (complex) degrees of freedom, and further discuss the Kodama state, the restriction to de Sitter background and the effect of reality conditions.

Fredholm-Regularity of Holomorphic Discs in Plane Bundles over Compact Surfaces

We study the space of holomorphic discs with boundary on a surface in a real 2-dimensional vector bundle over a compact 2-manifold. We prove that, if the ambient 4-manifold admits a fibre-preserving transitive holomorphic action, then a section with a single complex point has $C^{2,\alpha}$-close sections such that any (non-multiply covered) holomorphic disc with boundary in these sections are Fredholm regular. Fredholm regularity is also established when the complex surface is neutral K\"ahler, the action is both holomorphic and symplectic, and the section is Lagrangian with a single complex point.

Disk, interval, point: on constructions of quantum field theories with holomorphic action functionals

Bosonic quantum field theories with holomorphic action functionals are realized by two types of constructions involving supersymmetric quantum field theories, compactified on an interval in one type and compactified on a disk and deformed in the other. We establish the equivalence between the two types of constructions by reducing the disk to the interval and the interval to a point. As examples, we discuss constructions of zero-dimensional gauged sigma model, gauged quantum mechanics, gauged symplectic bosons in two dimensions, and Chern-Simons theory and its higher-dimensional variants.

Multiple D3-Instantons and Mock Modular Forms II

We analyze the modular properties of D3-brane instanton corrections to the hypermultiplet moduli space in type IIB string theory compactified on a Calabi-Yau threefold. In Part I, we found a necessary condition for the existence of an isometric action of S-duality on this moduli space: the generating function of DT invariants in the large volume attractor chamber must be a vector-valued mock modular form with specified modular properties. In this work, we prove that this condition is also sufficient at two-instanton order. This is achieved by producing a holomorphic action of SL(2,Z) on the twistor space which preserves the holomorphic contact structure. The key step is to cancel the anomalous modular variation of the Darboux coordinates by a local holomorphic contact transformation, which is generated by a suitable indefinite theta series. For this purpose we introduce a new family of theta series of signature (2,n-2), find their modular completion, and conjecture sufficient conditions for their convergence, which may be of independent mathematical interest.

Equivariant bundles and connections

Let X be a connected complex manifold equipped with a holomorphic action of a complex Lie group G. We investigate conditions under which a principal bundle on X admits a G-equivariance structure.

Meromorphic quotients for some holomorphic G-actions

Using mainly tools from [B.13] and [B.15] we give a necessary and sufficient condition in order that a holomorphic action of a connected complex Lie group $G$ on a reduced complex space $X$ admits a strongly quasi-proper meromorphic quotient. We apply this characterization to obtain a result which assert that, when $G = K.B$ with $B$ a closed complex subgroup of $G$ and $K$ a real compact subgroup of $G$, the existence of a strongly quasi-proper meromorphic quotient for the $B-$action implies, assuming moreover that there exists a $G-$invariant Zariski open dense subset in $X$ which is good for the $B-$action, the existence of a strongly quasi-proper meromorphic quotient for the $G-$action on $X$.

Representing Analytic Cohomology Groups of Complex Manifolds

Consider a holomorphic vector bundle L→X and an open cover 𝔘={U a :a∈A} of X, parametrized by a complex manifold A. We prove that the sheaf cohomology groups H q (X,L) can be computed from the complex C hol • (𝔘,L) of cochains (f a 0 ...a q ) a 0 ,...,a q ∈A that depend holomorphically on the a j , provided S={(a,x)∈A×X:x∈U a } is a Stein open subset of A×X. The result is proved in the setting of Banach manifolds, and is applied to study representations on cohomology groups induced by a holomorphic action of a complex reductive Lie group on L.

Levi problem and semistable quotients

A complex space X is in class 𝒬 G if it is a semistable quotient of the complement to an analytic subset of a Stein manifold by a holomorphic action of a reductive complex Lie group G. It is shown that every pseudoconvex unramified domain over X is also in 𝒬 G .

D3-instantons, mock theta series and twistors

Abstract The D-instanton corrected hypermultiplet moduli space of type II string theory compactified on a Calabi-Yau threefold is known in the type IIA picture to be determined in terms of the generalized Donaldson-Thomas invariants, through a twistorial construction. At the same time, in the mirror type IIB picture, and in the limit where only D3-D1-D(-1)-instanton corrections are retained, it should carry an isometric action of the S-duality group SL(2, $ \mathbb{Z} $ ). We prove that this is the case in the one-instanton approximation, by constructing a holomorphic action of SL(2, $ \mathbb{Z} $ ) on the linearized twistor space. Using the modular invariance of the D4-D2-D0 black hole partition function, we show that the standard Darboux coordinates in twistor space have modular anomalies controlled by period integrals of a Siegel-Narain theta series, which can be canceled by a contact transformation generated by a holomorphic mock theta series.

Moment maps associated with holomorphic isometric actions on Kähler manifolds

We give a new formula which writes down the moment map for holomorphic isometric action on a Kahler manifold using the motion of a charged particle when the action of is Hamiltonian.

Deformation of extremal metrics, complex manifolds and the relative Futaki invariant

Let (X,\Omega) be a closed polarized complex manifold, g be an extremal metric on X that represents the K\"ahler class \Omega, and G be a compact connected subgroup of the isometry group Isom(X,g). Assume that the Futaki invariant relative to G is nondegenerate at g. Consider a smooth family $(M \to B)$ of polarized complex deformations of (X,\Omega)\simeq (M_0,\Theta_0) provided with a holomorphic action of G with trivial action on B. Then for every t\in B sufficiently small, there exists an h^{1,1}(X)-dimensional family of extremal Kaehler metrics on M_t whose K\"ahler classes are arbitrarily close to \Theta_t. We apply this deformation theory to show that certain complex deformations of the Mukai-Umemura 3-fold admit Kaehler-Einstein metrics.

ON FIXED POINTS ON COMPACT RIEMANN SURFACES

A point of a Riemann surface X is said to be its fixed point if it is a fixed point of one of its nontrivial holomorphic automorphisms. We start this note by proving that the set Fix(X) of fixed points of Riemann surface X of genus g<TEX>${\geq}$</TEX>2 has at most 82(g-1) elements and this bound is attained just for X having a Hurwitz group of automorphisms, i.e., a group of order 84(g-1). The set of such points is invariant under the group of holomorphic automorphisms of X and we study the corresponding symmetric representation. We show that its algebraic type is an essential invariant of the topological type of the holomorphic action and we study its kernel, to find in particular some sufficient condition for its faithfulness.

Visible actions on the non-symmetric homogeneous space SO(8, ℂ)/G 2(ℂ)

A holomorphic action of a Lie group G on a connected complex manifold D is called strongly visible with slice S if G · S is open in D and if there exists an antiholomorphic diffeomorphism σ of D preserving each G -orbit in D such that σ | s = id s . This paper deals with the non-symmetric spherical variety SO(8, ℂ)/ G 2 (ℂ). We prove that a maximal compact subgroup of SO(8, ℂ) acts on SO(8, ℂ)/ G 2 (ℂ) in a strongly visible fashion. Moreover, we can take a slice to be of dimension three which coincides with the rank of this spherical variety.

Visible Actions on Reducible Multiplicity-Free Spaces

The notion of (strongly) visible actions was introduced by T. Kobayashi [10, 11] for the biholomorphic action on a complex manifold with (possibly) infinitely many orbits. A holomorphic action of a Lie group G on a complex manifold D is strongly visible with a slice S if D = G ⋅ S, and there exists an anti-holomorphic and orbit preserving diffeomorphism σ on D such that σ|S = idS. Generalizing our previous result [22] about strongly visible actions on irreducible multiplicity-free spaces à la Kac, we treat reducible multiplicity-free spaces classified by Benson–Ratcliff and Leahy and prove that any multiplicity-free space naturally admits a strongly visible action. Further, we give an explicit description of S and σ. Our result gives an evidence to Kobayashi's conjecture [12, Conjecture 3.2] in the case of linear actions, asserting that we can take S to have the same dimension with the rank of V .

A Hilbert–Mumford criterion for polystability in Kaehler geometry

Consider a Hamiltonian action of a compact Lie group K on a Kaehler manifold X with moment map μ: X → * . Assume that the action of K extends to a holomorphic action of the complexification G of K. We characterize which G-orbits in X intersect μ ―1 (0) in terms of the maximal weights lim t→∞ 〈μ(e its · x), s), where s ∈ . We do not impose any a priori restriction on the stabilizer of x. Under some mild restrictions on the action K X, we view the maximal weights as defining a collection of maps: for each x ∈ X, λ x : ∂ ∞ (K\G) → ℝ ∪ {∞}, where ∂ ∞ (K\G) is the boundary at infinity of the symmetric space K\G. We prove that G · x ∩ μ ―1 (0) ≠ 0 if: (1) λ x is everywhere nonnegative, (2) any boundary point y such that λ x (y) = 0 can be connected with a geodesic in K\G to another boundary point y' satisfying λ x (y') = 0. We also prove that the maximal weight functions are G-equivariant: for any g ∈ G and any y ∈ ∂ ∞ (K\G) we have λ g·x (y) = λ x (y · g).

Convexity properties of gradient maps

We consider the action of a real reductive group G on a Kähler manifold Z which is the restriction of a holomorphic action of a complex reductive group H. We assume that the action of a maximal compact subgroup U of H is Hamiltonian and that G is compatible with a Cartan decomposition of H. We have an associated gradient map μ p : Z → p where g = k ⊕ p is the Cartan decomposition of g . For a G-stable subset Y of Z we consider convexity properties of the intersection of μ p ( Y ) with a closed Weyl chamber in a maximal abelian subspace a of p . Our main result is a Convexity Theorem for real semi-algebraic subsets Y of Z = P ( V ) where V is a unitary representation of U.