Claude LeBrun
Building on previous results, we complete the classification of compact oriented Einstein 4-manifolds with det (W^+) > 0. There are, up to diffeomorphism, exactly 15 manifolds that carry such metrics, and, on each of these manifolds, such metrics sweep out exactly one connected component of the corresponding Einstein moduli space.
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We study anti-holomorphic involutions of the moduli space of principal $G$-Higgs bundles over a compact Riemann surface $X$, where $G$ is a complex semisimple Lie group. These involutions are defined by fixing anti-holomorphic involutions on both $X$ and $G$. We analyze the fixed point locus in the moduli space and their relation with representations of the orbifold fundamental group of $X$ equipped with the anti-holomorphic involution. We also study the relation with branes. This generalizes work by Biswas--Garc\'ıa-Prada--Hurtubise and Baraglia--Schaposnik.
Analytic torsion for irreducible holomorphic symplectic fourfolds with involution, I: Construction of an invariant
本文利用等变解析挠率，为具有反辛对合的 $K3^{[2]}$ 型不可约全纯辛流形构造了一个新不变量，并给出了该不变量对数的复 Hessian 公式。
Real Slices of Parabolic $\mathrm{SL}(r,\mathbb{C})$-Opers
研究抛物SL(r,C)-操在反共轭involution下的固定点集，证明不同描述下involution的一致性。
We study complex Lagrangian submanifolds of a compact hyper-Kähler manifold and prove two results: (a) that an involution of a hyper-Kähler manifold which is antiholomorphic with respect to one complex structure and which acts non-trivially on the corresponding symplectic form always has a fixed point locus which is complex Lagrangian with respect to one of the other complex structures, and (b) there exist Lagrangian submanifolds which are complex with respect to one complex structure and are not the fixed point locus of any involution which is anti-holomorphic with respect to one of the other…
In this note we prove that the space of linear anti-symplectic involutions is the homogenous space $Gl(n,\R)\Sp(n)$. This result is motivated by the study of symmetric periodic orbits in the restricted 3-body problem.
Our purpose in this article is to study anti-invariant statistical submersions from holomorphic statistical manifolds. Firstly we introduce holomorphic statistical submersions satisfying the certain condition, after we give anti-invariant statistical submersions satisfying the certain condition. And we supported our results with examples.
We introduce a class of $G$-invariant connections on a homogeneous principal bundle $Q$ over a hermitian symmetric space $M=G/K$. The parameter space carries the structure of normal variety and has a canonical anti-holomorphic involution. The fixed points of the anti-holomorphic involution are precisely the integrable invariant complex structures on $Q.$ This normal variety is closely related to quiver varieties and, more generally, to varieties of commuting matrix tuples modulo simultaneous conjugation.
We study the holomorphic sections of the Deligne-Hitchin moduli space of a compact Riemann surface that are invariant under the natural anti-holomorphic involutions of the moduli space. Their relationships with the harmonic maps are established. As a bi-product, a question of Simpson on such sections, posed in \cite{Si2}, is answered.
We study the non-embddability property for a class of real hypersurfaces, called real hypersurfaces of involution type, into the sphere in the low codimensional case, by making use of property of a naturally related Gauss curvature. We also study rigidity problems for conformal maps between a class of Kähler manifolds with pseud-conformally flat metrics.
The CR $δ$-invariant for CR-submanifolds was introduced in a recent article [B. Y. Chen, An optimal inequality for CR-warped products in complex space forms involving CR $δ$-invariant, Internat. J. Math. 23} (2012), no. 3, 1250045 (17 pages)]. In this paper, we prove two new optimal inequalities for anti-holomorphic submanifolds in complex space forms involving the CR $δ$-invariant. Moreover, we obtain some classification results for certain anti-holomorphic submanifolds in complex space forms which satisfy the equality case of either inequality.
We study anti-holomorphic semi-invariant submersions from Kählerian manifolds onto Riemannian manifolds. We prove that all distributions which are involved in the definition of the submersion are integrable. We also prove that the O'Neill's tensor $\mathcal{T}$ vanishes on the invariant vertical distribution. We give necessary and sufficient conditions for totally geodesicness and harmonicity of this type submersions. Moreover, we investigate the several curvatures of the total manifold and fibers and give a characterization theorem.
The mod 2 cohomology of fixed point sets of anti-symplectic involutions
设$M$是一个紧致的连通辛流形，其Hamilton作用在一个紧致$n$维环面$G=T^n$上.设$σ$是与$G$ -作用相容的反辛对合. $M$的实轨迹是$X$，即$σ$的不动点集。Duistermaat使用莫尔斯理论，根据$M$的上同调，给出了$X$的普通上同调的描述。在$X$上有一个剩余的$\G=(\Zt)^n$作用，我们可以利用Duistermaat的结果，以及关于等变上同调的一些一般事实，证明一个类似于Duistermaat定理的等变类似。在某些情况下，我们也可以将Goresky-Kottwitz-MacPherson和Goldin-Holm的定理推广到实轨迹。
Seiberg-Witten invariants and real curves
在具有方向保持对合c的紧致方向四流形上，我们计算了关于c对称的Seiberg-Witten方程的解，以构造“实”Seiberg-Witten不变量。利用Taubes的结果，我们证明了在具有反辛和反全纯对合的辛殆复流形上，这些不变量并不都是平凡的，并且典范丛可用实全纯曲线表示.
We studied the axiom of anti-invariant 2-spheres and the axiom of co-holomorphic $(2n+1)$-spheres. We proved that a nearly Kählerian manifold satisfying the axiom of anti-invariant 2-spheres is a space of constant holomorphic sectional curvature. We also showed that an almost Hermitian manifold $M$ of dimension $2m\geq6$ satisfying the axiom of co-holomorphic $(2n+1)$-spheres for some $n,$ where $(1\leq n\leq m-1)$, the manifold $M$ has pointwise constant type $α$ if and only if $M$ has pointwise constant anti-holomorphic sectional curvature $α$.
Anti-holomorphic twistor and Symplectic structure
众所周知，twister空间的截面twister是偶数维黎曼流形$X$上殆复结构的分类。本文证明了调和反全纯扭子与$X$上的辛结构等价。
We construct large families of new collapsing hyperkähler metrics on the K3 surface. The limit space is the quotient of a flat 3-torus by an involution. Away from finitely many exceptional points the collapse occurs with bounded curvature. There are at most 24 exceptional points where the curvature concentrates, which always contains the 8 fixed points of the involution on the 3-torus. The geometry around these points is modelled by ALF gravitational instantons: of dihedral type (Dk) for the fixed points of the involution on the 3-torus and of cyclic type (Ak) otherwise. The collapsing metrics…
We consider the connected-sum method of constructing compact Riemannian 7-manifolds with holonomy G_2 developed in math.DG/0012189. The method requires pairs of projective complex threefolds endowed with anticanonical K3 divisors, the latter `matching' via a certain non-holomorphic map. Suitable examples of threefolds were previously obtained in math.DG/0012189 by blowing up curves in Fano threefolds. In this paper, we give further suitable algebraic threefolds using theory of K3 surfaces with non-symplectic involution due to Nikulin. These threefolds are not obtainable from Fano threefolds, a…
Symplectic double groupoids and the generalized Kähler potential
本文解决了广义Kähler几何的基本自由度描述问题，证明其全纯结构为双辛广群间的全纯辛Morita双双模，度量由拉格朗日子流形决定，正面验证了广义Kähler势存在性猜想，并以紧偶维半单李群为例构造验证。
We study the relation between $J$-anti-invariant $2$-forms and pseudoholomorphic curves in this paper. We show the zero set of a closed $J$-anti-invariant $2$-form on an almost complex $4$-manifold supports a $J$-holomorphic subvariety in the canonical class. This confirms a conjecture of Draghici-Li-Zhang. A higher dimensional analogue is established. We also show the dimension of closed $J$-anti-invariant $2$-forms on an almost complex $4$-manifold is a birational invariant, in the sense that it is invariant under degree one pseudoholomorphic maps.
Wiedersehen metrics and exotic involutions of Euclidean spheres
我们提供明确的，简单的，几何公式的自由对合ρ欧几里德领域不共轭的对映对合。因此商S^n/rho是一个同伦等价但不与RP^n同胚的流形。我们利用这些公式构造了pi_1 Diff（S^5）和pi_1 Diff（S^13）中的显式非平凡元，并为群作用中的非消去现象提供了显式公式.
Crooked planes were defined by Drumm to bound fundamental polyhedra in Minkowski space for Margulis spacetimes. They were extended by Frances to closed polyhedral surfaces in the conformal compactification of Minkowski space (Einstein space) which we call crooked surfaces. The conformal model of anti-de Sitter space is the interior of the quotient of Einstein space by an involution fixing an Einstein plane. The purpose of this note is to show that the crooked planes defined in anti-de Sitter space recently by Danciger-Guéritaud-Kassel lift to restrictions of crooked surfaces in Einstein space …
Let X be a compact complex manifold whose anti-canonical line bundle is big. We show that X admits no non-trivial holomorphic vector fields if it is Gibbs stable (at any level). The proof is based on a vanishing result for measure preserving holomorphic vector fields on X of independent interest. As an application it shown that, in general, if the anti-canonical line bundle is big, there are no holomorphic vector fields on X that are tangent to a non-singular irreducible anti-canonical divisor S on X. More generally, the result holds for varieties with log terminal singularities and log pairs.…
On the holomorphicity of isometries of intrinsic metrics in complex analysis
设$\1$和$\2$是$\Cn$中的$\s$域，$f: \1 \rt \2$是Kobayashi或Carathéodory度量的等距。假设$f$作为$C^1$映射扩展到$ \bar \om_1$。然后证明了$f|_{\partial \1}: \partial \1 \rt \partial \2$是CR或反CR同构。因此，$\1$和$\2$必须是双全纯的或反双全纯的。主要工具是Pinchuk重新缩放技术的公制版本。
Through the action of anti-holomorphic involutions on a compact Riemann surface, we construct families of (A,B,A)-branes in the moduli spaces of G_c-Higgs bundles on the Riemann surface. We study the geometry of these (A,B,A)-branes in terms of spectral data and show they have the structure of real integrable systems.
Atiyah's classical work on circular symmetry and stationary phase shows how the $\hat{A}$-genus is obtained by formally applying the equivariant cohomology localization formula to the loop space of a simply connected spin manifold. The same technique, applied to a suitable ''antiholomorphic sector'' in the $\mathbb{C}/Λ$-equivariant cohomology of the conformal double loop space $\mathrm{Maps}(\mathbb{C}/Λ,X)$ of a rationally string manifold $X$ produces the Witten genus of $X$. This can be seen as an equivariant localization counterpart to Berwick-Evans supersymmetric localization derivation o…
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