Claus Hertling, Christian Sevenheck
We give an overview on the tt*-geometry defined for isolated hypersurface singularities and tame functions via Brieskorn lattices. We discuss nilpotent orbits in this context, as well as classifying spaces of Brieskorn lattices and (limits of) period maps.
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We investigate the pseudo-hyperkähler geometry of higher degree rational curves in the twistor space of a hyperkähler $4$-manifold.
A study is made of algebraic curves and surfaces in the flag manifold $\mathbb{F}=SU(3)/T^2$, and their configuration relative to the twistor projection $π$ from $\mathbb{F}$ to the complex projective plane $\mathbb{CP}^2$, defined with the help of an anti-holomorphic involution $j$. This is motivated by analogous studies of algebraic surfaces of low degree in the twistor space $\mathbb{CP}^3$ of $S^4$. Deformations of twistor fibres project to real surfaces in $\mathbb{CP}^2$, whose metric geometry is investigated. Attention is then focussed on toric Del Pezzo surfaces that are the simplest t…
We continue to study twistor spaces on the connected sum of four complex projective planes, whose anticanonical map is of degree two over the image. In particular, we determine the defining equation of the branch divisor of the anticanonical map in an explicit form. Together with previous two articles (arXiv:1009.3153 and arXiv:0705.0060), this completes explicit description of all such twistor spaces.
It is shown that there exists a twistor space on the $n$-fold connected sum of complex projective planes $n\mathbb{CP}^2$, whose algebraic dimension is one and whose general fiber of the algebraic reduction is birational to an elliptic ruled surface or a K3 surface. The former kind of twistor spaces are constructed over $n\mathbb{CP}^2$ for any $n\ge 5$, while the latter kind of example is constructed over $5\mathbb{CP}^2$. Both of these seem to be the first such example on $n\mathbb{CP}^2$. The algebraic reduction in these examples is induced by the anti-canonical system of the twistor spaces…
We study real lines on certain Moishezon threefolds which are potentially twistor spaces of 3CP^2. Here, line means a smooth rational curve whose normal bundle is O(1)^2 and the reality implies the invariance under an anti-holomorphic involution on the threefolds. Our threefolds are birational to double coverings of CP^3 branched along singular quartic surfaces. On these threefolds we find families of real lines in explicit form and prove that which families have to be chosen as twistor lines depend on how we take small resolutions of the double covering. This is a first step for determining t…
The paper studies explicitly the Hitchin system restricted to the Higgs fields on a fixed very stable rank 2 bundle in genus 2 and 3. The associated families of quadrics relate to both the geometry of Penrose's twistor spaces and several classical results.
Using examples of compact complex 3-manifolds which arise as twistor spaces, we show that the class of compact complex manifolds bimeromorphic to Kähler manifolds is not stable under small deformations of complex structure.
The twistor space of the sphere S^{2n} is an isotropic Grassmannian that fibers over S^{2n}. An orthogonal complex structure on a subdomain of S^{2n} (a complex structure compatible with the round metric) determines a section of this fibration with holomorphic image. In this paper, we use this correspondence to prove that any finite energy orthogonal complex structure on R^6 must be of a special warped product form, and we also prove that any orthogonal complex structure on R^{2n} that is asymptotically constant must itself be constant. We will also give examples defined on R^{2n} which have i…
Let $X$ be a quasi-projective curve, compactified to $(Y,D)$ with $X=Y-D$. We construct a Deligne-Hitchin twistor space out of moduli spaces of framed $λ$-connections of rank $2$ over $Y$ with logarithmic singularities and quasi-parabolic structure along $D$. To do this, one should divide by a Hecke-gauge groupoid. Tame harmonic bundles on $X$ give preferred sections, and the relative tangent bundle along a preferred section has a mixed twistor structure with weights $0,1,2$. The weight $2$ piece corresponds to the deformations of the KMS structure including parabolic weights and the residues …
In this note, we investigate pluri-half-anticanonical systems on the so called LeBrun twistor spaces. We determine its dimension, the base locus, structure of the associated rational map, and also structure of general members, in precise form. In particular, we show that if n>2 and m>1, the base locus of the system |mK^{-1/2}| on nCP^2 consists of two non-singular rational curves, along which any member has singularity, and that if we blow-up these curves, then the strict transform of a general members of |mK^{-1/2}| becomes an irreducible non-singular surface. We also show that if n>3 and m>n…
An exposition of the basic geometry of twistor integrals, intended for mathematicians.
Twistor sigma models for quaternionic geometry and graviton scattering
本文通过构造以四元数流形扭量空间为靶空间的新 sigma 模型，计算超凯勒与四元数-凯勒几何的标量势，并利用矩阵树定理从广义相对论直接推导出树级 MHV 引力子振幅的 Hodges 公式，进而推广到完整树级引力子 S 矩阵。
We study the geometry of the (generalized) twistor triangles $\triangle J_1J_2J_3$ in the period domain of compact complex tori of complex dimension $2n$ by the means of the representation theory of the algebras (of real dimension 8) generated by the complex structures $J_1,J_2,J_3$. Considering the period domain as the homogeneous space for $G=GL_{4n}(\mathbb{R})$, we introduce on it a $G$-invariant pseudometric and define pseudometric invariants, helping us to distinguish triangles from a reasonable class up to $G$-equivalence.
Joyce structures and their twistor spaces
本文介绍了源自Donaldson-Thomas不变量全纯生成函数的Joyce结构，重点阐述其相关扭量空间的几何性质，并证明了若干新的研究结果。
The Absolute Twistor Line and the Geometry of $\overline{\text{Spec}\, \mathbf Z}$
本研究通过合并仿射绝对曲线与阿基米德分量构造$\overline{\text{Spec}\mathbf{Z}}$的绝对代数几何，引入奇算术拓扑斯中的全局绝对曲线，揭示了其与扭量结构、霍普夫代数及几何L-函数局部因子的关联。
Geometry of the Donaldson-Friedman Pushout: Twistor degenerations and instanton charge
本文研究了Donaldson-Friedman构造中出现的奇异中心纤维，将其视为两个吹胀的Twistor空间沿例外四面体的Ferrand推出。通过显式代数几何模型，描述了与连通和构造相关的Twistor退化。并推导了半稳定光滑的组件特化公式，以及表面在双定位处的刚性粘合约束。
We associate to the resolved conifold an affine special Kähler (ASK) manifold of complex dimension 1, and an instanton corrected hyperkähler (HK) manifold of complex dimension 2. We describe these geometries explicitly, and show that the instanton corrected HK geometry realizes an Ooguri-Vafa-like smoothing of the semi-flat HK metric associated to the ASK geometry. On the other hand, the instanton corrected HK geometry associated to the resolved conifold can be described in terms of a twistor family of two holomorphic Darboux coordinates. We study a certain conformal limit of the twistor coord…
Given a path geometry on a surface $\mathcal{U}$, we construct a causal structure on a four-manifold which is the configuration space of non-incident pairs (point, path) on $\mathcal{U}$. This causal structure corresponds to a conformal structure if and only if $\mathcal{U}$ is a real projective plane, and the paths are lines. We give the example of the causal structure given by a symmetric sextic, which corresponds on an ${\rm SL}(2,{\mathbb R})$-invariant projective structure where the paths are ellipses of area $π$ centred at the origin. We shall also discuss a causal structure on a seven-d…
Applications of singularity theory in applied algebraic geometry and algebraic statistics
本文综述了拓扑学与奇点理论在研究应用代数几何和代数统计中具体优化问题的代数复杂度方面的近期应用。
We elaborate on the quantization of toric varieties by combining techniques from toric geometry, isospectral deformations and noncommutative geometry in braided monoidal categories, and the construction of instantons thereon by combining methods from noncommutative algebraic geometry and a quantised twistor theory. We classify the real structures on a toric noncommutative deformation of the Klein quadric and use this to derive a new noncommutative four-sphere which is the unique deformation compatible with the noncommutative twistor correspondence. We extend the computation of equivariant inst…
We introduce a geometric realization of noncommutative singularity resolutions. To do this, we first present a new conjectural method of obtaining conventional resolutions using coordinate rings of matrix-valued functions. We verify this conjecture for all cyclic quotient surface singularities, the Kleinian D_n and E_6 surface singularities, the conifold singularity, and a non-isolated singularity, using appropriate quiver algebras. This conjecture provides a possible new generalization of the classical McKay correspondence. Then, using symplectic reduction within these rings, we obtain new, n…
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