arXiv research

Study on Hermitian Calabi functional in complexified orbits of symplectic manifolds.

The paper describes orbits of parabolic subgroups in complexified actions.

We first investigate the geometry of orbits of the isotropy action on a semi-simple pseudo-Riemannian symmetric space by investigating the complexified action. Next we investigate the geometry of the orbits of Hermann type actions on the symmetric spaces. By considering two special Hermann type actions on the symmetric…

This paper explores hyperkähler structures on specific orbits using different methods.

Study of complexified Hermitian symmetric spaces and their structures.

Representations of $C^*$-algebras are realized on section spaces of holomorphic homogeneous vector bundles. The corresponding section spaces are investigated by means of a new notion of reproducing kernel, suitable for dealing with involutive diffeomorphisms defined on the base spaces of the bundles. Applications of th…

Proves volume conjecture for double twist knots using complexified tetrahedrons.

Study special complexified Kähler forms in mirror symmetry.

Study norm-squared of momentum map in infinite dimensions with applications to symplectic geometry.

Extends K-energy to complexified Kähler classes for scalar curvature study.

Let M be a Kaehler manifold with a free, holomorphic and Hamiltonian action of the standard n-torus T. We give a simple, explicit and canonical formula for the Kaehler potential on the Kaehler reduction of M. As a consequence we can derive improvements of several classical results known for more general Hamiltonian red…

We prove the $K(π,1)$ conjecture for affine Artin groups: the complexified complement of an affine reflection arrangement is a classifying space. This is a long-standing problem, due to Arnol'd, Pham, and Thom. Our proof is based on recent advancements in the theory of dual Coxeter and Artin groups, as well as on sever…

We shall construct a natural Higgs bundle structure on the complexified Kähler cone of a compact Kähler manifold, which can be seen as an analogy of the classical Higgs bundle structure associated to a variation of Hodge structure. In the proof of the flat-ness of our Higgs bundle, we find a commutator identity that ca…

New combinatorial model for Milnor fibration using oriented matroids.

Derives scalar curvature formula in generalized Kähler geometry.

The paper constructs homotopically non-trivial spheres in complexified spaces.

It is known that there exist Calabi-Yau structures on the complexifications of symmetric spaces of compact type. In this paper, we describe the Calabi-Yau structures of the complexified symmetric spaces in terms of the Schwarz's theorem in detail. We consider the case where the Calabi-Yau structure arises from the Riem…

We study the Kaehler metric given by the logarithm of a cubic form on its complexified index cone. Under mirror symmetry, this metric should asymptotically correspond to the Weil-Petersson metric. Using the theory of special Kaehler manifolds, a proof of a curvature formula for this metric is given.

Gradient maps of real reductive group actions on manifolds studied.

For a real oriented hyperplane arrangement, we show that the corresponding Salvetti complex is homotopy equivalent to the complement of the complexified arrangement. This result was originally proved by M. Salvetti. Our proof follows the framework of a proof given by L. Paris and relies heavily on the notation of orien…

We study the space of conformal immersions of a 2-torus into the 4-sphere. The moduli space of generalized Darboux transforms of such an immersed torus has the structure of a Riemann surface, the spectral curve. This Riemann surface arises as the zero locus of the determinant of a holomorphic family of Dirac type opera…

Constructs simplified or complexified simplicial complexes.

In this paper, we first define the complexification of a real analytic map between real analytic Koszul manifolds and show that the complexified map is the holomorphic extension of the original map. Next we define an anti-Kaehler metric compatible with the adapted complex structure on the complexification of a real ana…

Let $V$ be a finite dimensional complex vector space and $W\subseteq \GL(V)$ be a finite complex reflection group. Let $V^{\reg}$ be the complement in $V$ of the reflecting hyperplanes. We prove that $V^{\reg}$ is a $K(π,1)$ space. This was predicted by a classical conjecture, originally stated by Brieskorn for complex…

Subharmonicity of Dirichlet energy proven for Kähler manifolds.

Adds layers to NNs to protect them from reverse engineering.

Criterion for polystability in Lie group actions on manifolds.

Let $V$ be a finite dimensional complex vector space and $W\subset \GL(V)$ be a finite complex reflection group. Let $V^{\reg}$ be the complement in $V$ of the reflecting hyperplanes. A classical conjecture predicts that $V^{\reg}$ is a $K(pi,1)$ space. When $W$ is a complexified real reflection group, the conjecture f…

The paper constructs invariant Calabi-Yau structures on complexified symmetric spaces.

Reconstructs supersymmetry and supergravity in complexified Z/2-graded Algebraic Geometry.

In this note we show that any real exact G-invariant (1,1)-form is the Ricci form of a Kaehler metric on the complexification of an irreducible compact symmetric space G/K.

We prove the existence of lattice isomorphic line arrangements having $π_1$-equivalent or homotopy-equivalent complements and non homeomorphic embeddings in the complex projective plane. We also provide two explicit examples, one is formed by real-complexified arrangements while the second is not.

We present a cocycle model for elliptic cohomology with complex coefficients in which methods from 2-dimensional quantum field theory can be used to rigorously construct cocycles. For example, quantizing a theory of vector bundle-valued fermions yields a cocycle representative of the elliptic Thom class. This construct…

Every finite dimensional real representation of a compact real semisimple Lie algebra determines a metric 2-step nilpotent Lie algebra and a corresponding simply connected metric 2-step nilpotent Lie group N. We study the differential geometry of N using representation theory of the complexified complex semisimple Lie …

We study the geometry of complexified moduli spaces of special Lagrangian submanifolds in the complement of an anticanonical divisor in a compact Kahler manifold. In particular, we explore the connections between T-duality and mirror symmetry in concrete examples, and show how quantum corrections arise in this context.

We represent algebraic curves via commuting matrix polynomials. This allows us to show that the Hilbert scheme of cohomologically stable twisted rational curves of degree $d$ in ${\Bbb P}^3\backslash {\Bbb P}^1$ is isomorphic to a complexified hyperkähler quotient of an open subset of a vector space by a non-reductive …

Constructs maps from field theories to complexified K-theory and elliptic cohomology.

The complexified ${\Bbb Z}/2$-graded $C^\infty$-Algebraic Geometry aspect of a superspace(-time) $\widehat{X}$ in Sec.\,1 of D(14.1) (arXiv:1808.05011 [math.DG]) together with the Spin-Statistics Theorem in Quantum Field Theory, which requires fermionic components of a superfield be anticommuting, lead us to the notion…

Defines Chern-Robinson connection on Robinson manifolds.

We prove that a K-contact Lie group of dimension five or greater is the central extension of a symplectic Lie group by complexifying the Lie algebra and applying a result from complex contact geometry, namely, that, if the adjoint action of the complex Reeb vector field on a complex contact Lie algebra is diagonalizabl…

We identify two Frobenius manifolds obtained from two different differential Gerstenhaber-Batalin-Vilkovisky algebras on a compact Kaehler manifold. One is constructed on the Dolbeault cohomology, and the other on the de Rham cohomology. Our result can be considered as a generalization of the identification of the Dolb…

We present the alternative topological twisting of N=4 Yang-Mills, in which the path integral is dominated not by instantons, but by flat connections of the COMPLEXIFIED gauge group. The theory is nontrivial on compact orientable four-manifolds with nonpositive Euler number, which are necessarily not simply connected. …

By using computer assistance, we prove that the fundamental group of the complement of a real complexified line arrangement is not determined by its intersection lattice, providing a counter-example for a problem of Falk and Randell. We also deduce that the torsion of the lower central series quotients is not combinato…

This paper describes results characterizing the range of the time-t heat operator on various manifolds, including Euclidean spaces, spheres, and hyperbolic spaces. The guiding principle behind these results is this: The functions in the range of the heat operator should be, roughly, those functions having an analytic c…

Study limits of adjoint orbits for Lie groups, describing nilpotent orbits.

If $X$ is a connected complex manifold with $d_X = 2$ that admits the holomorphic and transitive action of a (connected) Lie group $G$, then the action extends to an action of the complexification $\hat{G}$ of $G$ on $X$ except when either the unit disk or else a strictly pseudoconcave homogeneous complex manifold is i…

For certain problems involving vector fields, it is possible to find an associated imaginary field that, in conjunction with the first, forms a complex field for which the equation can be solved. This result is generalized to arbitrary Clifford algebras, followed by quaternionic vectors as a special case. All results a…

We study Milnor fibers of complexified real line arrangements. We give a new algorithm computing monodromy eigenspaces of the first cohomology. The algorithm is based on the description of minimal CW-complexes homotopic to the complements, and uses the real figure, that is, the adjacency relations of chambers. It enabl…