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Masafumi Okumura - One of the best experts on this subject based on the ideXlab platform.
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normal curvature of cr submanifolds of maximal cr dimension of the complex Projective Space
Acta Mathematica Hungarica, 2018Co-Authors: Mirjana Djoric, Masafumi OkumuraAbstract:We prove that there do not exist CR submanifolds Mn of maximal CR dimension of a complex Projective Space \({\mathbf{P}^{\frac{n+p}{2}}(\mathbf{C})}\) with flat normal connection D of M, when the distinguished normal vector field is parallel with respect to D. If D is lift-flat, then there exists a totally geodesic complex Projective subSpace \({\mathbf{P}^{\frac{n+1}{2}}(\mathbf{C})}\) of \({\mathbf{P}^{\frac{n+p}{2}}(\mathbf{C})}\) such that M is a real hypersurface of \({\mathbf{P}^{\frac{n+1}{2}}(\mathbf{C})}\).
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cr submanifolds of complex Projective Space
2010Co-Authors: Mirjana Djoric, Masafumi OkumuraAbstract:1. Complex manifold.- 2. Almost complex structure.- 3. Complex vector Space complexification.- 4. Kahler manifold.- 5. Structure equations of a submanifold.- 6. Submanifolds of a Euclidean Space.- 7. Submanifolds of a complex manifold.- 8. The Levi form.- 9. The principal circle bundle S^{2n+1}({\bf P}^n({\bf C}),S^1).- 10. Submersion and immersion.- 11. Hypersurfaces of a Riemannian manifold of constant curvature.- 12. Hypersurfaces of a sphere S^{n+1}(1/a).- 13. Hypersurfaces of a sphere with parallel shape operator.- 14. Codimension reduction of a submanifold.- 15. CR submanifolds of maximal CR dimension.- 16. Real hypersurfaces of a complex Projective Space.- 17. Tubes around submanifolds.- 18. Levi form of CR submanifolds of maximal CR dimension of a complex Space form.- 19. Eigenvalues of the shape operator A of CR submanifolds of maximal CR dimension of a complex Space form.- 20. CR submanifolds of maximal CR dimension satisfying the condition h(FX,Y)+h(X,FY)=0.- 21. Contact CR submanifolds of maximal CR dimension.- 22. Invariant submanifolds of real hypersurfaces of complex Space forms.- 23. The scalar curvature of CR submanifolds of maximal CR dimension.
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certain condition on the second fundamental form of cr submanifolds of maximal cr dimension of complex Projective Space
Israel Journal of Mathematics, 2009Co-Authors: Mirjana Djoric, Masafumi OkumuraAbstract:We study m-dimensional real submanifolds M with (m − 1)-dimensional maximal holomorphic tangent subSpace in complex Projective Space. On these manifolds there exists an almost contact structure F which is naturally induced from the ambient Space and in this paper we study the condition h(FX, Y) − h(X, FY) = g(FX, Y)η, η ∈ T⊥(M), on the almost contact structure F and on the second fundamental form h of these submanifolds and we characterize certain model Spaces in complex Projective Space.
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the scalar curvature of cr submanifolds of maximal cr dimension of complex Projective Space
Monatshefte für Mathematik, 2008Co-Authors: Mirjana Djoric, Masafumi OkumuraAbstract:We treat n-dimensional compact minimal submanifolds of complex Projective Space when the maximal holomorphic tangent subSpace is (n − 1)-dimensional and we give a sufficient condition for such submanifolds to be tubes over totally geodesic complex subSpaces.
Mirjana Djoric - One of the best experts on this subject based on the ideXlab platform.
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normal curvature of cr submanifolds of maximal cr dimension of the complex Projective Space
Acta Mathematica Hungarica, 2018Co-Authors: Mirjana Djoric, Masafumi OkumuraAbstract:We prove that there do not exist CR submanifolds Mn of maximal CR dimension of a complex Projective Space \({\mathbf{P}^{\frac{n+p}{2}}(\mathbf{C})}\) with flat normal connection D of M, when the distinguished normal vector field is parallel with respect to D. If D is lift-flat, then there exists a totally geodesic complex Projective subSpace \({\mathbf{P}^{\frac{n+1}{2}}(\mathbf{C})}\) of \({\mathbf{P}^{\frac{n+p}{2}}(\mathbf{C})}\) such that M is a real hypersurface of \({\mathbf{P}^{\frac{n+1}{2}}(\mathbf{C})}\).
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cr submanifolds of complex Projective Space
2010Co-Authors: Mirjana Djoric, Masafumi OkumuraAbstract:1. Complex manifold.- 2. Almost complex structure.- 3. Complex vector Space complexification.- 4. Kahler manifold.- 5. Structure equations of a submanifold.- 6. Submanifolds of a Euclidean Space.- 7. Submanifolds of a complex manifold.- 8. The Levi form.- 9. The principal circle bundle S^{2n+1}({\bf P}^n({\bf C}),S^1).- 10. Submersion and immersion.- 11. Hypersurfaces of a Riemannian manifold of constant curvature.- 12. Hypersurfaces of a sphere S^{n+1}(1/a).- 13. Hypersurfaces of a sphere with parallel shape operator.- 14. Codimension reduction of a submanifold.- 15. CR submanifolds of maximal CR dimension.- 16. Real hypersurfaces of a complex Projective Space.- 17. Tubes around submanifolds.- 18. Levi form of CR submanifolds of maximal CR dimension of a complex Space form.- 19. Eigenvalues of the shape operator A of CR submanifolds of maximal CR dimension of a complex Space form.- 20. CR submanifolds of maximal CR dimension satisfying the condition h(FX,Y)+h(X,FY)=0.- 21. Contact CR submanifolds of maximal CR dimension.- 22. Invariant submanifolds of real hypersurfaces of complex Space forms.- 23. The scalar curvature of CR submanifolds of maximal CR dimension.
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certain condition on the second fundamental form of cr submanifolds of maximal cr dimension of complex Projective Space
Israel Journal of Mathematics, 2009Co-Authors: Mirjana Djoric, Masafumi OkumuraAbstract:We study m-dimensional real submanifolds M with (m − 1)-dimensional maximal holomorphic tangent subSpace in complex Projective Space. On these manifolds there exists an almost contact structure F which is naturally induced from the ambient Space and in this paper we study the condition h(FX, Y) − h(X, FY) = g(FX, Y)η, η ∈ T⊥(M), on the almost contact structure F and on the second fundamental form h of these submanifolds and we characterize certain model Spaces in complex Projective Space.
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the scalar curvature of cr submanifolds of maximal cr dimension of complex Projective Space
Monatshefte für Mathematik, 2008Co-Authors: Mirjana Djoric, Masafumi OkumuraAbstract:We treat n-dimensional compact minimal submanifolds of complex Projective Space when the maximal holomorphic tangent subSpace is (n − 1)-dimensional and we give a sufficient condition for such submanifolds to be tubes over totally geodesic complex subSpaces.
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cr submanifolds of maximal cr dimension in complex Projective Space and its holomorphic sectional curvature
2003Co-Authors: Mirjana DjoricAbstract:Let M be a CR submanifold of maximal CR dimension in a complex Projective Space such that the distinguished vector fleld » is parallel with respect to the normal connection. In this article we treat the special case when the shape operator with respect to this vector fleld has exactly two distinct eigenvalues and we give another su-cient condition for M to be an open subset of a geodesic sphere by discussing its holomorphic sectional curvature.
Nick Sheridan - One of the best experts on this subject based on the ideXlab platform.
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homological mirror symmetry for calabi yau hypersurfaces in Projective Space
Inventiones Mathematicae, 2015Co-Authors: Nick SheridanAbstract:We prove Homological Mirror Symmetry for a smooth $$d$$ -dimensional Calabi–Yau hypersurface in Projective Space, for any $$d \ge 3$$ (for example, $$d=3$$ is the quintic threefold). The main techniques involved in the proof are: the construction of an immersed Lagrangian sphere in the ‘ $$d$$ -dimensional pair of pants’; the introduction of the ‘relative Fukaya category’, and an understanding of its grading structure; a description of the behaviour of this category with respect to branched covers (via an ‘orbifold’ Fukaya category); a Morse–Bott model for the relative Fukaya category that allows one to make explicit computations; and the introduction of certain graded categories of matrix factorizations mirror to the relative Fukaya category.
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homological mirror symmetry for calabi yau hypersurfaces in Projective Space
arXiv: Symplectic Geometry, 2011Co-Authors: Nick SheridanAbstract:We prove Homological Mirror Symmetry for a smooth d-dimensional Calabi-Yau hypersurface in Projective Space, for any d > 2 (for example, d = 3 is the quintic three-fold). The main techniques involved in the proof are: the construction of an immersed Lagrangian sphere in the `d-dimensional pair of pants'; the introduction of the `relative Fukaya category', and an understanding of its grading structure; a description of the behaviour of this category with respect to branched covers (via an `orbifold' Fukaya category); a Morse-Bott model for the relative Fukaya category that allows one to make explicit computations; and the introduction of certain graded categories of matrix factorizations mirror to the relative Fukaya category.
Alexander Varchenko - One of the best experts on this subject based on the ideXlab platform.
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Equivariant quantum differential equation, Stokes bases, and K-theory for a Projective Space
European Journal of Mathematics, 2021Co-Authors: V Tarasov, Alexander VarchenkoAbstract:We consider the equivariant quantum differential equation for the Projective Space $$P^{n-1}$$ P n - 1 and introduce a compatible system of difference equations. We prove an equivariant gamma theorem for $$P^{n-1}$$ P n - 1 , which describes the asymptotics of the differential equation at its regular singular point in terms of the equivariant characteristic gamma class of the tangent bundle of $$P^{n-1}$$ P n - 1 . We describe the Stokes bases of the differential equation at its irregular singular point in terms of the exceptional bases of the equivariant K -theory algebra of $$P^{n-1}$$ P n - 1 and a suitable braid group action on the set of exceptional bases. Our results are an equivariant version of the well-known results of Dubrovin and Guzzetti..
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equivariant quantum differential equation and qkz equations for a Projective Space stokes bases as exceptional collections stokes matrices as gram matrices and b theorem
arXiv: Algebraic Geometry, 2019Co-Authors: Giordano Cotti, Alexander VarchenkoAbstract:In arXiv:1901.02990v1 the equivariant quantum differential equation ($qDE$) for a Projective Space was considered and a compatible system of difference $qKZ$ equations was introduced; the Space of solutions to the joint system of the $qDE$ and $qKZ$ equations was identified with the Space of the equivariant $K$-theory algebra of the Projective Space; Stokes bases in the Space of solutions were identified with exceptional bases in the equivariant $K$-theory algebra. This paper is a continuation of arXiv:1901.02990v1. We describe the relation between solutions to the joint system of the $qDE$ and $qKZ$ equations and the topological-enumerative solution to the $qDE$ only, defined as a generating function of equivariant descendant Gromov-Witten invariants. The relation is in terms of the equivariant graded Chern character on the equivariant $K$-theory algebra, the equivariant Gamma class of the Projective Space, and the equivariant first Chern class of the tangent bundle of the Projective Space. We consider a Stokes basis, the associated exceptional basis in the equivariant $K$-theory algebra, and the associated Stokes matrix. We show that the Stokes matrix equals the Gram matrix of the equivariant Grothendieck-Euler-Poincar\'{e} pairing wrt to the basis, which is the left dual to the associated exceptional basis. We identify the Stokes bases in the Space of solutions with explicit full exceptional collections in the equivariant derived category of coherent sheaves on the Projective Space, where the elements of those exceptional collections are just line bundles on the Projective Space and exterior powers of the tangent bundle of the Projective Space. These statements are equivariant analogs of results of G. Cotti, B. Dubrovin, D. Guzzetti, and S. Galkin, V. Golyshev, H. Iritani.
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equivariant quantum differential equation stokes bases and k theory for a Projective Space
arXiv: Algebraic Geometry, 2019Co-Authors: V Tarasov, Alexander VarchenkoAbstract:We consider the equivariant quantum differential equation for the Projective Space $P^{n-1}$. We prove an equivariant gamma theorem for $P^{n-1}$, which describes the asymptotics of the differential equation at its regular singular point in terms of the equivariant characteristic gamma class of the tangent bundle of $P^{n-1}$. We describe the Stokes bases of the differential equation at its irregular singular point in terms of the exceptional bases of the equivariant K-theory algebra of $P^{n-1}$ and a suitable braid group action on the set of exceptional bases. Our results are an equivariant version of the well-know results of B. Dubrovin and D. Guzzetti.
Juan De Dios Perez - One of the best experts on this subject based on the ideXlab platform.
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lie derivatives of the shape operator of a real hypersurface in a complex Projective Space
Mediterranean Journal of Mathematics, 2021Co-Authors: Juan De Dios Perez, David PerezlopezAbstract:We consider real hypersurfaces M in complex Projective Space equipped with both the Levi-Civita and generalized Tanaka–Webster connections. Associated with the generalized Tanaka–Webster connection we can define a differential operator of first order. For any nonnull real number k and any symmetric tensor field of type (1,1) B on M, we can define a tensor field of type (1,2) on M, $$B^{(k)}_T$$ , related to Lie derivative and such a differential operator. We study symmetry and skew symmetry of the tensor $$A^{(k)}_T$$ associated with the shape operator A of M.
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commutativity of cho and structure jacobi operators of a real hypersurface in a complex Projective Space
Annali di Matematica Pura ed Applicata, 2015Co-Authors: Juan De Dios PerezAbstract:We consider real hypersurfaces $$M$$ in complex Projective Space equipped with both the Levi-Civita and generalized Tanaka-Webster connections. For any non-null constant $$k$$ and any vector field $$X$$ tangent to $$M$$ , we can define an operator on $$M$$ , $$F_X^{(k)}$$ , related to both connections. We study commutativity problems of these operators and the structure Jacobi operator of $$M$$ .
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real hypersurfaces in complex Projective Space with recurrent structure jacobi operator
Differential Geometry and Its Applications, 2008Co-Authors: Juan De Dios Perez, Florentino G SantosAbstract:Abstract We classify real hypersurfaces of complex Projective Space C P m , m ⩾ 3 , with D -recurrent structure Jacobi operator and apply this result to prove the non-existence of such hypersurfaces with recurrent structure Jacobi operator.
