scholarly journals Noncommutative homological projective duality

2019 ◽  
Vol 350 ◽  
pp. 877-972 ◽  
Author(s):  
Alexander Perry
Keyword(s):  
Projective Duality

scholarly journals Homological projective duality for determinantal varieties

2016 ◽  
Vol 296 ◽  
pp. 181-209 ◽  
Author(s):  
Marcello Bernardara ◽  
Michele Bolognesi ◽  
Daniele Faenzi
Keyword(s):  
Projective Duality ◽  
Determinantal Varieties

scholarly journals ABEL–JACOBI MAPS FOR HYPERSURFACES AND NONCOMMUTATIVE CALABI–YAU'S

2010 ◽  
Vol 12 (03) ◽  
pp. 373-416 ◽  
Author(s):  
A. KUZNETSOV ◽  
L. MANIVEL ◽  
D. MARKUSHEVICH
Keyword(s):  
Moduli Space ◽  
Hochschild Homology ◽  
Volume Form ◽  
Projective Duality ◽  
Special Case

It is well known that the Fano scheme of lines on a cubic 4-fold is a symplectic variety. We generalize this fact by constructing a closed (2n - 4)-form on the Fano scheme of lines on a (2n - 2)-dimensional hypersurface Yn of degree n. We provide several definitions of this form — via the Abel–Jacobi map, via Hochschild homology, and via the linkage class — and compute it explicitly for n = 4. In the special case of a Pfaffian hypersurface Yn we show that the Fano scheme is birational to a certain moduli space of sheaves of a (2n - 4)-dimensional Calabi–Yau variety X arising naturally in the context of homological projective duality, and that the constructed form is induced by the holomorphic volume form on X. This remains true for a general non-Pfaffian hypersurface but the dual Calabi–Yau becomes noncommutative.

scholarly journals The homological projective dual of

2020 ◽  
Vol 156 (3) ◽  
pp. 476-525
Author(s):  
Jørgen Vold Rennemo
Keyword(s):  
Complete Intersection ◽  
Invariant Theory ◽  
Clifford Algebras ◽  
Derived Categories ◽  
Derived Category ◽  
Module Category ◽  
Geometric Invariant ◽  
Projective Duality ◽  
Clifford Module ◽  
Category Of Modules

We study the derived category of a complete intersection $X$ of bilinear divisors in the orbifold $\operatorname{Sym}^{2}\mathbb{P}(V)$. Our results are in the spirit of Kuznetsov’s theory of homological projective duality, and we describe a homological projective duality relation between $\operatorname{Sym}^{2}\mathbb{P}(V)$ and a category of modules over a sheaf of Clifford algebras on $\mathbb{P}(\operatorname{Sym}^{2}V^{\vee })$. The proof follows a recently developed strategy combining variation of geometric invariant theory (VGIT) stability and categories of global matrix factorisations. We begin by translating $D^{b}(X)$ into a derived category of factorisations on a Landau–Ginzburg (LG) model, and then apply VGIT to obtain a birational LG model. Finally, we interpret the derived factorisation category of the new LG model as a Clifford module category. In some cases we can compute this Clifford module category as the derived category of a variety. As a corollary we get a new proof of a result of Hosono and Takagi, which says that a certain pair of non-birational Calabi–Yau 3-folds have equivalent derived categories.

scholarly journals Derived categories of quintic del Pezzo fibrations

2021 ◽  
Vol 27 (1) ◽  
Author(s):  
Fei Xie
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Derived Categories ◽  
Derived Category ◽  
The Other ◽  
Del Pezzo Surfaces ◽  
Linear Section ◽  
Projective Duality ◽  
Del Pezzo ◽  
Space Approach

AbstractWe provide a semiorthogonal decomposition for the derived category of fibrations of quintic del Pezzo surfaces with rational Gorenstein singularities. There are three components, two of which are equivalent to the derived categories of the base and the remaining non-trivial component is equivalent to the derived category of a flat and finite of degree 5 scheme over the base. We introduce two methods for the construction of the decomposition. One is the moduli space approach following the work of Kuznetsov on the sextic del Pezzo fibrations and the components are given by the derived categories of fine relative moduli spaces. The other approach is that one can realize the fibration as a linear section of a Grassmannian bundle and apply homological projective duality.

scholarly journals Big bang maps and color charge force

2021 ◽  
Vol 12 (2) ◽  
pp. 164-172
Author(s):  
Gudrun Kalmbach H.E.
Keyword(s):  
Hilbert Space ◽  
Gravity Field ◽  
Fiber Bundle ◽  
Big Bang ◽  
Color Charge ◽  
Orthogonal Projections ◽  
3 Dimensional ◽  
Projective Duality ◽  
S Factor ◽  
Hopf Map

The Planck and other natural numbers are used for units of forces. They arise also as weights of Gleason operators, defined by 3-dimensional spin-like base triples GF and their weigths. The spin lengths are the spin GF weights for instance. The measuring GF operator triples arise by projective duality from 1-dimensional force vectors in projective to R5 extended Hilbert space H4. Color charges are set as a separate force, using a G-compass (figure 2). For the universes evolution after a big bang several maps are introduced, mostly belonging to the gravity field quantum rgb-graviton. It presents the neutral color charge of nucleons. Orthogonal projections of H4, also in spiralic and angular form, central or stereographic projective maps belong to them. They project also the S³ factor of the strong interation geometry S³xS5 down to the SU(2) geometry S³ of the Hopf map. Fiber bundle maps are added also to S5 with the same fiber S1 to the base space CP² for nucleons and atomic kernels. In octonian coordinates, listed by indices, 01234567, there are three projections from the energy space 123456 of SI to complex quaternionic 2x2-matrix presentations of spacetime 1234, of CP² as 3456 and of GR with mass and rgb-gravitons 1256. GR and CP² are projected into 1234 as the universes spacetime, observable as bubbles for atoms and matter 3456 and GR potentials and actions about and for mass carrying systems 1256.

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