Spherical and Exceptional Objects

2007 ◽  
Author(s):  
D. Huybrechts
Keyword(s):  
Projective Space ◽  
Spectral Sequence ◽  
Mirror Symmetry ◽  
Final Section ◽  
Derived Category ◽  
Braid Groups ◽  
Spherical Object ◽  
Geometric Context ◽  
Orthogonal Decompositions ◽  
Exceptional Sequences

Spherical objects — motivated by considerations in the context of mirror symmetry — are used to construct special autoequivalences. Their action on cohomology can be described precisely, considering more than one spherical object often leads to complicated (braid) groups acting on the derived category. The results related to Beilinson are almost classical. Section 3 of this chapter gives an account of the Beilinson spectral sequence and how it is used to deduce a complete description of the derived category of the projective space. This will use the language of exceptional sequences and semi-orthogonal decompositions encountered here. The final section gives a simplified account of the work of Horja, which extends the theory of spherical objects and their associated twists to a broader geometric context.

scholarly journals A Spectral Sequence for the Hochschild Cohomology of a Coconnective DGA

2013 ◽  
Vol 112 (2) ◽  
pp. 182 ◽  
Author(s):  
Shoham Shamir
Keyword(s):  
Spectral Sequence ◽  
Loop Space ◽  
Hochschild Cohomology ◽  
Derived Category ◽  
Orientable Manifold ◽  
Support Varieties ◽  
Mod P

A spectral sequence for the computation of the Hochschild cohomology of a coconnective dga over a field is presented. This spectral sequence has a similar flavour to the spectral sequence presented in [7] for the computation of the loop homology of a closed orientable manifold. Using this spectral sequence we identify a class of spaces for which the Hochschild cohomology of their mod-$p$ cochain algebra is Noetherian. This implies, among other things, that for such a space the derived category of mod-$p$ chains on its loop-space carries a theory of support varieties.

scholarly journals Homological mirror symmetry for higher-dimensional pairs of pants

2020 ◽  
Vol 156 (7) ◽  
pp. 1310-1347
Author(s):  
Yankı Lekili ◽  
Alexander Polishchuk
Keyword(s):  
Mirror Symmetry ◽  
Derived Category ◽  
Affine Variety ◽  
Fukaya Category ◽  
Homological Mirror Symmetry ◽  
Endomorphism Algebra ◽  
Generating Set ◽  
Fukaya Categories ◽  
Higher Dimensional ◽  
Abelian Covers

Using Auroux’s description of Fukaya categories of symmetric products of punctured surfaces, we compute the partially wrapped Fukaya category of the complement of $k+1$ generic hyperplanes in $\mathbb{CP}^{n}$, for $k\geqslant n$, with respect to certain stops in terms of the endomorphism algebra of a generating set of objects. The stops are chosen so that the resulting algebra is formal. In the case of the complement of $n+2$ generic hyperplanes in $\mathbb{C}P^{n}$ ($n$-dimensional pair of pants), we show that our partial wrapped Fukaya category is equivalent to a certain categorical resolution of the derived category of the singular affine variety $x_{1}x_{2}\ldots x_{n+1}=0$. By localizing, we deduce that the (fully) wrapped Fukaya category of the $n$-dimensional pair of pants is equivalent to the derived category of $x_{1}x_{2}\ldots x_{n+1}=0$. We also prove similar equivalences for finite abelian covers of the $n$-dimensional pair of pants.

Projective Geometry in the One-Dimensional Affine Group

1964 ◽  
Vol 16 ◽  
pp. 683-700 ◽  
Author(s):  
Hans Schwerdtfeger
Keyword(s):  
Projective Space ◽  
Projective Plane ◽  
Simple Group ◽  
Final Section ◽  
Theoretical Interpretation ◽  
Special Group ◽  
One Dimensional ◽  
The One ◽  
Straight Lines ◽  
Geometrical Investigation

The idea of considering the set of the elements of a group as a space, provided with a topology, measure, or metric, connected somehow with the group operation, has been used often in the work of E. Cartan and others. In the present paper we shall study a very special group whose space can be embedded naturally into a projective plane and where the straight lines have a very simple group-theoretical interpretation. It remains to be seen whether this geometrical embedding in a projective space can be extended to other classes of groups and whether the method could become an instrument of geometrical investigation, like co-ordinates or reflections. In the final section it is shown how a geometrical theorem may lead to relations within the group.

Where to Go from Here

2007 ◽  
Author(s):  
D. Huybrechts
Keyword(s):  
Mirror Symmetry ◽  
Derived Categories ◽  
Derived Category ◽  
Vector Spaces ◽  
Crepant Resolution ◽  
Hilbert Schemes ◽  
Mckay Correspondence ◽  
Homological Mirror Symmetry ◽  
Twisted Sheaves ◽  
A Finite Group

This chapter gives pointers for more advanced topics, which require prerequisites that are beyond standard introductions to algebraic geometry. The Mckay correspondence relates the equivariant-derived category of a variety endowed with the action of a finite group and the derived category of a crepant resolution of the quotient. This chapter gives the results from Bridgeland, King, and Reid for a special crepant resolution provided by Hilbert schemes and of Bezrukavnikov and Kaledin for symplectic vector spaces. A brief discussion of Kontsevich's homological mirror symmetry is included, as well as a discussion of stability conditions on triangulated categories. Twisted sheaves and their derived categories can be dealt with in a similar way, and some of the results in particular for K3 surfaces are presented.

scholarly journals Toric systems and mirror symmetry

2013 ◽  
Vol 149 (11) ◽  
pp. 1839-1855 ◽  
Author(s):  
Raf Bocklandt
Keyword(s):  
Mirror Symmetry ◽  
Line Bundles ◽  
Del Pezzo Surfaces ◽  
Toric Surface ◽  
Exceptional Sequence ◽  
Del Pezzo Surface ◽  
Rational Surfaces ◽  
Exceptional Sequences ◽  
Del Pezzo ◽  
Compositio Math

AbstractIn their paper [Exceptional sequences of invertible sheaves on rational surfaces, Compositio Math. 147 (2011), 1230–1280], Hille and Perling associate to every cyclic full strongly exceptional sequence of line bundles on a toric weak del Pezzo surface a toric system, which defines a new toric surface. We interpret this construction as an instance of mirror symmetry and extend it to a duality on the set of toric weak del Pezzo surfaces equipped with a cyclic full strongly exceptional sequence.

scholarly journals Fukaya categories of surfaces, spherical objects and mapping class groups

2021 ◽  
Vol 9 ◽  
Author(s):  
Denis Auroux ◽  
Ivan Smith
Keyword(s):  
Mirror Symmetry ◽  
Homology Class ◽  
Simple Closed Curve ◽  
Mapping Class Groups ◽  
Mapping Class ◽  
Continuous Group ◽  
Fukaya Category ◽  
Class Groups ◽  
Spherical Object ◽  
Symplectic Mapping

Abstract We prove that every spherical object in the derived Fukaya category of a closed surface of genus at least $2$ whose Chern character represents a nonzero Hochschild homology class is quasi-isomorphic to a simple closed curve equipped with a rank $1$ local system. (The homological hypothesis is necessary.) This largely answers a question of Haiden, Katzarkov and Kontsevich. It follows that there is a natural surjection from the autoequivalence group of the Fukaya category to the mapping class group. The proofs appeal to and illustrate numerous recent developments: quiver algebra models for wrapped categories, sheafifying the Fukaya category, equivariant Floer theory for finite and continuous group actions and homological mirror symmetry. An application to high-dimensional symplectic mapping class groups is included.

Calculation of Lin's Ext groups

1980 ◽  
Vol 87 (3) ◽  
pp. 459-469 ◽  
Author(s):  
W. H. Lin ◽  
D. M. Davis ◽  
M. E. Mahowald ◽  
J. F. Adams
Keyword(s):  
Projective Space ◽  
Spectral Sequence ◽  
Adams Spectral Sequence ◽  
Stable Homotopy ◽  
Real Projective Space ◽  
Original Proof ◽  
Ext Groups ◽  
Fourth Author

The first-named author has proved interesting results about the stable homotopy and cohomotopy of spaces related to real projective space RP∞; these are presented in an accompanying paper (6). His proof is by the Adams spectral sequence, and so depends on the calculation of certain Ext groups. The object of this paper is to prove the required result about Ext groups. The proof to be given is not Lin's original proof, which involved substantial calculation; it follows an idea of the second and third authors. The version to be given incorporates modifications suggested later by the fourth author.

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