Variation of geometric invariant theory quotients and derived categories

Ballard, Matthew; Favero, David; Katzarkov, Ludmil

doi:10.48550/arxiv.1203.6643

Cited by 48 publications

(133 citation statements)

References 0 publications

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124

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“…Combining these with the previous paragraph, we conclude that the two positive real roots at r = 0 come together on the positive real axis for the first time at r = b. 2 Moreover, using from two paragraphs ago, it follows that the critical points above the two positive real critical values of w r (as elements of C n ) come together at the unique singular point p = (p 1 , . .…”

Section: 3supporting

confidence: 64%

“…As was observed in [7], there is a natural such functor arising from the VGIT machinery of Ballard-Favero-Katzarkov [2]. The next step, based on explicit computations with matrix factorizations, is the identification of the image under the above functor of the exceptional collection e a− with the left dual of the initial segment of the exceptional collection e a .…”

Section: Conjecture 122mentioning

confidence: 91%

“…Here we record a specialization of the construction in [7], which itself is a particular case of the general VGIT construction in [2].…”

Section: Vgit Embeddingmentioning

confidence: 99%

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On homological mirror symmetry for chain type polynomials

Polishchuk¹,

Varolgunes²

2021

Preprint

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We consider Takahashi's categorical interpretation of the Berglund-Hubsch mirror symmetry conjecture for invertible polynomials in the case of chain polynomials. Our strategy is based on a stronger claim that the relevant categories satisfy a recursion of directed A ∞ -categories, which may be of independent interest. We give a full proof of this claim on the B-side. On the A-side we give a detailed sketch of an argument, which falls short of a full proof because of certain missing foundational results in Fukaya-Seidel categories, most notably a generation statement.

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Section: 3supporting

confidence: 64%

Section: Conjecture 122mentioning

confidence: 91%

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On homological mirror symmetry for chain type polynomials

Polishchuk¹,

Varolgunes²

2021

Preprint

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“…Finally we show the schober conditions. The required equivalences between D(X − ) and D(X + ) are explained above (2). For the equivalences between the orthogonals, these are generated by single objects, and therefore by symmetry it suffice to prove the following lemma.…”

Section: B-side Constructionsmentioning

confidence: 97%

“…We use a general construction of Coates, Iritani, Jiang, and Segal [9] to realize the blowup p − as a variation of GIT. We then use standard technology which relates derived categories under variation of GIT [2,19] to obtain a semi-orthogonal decomposition. The semi-orthogonal decomposition for p + follows by an appropriate adaptation of this argument.…”

Section: B-side Constructionsmentioning

confidence: 99%

Mirror symmetry for perverse schobers from birational geometry

Donovan

Kuwagaki

2019

Preprint

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Perverse schobers are categorical analogs of perverse sheaves. Examples arise from varieties admitting flops, determined by diagrams of derived categories of coherent sheaves associated to the flop: in this paper we construct mirror partners to such schobers, determined by diagrams of Fukaya categories with stops, for examples in dimensions 2 and 3. Interpreting these schobers as supported on loci in mirror moduli spaces, we prove homological mirror symmetry equivalences between them. Our construction uses the coherent-constructible correspondence and a recent result of Ganatra-Pardon-Shende [17] to relate the schobers to certain categories of constructible sheaves. As an application, we obtain new mirror symmetry proofs for singular varieties associated to our examples, by evaluating the categorified cohomology operators of Bondal-Kapranov-Schechtman [6] on our mirror schobers.

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Categories of massless D-branes and del Pezzo surfaces

Addington

Aspinwall

2013

J. High Energ. Phys.

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In analogy with the physical concept of a massless D-brane, we define a notion of "Qmasslessness" for objects in the derived category. This is defined in terms of monodromy around singularities in the stringy Kähler moduli space and is relatively easy to study using "spherical functors". We consider several examples in which del Pezzo surfaces and other rational surfaces in Calabi-Yau threefolds are contracted. For precisely the del Pezzo surfaces that can be written as hypersurfaces in weighted P 3 , the category of Q-massless objects is a "fractional Calabi-Yau" category of graded matrix factorizations.arXiv:1305.5767v2 [hep-th]

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Variation of geometric invariant theory quotients and derived categories

Cited by 48 publications

References 0 publications

On homological mirror symmetry for chain type polynomials

On homological mirror symmetry for chain type polynomials

Mirror symmetry for perverse schobers from birational geometry

Categories of massless D-branes and del Pezzo surfaces

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