Derived algebraic geometry, determinants of perfect complexes, and applications to obstruction theories for maps and complexes

Schürg, Timo; Toën, Bertrand; Vezzosi, Gabriele

doi:10.1515/crelle-2013-0037

Cited by 56 publications

(54 citation statements)

References 26 publications

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“…By considering a derived extension of the morphism f (2.9), the first two terms in (2.10) are the restriction of cotangent complexes of the corresponding derived schemes to the classical underlying schemes. They are obstruction theories (see [35,Sect. 1.2]), which fit into a commutative diagram…”

Section: Quintic Fibrationmentioning

confidence: 99%

Genus zero Gopakumar–Vafa type invariants for Calabi–Yau 4-folds

Cao

Maulik

Toda

2018

Advances in Mathematics

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As an analogy to Gopakumar-Vafa conjecture on CY 3-folds, Klemm-Pandharipande defined GV type invariants on CY 4-folds using GW theory and conjectured their integrality. In this paper, we define stable pair type invariants on CY 4-folds and use them to interpret these GV type invariants. Examples are computed for both compact and non-compact CY 4-folds to support our conjectures. 1 0.4. Verifications of the conjecture I: compact examples. We first prove our conjectures for some special compact Calabi-Yau 4-folds.Sextic 4-folds. Let X ⊆ P 5 be a degree six smooth hypersurface and [l] ∈ H 2 (X, Z) ∼ = H 2 (P 5 , Z) be the line class. We check our conjectures for β = [l] and 2[l]. Proposition 0.3. (Proposition 2.1, 2.2) Let X be a smooth sextic 4-fold and [l] ∈ H 2 (X, Z) be the line class. Then Conjecture 0.1 and 0.2 are true for β = [l] and 2[l].Elliptic fibrations. We consider a projective CY 4-fold X which admits an elliptic fibration π : X → P 3 ,given by a Weierstrass model (2.1). Let f be a general fiber of π and h be a hyperplane in P 3 , set B = π * h, E = ι(P 3 ) ∈ H 6 (X, Z), where ι is a section of π. Then we have

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Section: Quintic Fibrationmentioning

confidence: 99%

Genus zero Gopakumar–Vafa type invariants for Calabi–Yau 4-folds

Cao

Maulik

Toda

2018

Advances in Mathematics

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“…Let L X denote the cotangent complex on a space X. Let E • → L M be the reduced perfect obstruction theory on M , and let F • be the cone of the map We recall the construction of E • , see [MP13,STV11]. Consider the semi-regularity map…”

Section: The Virtual Classmentioning

confidence: 99%

Gromov–Witten invariants of the Hilbert schemes of points of a K3 surface

Oberdieck

2017

Geom. Topol.

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We study the enumerative geometry of rational curves on the Hilbert schemes of points of a K3 surface.Let S be a K3 surface and let Hilb d (S) be the Hilbert scheme of d points of S. In case of elliptically fibered K3 surfaces S → P 1 , we calculate genus 0 Gromov-Witten invariants of Hilb d (S), which count rational curves incident to two generic fibers of the induced Lagrangian fibration Hilb d (S) → P d . The generating series of these invariants is the Fourier expansion of a power of the Jacobi theta function times a modular form, hence of a Jacobi form.We also prove results for genus 0 Gromov-Witten invariants of Hilb d (S) for several other natural incidence conditions. In each case, the generating series is again a Jacobi form. For the proof we evaluate Gromov-Witten invariants of the Hilbert scheme of 2 points of P 1 × E, where E is an elliptic curve.Inspired by our results, we conjecture a formula for the quantum multiplication with divisor classes on Hilb d (S) with respect to primitive curve classes. The conjecture is presented in terms of natural operators acting on the Fock space of S. We prove the conjecture in the first non-trivial case Hilb 2 (S). As a corollary, we find that the full genus 0 Gromov-Witten theory of Hilb 2 (S) in primitive classes is governed by Jacobi forms.We present two applications. A conjecture relating genus 1 invariants of Hilb d (S) to the Igusa cusp form was proposed in joint work with R. Pandharipande in [OP14]. Our results prove the conjecture in case d = 2. Finally, we present a conjectural formula for the number of hyperelliptic curves on a K3 surface passing through 2 general points.

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“…We formulate the last equation of Section 6.9 as the following result. 42 The integration over W (n 2 , β 2 ) is well-defined since the insertions τ 0 (δ) yields a complete cycle as before.…”

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confidence: 95%

The Katz–klemm–vafa Conjecture for Surfaces

Pandharipande

Thomas

2016

Forum of Mathematics, Pi

101

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We prove the KKV conjecture expressing Gromov-Witten invariants of K 3 surfaces in terms of modular forms. Our results apply in every genus and for every curve class. The proof uses the Gromov-Witten/Pairs correspondence for K 3-fibered hypersurfaces of dimension 3 to reduce the KKV conjecture to statements about stable pairs on (thickenings of) K 3 surfaces. Using degeneration arguments and new multiple cover results for stable pairs, we reduce the KKV conjecture further to the known primitive cases. Our results yield a new proof of the full YauZaslow formula, establish new Gromov-Witten multiple cover formulas, and express the fiberwise Gromov-Witten partition functions of K 3-fibered 3-folds in terms of explicit modular forms.

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Derived algebraic geometry, determinants of perfect complexes, and applications to obstruction theories for maps and complexes

Cited by 56 publications

References 26 publications

Genus zero Gopakumar–Vafa type invariants for Calabi–Yau 4-folds

Genus zero Gopakumar–Vafa type invariants for Calabi–Yau 4-folds

Gromov–Witten invariants of the Hilbert schemes of points of a K3 surface

The Katz–klemm–vafa Conjecture for Surfaces

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