Classification of Boundary Lefschetz Fibrations over the Disc

Behrens, Stefan; Cavalcanti, Gil R.; Klaasse, Ralph L.

doi:10.1093/oso/9780198802020.003.0015

Cited by 3 publications

(4 citation statements)

References 18 publications

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“…The remaining diagram collapses to a single one‐handle after two further handle slides, which shows that

\tilde{X} = S^{1} \times S^{3} # {\bar{double-struckC P}}^{2}

. For more details, see .…”

Section: Examples and Applicationsmentioning

confidence: 91%

“…In this section we discuss some applications of the results obtained in this paper. For more examples we refer to , which in particular contains the classification theorem for boundary Lefschetz fibrations over

D^{2}

degenerating over its boundary (see Theorem ). We start with the following immediate consequence of the results in Section 6 and Section 7.…”

Section: Examples and Applicationsmentioning

confidence: 99%

“…Here relative minimality refers to the nonexistence of spheres with self‐intersection

- 1

in the fibers, nor of those vanishing cycles which are isotopic to the cycle specifying the boundary monodromy of the boundary Lefschetz fibration (see [, Definition 3.3]). This result can be combined with Theorem to equip the manifolds

S^{1} \times S^{3} # n \bar{C P^{2}}

# false(2 m + 1 false) C P^{2} # n \bar{C P^{2}}

and

# false(2 m + 1 false) S^{2} \times S^{2}

with a stable generalized complex structure whose type‐change locus has a single component.…”

Section: Introductionmentioning

confidence: 99%

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Fibrations and stable generalized complex structures

Cavalcanti

Klaasse

2018

Proc. London Math. Soc.

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A generalized complex structure is called stable if its defining anticanonical section vanishes transversally, on a codimension-two submanifold. Alternatively, it is a zero elliptic residue symplectic structure in the elliptic tangent bundle associated to this submanifold. We develop Gompf-Thurston symplectic techniques adapted to Lie algebroids, and use these to construct stable generalized complex structures out of log-symplectic structures. In particular we introduce the notion of a boundary Lefschetz fibration for this purpose and describe how they can be obtained from genus one Lefschetz fibrations over the disc. ContentsLet X be a 2n-dimensional manifold equipped with a closed three-form H ∈ Ω 3 cl (X). Recall that the double tangent bundle TX := T X ⊕ T * X is a Courant algebroid whose anchor is the projection p : TX → T X. It carries a natural pairing V + ξ,This takes (TX, H) to (TX, H + dB), leading to theŠevera class [H] ∈ H 3 (X; R) determining TX up to Courant isomorphism. The Courant automorphisms of TX are generated by the diffeomorphisms and closed B-field transformations.Definition 2.1. A generalized complex structure on (X, H) is a complex structure J on TX that is orthogonal with respect to ·, · , and whose +i-eigenbundle is involutive underThere is an alternative definition of a generalized complex structure using spinors. To state it, recall that sections v = V + ξ ∈ Γ(TX) of the double tangent bundle act on differential forms via Clifford multiplication, given byDefinition 2.2. A generalized complex structure on (X, H) is given by a complex line bundle K J ⊂ ∧ • T * C X pointwise generated by a differential form ρ = e B+iω ∧ Ω with Ω a decomposable complex k-form, satisfying Ω ∧ Ω ∧ ω n−k = 0, and such that dρ + H ∧ ρ = v · ρ for any local section ρ ∈ Γ(K J ) and some v ∈ Γ(TX).Both definitions are related using that K J = Ann(E J ) is the annihilator under the Clifford action of E J , the +i-eigenbundle of J . The bundle K J is called the canonical bundle of J . For later use, we further introduce the analogue of a Calabi-Yau manifold in generalized geometry. Denote by d H = d + H∧ the H-twisted de Rham differential.Example 2.4. The following provide examples of generalized complex structures on (X, 0).• Let ω be a symplectic structure on X. Then K Jω := e iω defines a generalized complex structure J ω . • Let J be a complex structure on X with canonical bundle K J = ∧ n,0 T * X. Then K JJ := K J defines a generalized complex structure J J . • Let P ∈ Γ(∧ 2,0 T X) a holomorphic Poisson structure with respect to a complex structure J. Then K JP,J := e P K J defines a generalized complex structure J P,J .The automorphisms J ω , J J and J P,J are given by, with π = Im(P ):We next introduce the type of a generalized complex structure J , which colloquially provides a measure for how many complex directions there are. The type is an integer-valued upper semicontinuous function on X whose parity is locally constant.Definition 2.5. Let J be a generalized complex structure on X. The type of J is a m...

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“…The remaining diagram collapses to a single one‐handle after two further handle slides, which shows that

\tilde{X} = S^{1} \times S^{3} # {\bar{double-struckC P}}^{2}

. For more details, see .…”

Section: Examples and Applicationsmentioning

confidence: 91%

D^{2}

degenerating over its boundary (see Theorem ). We start with the following immediate consequence of the results in Section 6 and Section 7.…”

Section: Examples and Applicationsmentioning

confidence: 99%

“…Here relative minimality refers to the nonexistence of spheres with self‐intersection

- 1

S^{1} \times S^{3} # n \bar{C P^{2}}

# false(2 m + 1 false) C P^{2} # n \bar{C P^{2}}

and

# false(2 m + 1 false) S^{2} \times S^{2}

with a stable generalized complex structure whose type‐change locus has a single component.…”

Section: Introductionmentioning

confidence: 99%

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Fibrations and stable generalized complex structures

Cavalcanti

Klaasse

2018

Proc. London Math. Soc.

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“…Note that elliptic symplectic structures (of zero elliptic residue) can exist on A |D| both in cases when X is and is not almost-complex (c.f. [4]), depending on the coorientability of D as measured by w 1 (N D) ∈ H 1 (X; Z 2 ). We see there is nontrivial dependence on the locus D in this case.…”

Section: 4mentioning

confidence: 99%

Obstructions for Symplectic Lie Algebroids

Klaasse

2018

Preprint

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Several generically-nondegenerate Poisson structures can be effectively studied as symplectic structures on naturally associated Lie algebroids. Relevant examples of this phenomenon include log-, elliptic, b k -, scattering and elliptic-log Poisson structures.In this paper we discuss topological obstructions to the existence of such Poisson structures (obtained through their symplectic Lie algebroids) of several different flavors, namely coming from cohomology, characteristic classes, and Seiberg-Witten theory.

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Obstructions for Symplectic Lie Algebroids

Klaasse¹

2020

SIGMA

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Several types of generically-nondegenerate Poisson structures can be effectively studied as symplectic structures on naturally associated Lie algebroids. Relevant examples of this phenomenon include log-, elliptic, b<sup>k</sup>-, scattering and elliptic-log Poisson structures. In this paper we discuss topological obstructions to the existence of such Poisson structures, obtained through the characteristic classes of their associated symplectic Lie algebroids. In particular we obtain the full obstructions for surfaces to carry such Poisson structures.

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Classification of Boundary Lefschetz Fibrations over the Disc

Cited by 3 publications

References 18 publications

Fibrations and stable generalized complex structures

Fibrations and stable generalized complex structures

Obstructions for Symplectic Lie Algebroids

Obstructions for Symplectic Lie Algebroids

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