Semiregularity and obstructions of complete intersections

Iacono, Donatella; Manetti, Marco

doi:10.1016/j.aim.2012.11.011

Cited by 24 publications

(21 citation statements)

References 25 publications

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“…curves -has proved difficult; there is a hotchpotch of results in different cases [BF2,BL1,Blo,BuF,Don,IM,KL,Lee,Li,Liu,Man,MP,MPT,OP,Ran,Ros,Sch,STV]. Here we give quite a general construction using a mixture of some of these methods.…”

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confidence: 99%

Reduced classes and curve counting on surfaces I: theory

Kool¹,

Thomas²

2014

Alg. Geom.

130

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Abstract. We develop a theory of reduced Gromov-Witten and stable pair invariants of surfaces and their canonical bundles.We show that classical Severi degrees are special cases of these invariants. This proves a special case of the MNOP conjecture, and allows us to generalise the Göttsche conjecture to the non-ample case. In a sequel we prove this generalisation.We prove a remarkable property of the moduli space of stable pairs on a surface. It is the zero locus of a section of a bundle on a smooth compact ambient space, making calculation with the reduced virtual cycle possible. IntroductionMotivation. Fix a nonsingular projective surface S and a homology class β ∈ H 2 (S, Z). There are various ways of counting holomorphic curves in S in class β; in this paper we focus on Gromov-Witten invariants [Beh, LT] and stable pairs [PT1,Ott]. Since these are deformation invariant they must vanish in class β if there exists a deformation of S for which the Hodge type of β is not (1, 1). We can see the origin of this vanishing without deforming S as follows. For simplicity work in the simplest case of an embedded curve C ⊂ S with normal bundle N C = O C (C). As a Cartier divisor, C is the zero locus of a section s C of a line bundle L := O S (C), giving the exact sequenceThe resulting long exact sequence describes the relationship between first order deformations and obstructions H 0 (N C ), H 1 (N C ) of C ⊂ S, and the deformations and obstructionsThe resulting "semi-regularity map"S) takes obstructions to deforming C to the "cohomological part" of these obstructions. Roughly speaking, if we deform S, we get an associated obstruction in H 1 (N C ) to deforming C with it; its image in H 0,2 (S) is the (0, 2)-part of the cohomology class β ∈ H 2 (S) in the deformed complex structure. Thus it gives the obvious cohomological obstruction to deforming C: that β must remain of type (1, 1) in the deformed complex structure on S.In particular, when S is fixed, obstructions lie in the kernel ofMore generally, if we only consider deformations of S for which β remains (1, 1) then the same is true. And when h 0,2 (S) > 0 but H 2 (L) = 0, the existence of this trivial H 0,2 (S) piece of the obstruction sheaf guarantees that the virtual class vanishes.So it would be nice to restrict attention to surfaces and classes (S, β) inside the Noether-Lefschetz locus, 1 defining a new obstruction theory using only the kernel of the semi-regularity map.2 Checking that this kernel really defines an obstruction theory in the generality needed to define a virtual cycle -i.e. for deformations to all orders, over an arbitrary base, of possibly non-embedded 1 The locus of surfaces S for which β ∈ H 2 (S) has type (1, 1); for more details see [Voi, MP]. 2For embedded curves this means we use the obstruction space H 1 (L) to deforming sections of L. We have been able to remove the obstructions H 2 (O S ) to deforming L since the space of line bundles is smooth over the Noether-Lefschetz locus. Here we give quite a general construction using a mixture ...

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confidence: 99%

Reduced classes and curve counting on surfaces I: theory

Kool¹,

Thomas²

2014

Alg. Geom.

130

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“…[12], and the codomain of AJ 2 as the codomain of the semiregularity map, cf. [19]. A closer inspection reveals that the maps AJ 1 and AJ 2 are indeed the usual infinitesimal Abel-Jacobi and semiregularity maps, respectively.…”

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confidence: 86%

“…When the Hodge to de Rham spectral sequence degenerate at E 1 , e.g. if X is compact Kähler, then (9.1) and (9.2) are injective maps and then we recover the differential AJ 1 as the usual infinitesimal Abel-Jacobi map, see either [12, pag. 28] 3 or [27, Lemme 12.6] H 0 (Z; N X/Z ) → H p (X, Ω p−1 X ), and the obstruction map as the usual semiregularity map [2,19,23].…”

Section: The Infinitesimal Abel-jacobi Map Of a Submanifoldmentioning

confidence: 99%

“…A closer inspection reveals that the maps AJ 1 and AJ 2 are indeed the usual infinitesimal Abel-Jacobi and semiregularity maps, respectively. A pleasant consequence of the unobstructedness of the Grassmannian is then that the obstruction space of Hilb Z|X is contained in the kernel of AJ 2 , and one gets a proof of the classical principle "semiregularity kills obstructions" [2,19].…”

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confidence: 99%

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Formal Abel–Jacobi Maps

Fiorenza

Manetti

2018

International Mathematics Research Notices

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“…For the definition and the main properties of Der * A (R, R) and Hom * A (C, C) the reader may consult e.g., [15,Section 1]. Notice that the differential is the internal derivation δ = [d, −] and Der i A (R, R) = 0 for every i = 0: this implies that D i A (R, C) = Hom i R (C, C) for every i = 0, and D 0…”

Section: Consider Now the Map W H : Hommentioning

confidence: 99%