Local topological obstruction for divisors
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Local topological obstruction for divisors Indranil Biswas1 · Ananyo Dan2 Received: 5 November 2018 / Accepted: 3 November 2020 © Universidad Complutense de Madrid 2020
Abstract Given a smooth, projective variety X and an effective divisor D ⊆ X , it is well-known that the (topological) obstruction to the deformation of the fundamental class of D as a Hodge class, lies in H 2 (O X ). In this article, we replace H 2 (O X ) by H D2 (O X ) and give an analogous topological obstruction theory. We compare the resulting local topological obstruction theory with the geometric obstruction theory (i.e., the obstruction to the deformation of D as an effective Cartier divisor of a first order infinitesimal deformations of X ). We apply this to study the jumping locus of families of linear systems and the Noether–Lefschetz locus. Finally, we give examples of first order deformations X t of X for which the cohomology class [D] deforms as a Hodge class but D does not lift as an effective Cartier divisor of X t . Keywords Obstruction theories · Hodge locus · Semi-regularity map · Deformation of linear systems · Noether–Lefschetz locus Mathematics Subject Classification 14B10 · 14B15 · 14C30 · 14C20 · 14C25 · 14D07
1 Introduction The base field k is always assumed to be algebraically closed of characteristic zero. Consider a family of smooth, projective varieties π : X −→ B
B
Ananyo Dan [email protected] Indranil Biswas [email protected]
1
School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India
2
BCAM – Basque Centre for Applied Mathematics, Alameda de Mazarredo 14, 48009 Bilbao, Spain
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I. Biswas, A. Dan
parameterized by a complex manifold B. Fix a closed point o ∈ B and denote by X := π −1 (o) the fiber over o. Let D ⊆ X be an effective divisor. Recall that, by Hodge decomposition, H 2 (X , C) decomposes as a direct sum of sub-vector spaces H i,2−i (X ) for 0 ≤ i ≤ 2. The integral elements of H 1,1 (X ), meaning the elements of H 1,1 (X )∩(H 2 (X , Z)/(torsion)), are known as Hodge classes. It is well-known that the fundamental class of a divisor, given by the first Chern class of the associated line bundle, is a Hodge class, which is proved using the exponential short exact sequence. There is an obvious “topological” obstruction for an effective divisor D ⊆ X to deform along with the variety X : the fundamental class [D] of the divisor D is needed to deform as a Hodge class. Using the Lefschetz (1, 1)-theorem, this topological obstruction is precisely the obstruction to the deformation of the invertible sheaf O X (D). Using the exponential short exact sequence, one can then check that this obstruction actually lies in H 2 (X , O X ). We will say that the fundamental class of D deforms as a Hodge class along t ∈ To B if the invertible sheaf O X (D) deforms to an invertible sheaf on the first order infinitesimal deformation X t of X , in the direction of t. If the obstruction vanishes, we may further ask whether the divisor D deforms as an effective Cartier divisor
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