Abstract:
The lectures will be devoted to the study of deformations
of pairs consisting of a smooth complex projective variety
X and a subscheme Z of X, say smooth or lci.
This deformation problem is obviously related to the
problem of deforming X keeping the fundamental class of Z a Hodge class.
This second problem is understood via the study of variation of Hodge structure of X.
We will explain the semi-regularity criterion due to Bloch and a generalization of it due to Ran.
Both results are now well understood using the so-called T1-lifting principle, which is fundamental in deformation theory.
Suggested reading:
1. Bloch, Spencer: Semi-regularity and de Rham cohomology. Invent. Math. 17 (1972), 51–66.
2. Ran, Ziv: Deformations of manifolds with torsion or negative canonical bundle. J. Algebraic Geom. 1 (1992), no. 2, 279–291.
3. Ran, Ziv: Hodge theory and the Hilbert scheme. J. Differential Geom. 37 (1993), no. 1, 191–198.
4. Kawamata, Yujiro: Unobstructed deformations. A remark on a paper of Z. Ran: "Deformations of manifolds with torsion or negative canonical bundle'' [J. Algebraic Geom. 1 (1992), no. 2, 279--291
5. Voisin, Claire: Sur la stabilité des sous-variétés lagrangiennes des variétés symplectiques holomorphes. in Complex projective geometry (Trieste, 1989/Bergen, 1989), 294–303, London Math. Soc. Lecture Note Ser., 179, Cambridge Univ. Press, Cambridge, 1992 (see also my http://www.math.jussieu.fr/~voisin/Articlesweb/Lagrangian1.pdf).
6. Voisin, Claire: Hodge Theory and complex algebraic geometry I and II, Cambridge University Press 2002-3
Volume I, chapters 9.1, 9.2, 10.1, 10.2
Volume II, chapter 5.3