Goal
The cohomology of a complex-algebraic variety comes with additional structure that is respected by morphisms and varies in a sense holomorphically on parameters. For a nonsingular projective variety this is the classical Hodge structure. In general it is what is called a mixed Hodge structure. We shall recall or define these notions, state and derive their main properties and pay particular attention to their behaviour in families.
Literature (to be expanded):
Topics in transcendental algebraic geometry, (Ph. A. Griffiths, ed.) Ann. of Math. Stud. 106, Princeton Univ. Press, Princeton, NJ, 1984.
P. Deligne: Théorie de Hodge, I. Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 1, pp. 425–430 (1971), Théorie de Hodge II. Inst. Hautes Études Sci. Publ. Math. 40 (1971), 5–57, Théorie de Hodge, III. Inst. Hautes Études Sci. Publ. Math. 44 (1974), 5–77.
P. Deligne: Poids dans la cohomologie des variétés algébriques, Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974), Vol. 1, pp. 79–85. Canad. Math. Congress, Montreal, Que., 1975.
C.A.M. Peters, J.H.M. Steenbrink: Mixed Hodge structures, Erg. der Math. und ihrer Grenzgeb. 3. Folge 52, Springer-Verlag, Berlin, 2008. xiv+470 pp.
Time and Place
Wednesdays 19:00-21:00, except on public holidays in Lecture Room 3 (2nd floor) of the Jin Chun Yuan West Building.
Program and covered material
March 1: Eduard Looijenga: Overview of (mainly) classical Hodge theory.
March 8: Bin Wang: Variation of Hodge structure: Kaehler manifolds and Hodge decomposition.
March 15: Zhiwei Zheng: Torelli maps, examples (mainly K3 surfaces).
March 22: Yi Wang: Chapter 3, Infinitesimal Variation of Hodge structure.
March 29: Chen Bingyi: Proof of the Hodge decomposition.
April 5: Chen Bingyi: Sketch of the proof of the Hard Lefschetz theorem.
Zi Yunpeng: Chapter 4, Asymptotic Behavior of a variation of Hodge structure.
April 12: Zi Yunpeng: continuation of talk about Chapter 4
April 19: Zheng Zhiwei: Limiting Mixed Hodge Structures: weight one case.
April 26: Zheng Zhiwei: finishes Chapter 5.
May 10: Xu Kai: Geometric case: Mixed Hodge Structure of smooth quasi-projective varieties
and of singular varieties.
May 17: Xu Kai: Clemens-Schmid exact sequence.
May 31: Eduard Looijenga: Polarized Hodge structured parametrized by bounded symmetric domains.
Xu Kai: Clemens-Schmid exact sequence.
Contact and in charge of the organization: 王彬