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Algebraic Geometry Seminar
Seminar information archive ～05/02｜Next seminar｜Future seminars 05/03～
| Date, time & place | Friday 13:30 - 15:00 118Room #118 (Graduate School of Math. Sci. Bldg.) |
|---|---|
| Organizer(s) | GONGYO Yoshinori, KAWAKAMI Tatsuro, ENOKIZONO Makoto |
2016/06/20
16:30-18:00 Room #123 (Graduate School of Math. Sci. Bldg.)
De-Qi Zhang (National University of Singapore)
BUILDING BLOCKS OF POLARIZED ENDOMORPHISMS OF NORMAL PROJECTIVE VARIETIES (English)
http://www.math.nus.edu.sg/~matzdq/
De-Qi Zhang (National University of Singapore)
BUILDING BLOCKS OF POLARIZED ENDOMORPHISMS OF NORMAL PROJECTIVE VARIETIES (English)
[ Abstract ]
An endomorphism f of a normal projective variety X is polarized if f∗H ∼ qH for some ample Cartier divisor H and integer q > 1.
We first assert that a suitable maximal rationally connected fibration (MRC) can be made f-equivariant using a construction of N. Nakayama, that f descends to a polarized endomorphism of the base Y of this MRC and that this Y is a Q-abelian variety (quasi- ́etale quotient of an abelian variety). Next we show that we can run the minimal model program (MMP) f-equivariantly for mildly singular X and reach either a Q-abelian variety or a Fano variety of Picard number one.
As a consequence, the building blocks of polarized endomorphisms are those of Q- abelian varieties and those of Fano varieties of Picard number one.
Along the way, we show that f always descends to a polarized endomorphism of the Albanese variety Alb(X) of X, and that a power of f acts as a scalar on the Neron-Severi group of X (modulo torsion) when X is smooth and rationally connected.
Partial answers about X being of Calabi-Yau type or Fano type are also given with an extra primitivity assumption on f which seems necessary by an example.
This is a joint work with S. Meng.
[ Reference URL ]An endomorphism f of a normal projective variety X is polarized if f∗H ∼ qH for some ample Cartier divisor H and integer q > 1.
We first assert that a suitable maximal rationally connected fibration (MRC) can be made f-equivariant using a construction of N. Nakayama, that f descends to a polarized endomorphism of the base Y of this MRC and that this Y is a Q-abelian variety (quasi- ́etale quotient of an abelian variety). Next we show that we can run the minimal model program (MMP) f-equivariantly for mildly singular X and reach either a Q-abelian variety or a Fano variety of Picard number one.
As a consequence, the building blocks of polarized endomorphisms are those of Q- abelian varieties and those of Fano varieties of Picard number one.
Along the way, we show that f always descends to a polarized endomorphism of the Albanese variety Alb(X) of X, and that a power of f acts as a scalar on the Neron-Severi group of X (modulo torsion) when X is smooth and rationally connected.
Partial answers about X being of Calabi-Yau type or Fano type are also given with an extra primitivity assumption on f which seems necessary by an example.
This is a joint work with S. Meng.
http://www.math.nus.edu.sg/~matzdq/
