on projective morphisms of varieties with nef - MOspace

65 Pages·2012·0.27 MB·English

ON PROJECTIVE MORPHISMS OF VARIETIES
WITH NEF ANTICANONICAL DIVISOR
A Dissertation presented to
the Faculty of the Graduate School
at the University of Missouri
In Partial Fulfillment
of the Requirements for the Degree
Doctor of Philosophy
by
LUNHAO AO
Dr. Qi Zhang, Dissertation Supervisor
JULY 2012
(cid:13)c Copyright by Lunhao Ao 2012
All Rights Reserved
The undersigned, appointed by the Dean of the Graduate School, have examined
the dissertation entitled:
ON PROJECTIVE MORPHISMS OF VARIETIES
WITH NEF ANTICANONICAL DIVISOR
presented by Lunhao Ao,
a candidate for the degree of Doctor of Philosophy and hereby certify that, in their
opinion, it is worthy of acceptance.
Dr. Qi Zhang
Dr. S. Dale Cutkosky
Dr. Zhenbo Qin
Dr. Jianguo Sun
ACKNOWLEDGEMENTS
Looking back, I am very grateful for what I have received from the mathematics
department of University of Missouri-Columbia throughout the past four years. All
these years of studies are full of excitement and challenge.
Firstly I wish to express my sincere, heartily, and deepest gratitude to my adviser
Professor Qi Zhang for his constant help, guidance, support and encouragement.
Secondly, I wish to express my sincere and heartily gratitude to Professor S. Dale
Cutkosky and Professor Zhenbo Qin for their teaching, guidance and support. I also
would like to thank my doctoral committee member Professor Jianguo Sun for his
interest and help.
I am grateful to Professor Dan Edidin, Professor Charles Li, Professor Hema
Srinivasan,ProfessorJanSegert,ProfessorShuguangWangfortheirhelpandsupport.
Finally, I am thankful for the financial support from the Department of mathe-
matics, University of Missouri-Columbia.
ii
TABLE OF CONTENTS
ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . ii
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
CHAPTER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Basic properties of ample and nef divisors and some statements . 3
2.1 Ample Divisors and Nef Divisors . . . . . . . . . . . . . . . . . . . . 3
2.2 Ample and Nef Vector bundles . . . . . . . . . . . . . . . . . . . . . . 7
3 Some results about mod p reduction . . . . . . . . . . . . . . . . . 9
3.1 Some early statement for mod p reduction . . . . . . . . . . . . . . . 9
3.2 The “bend-and-break” lemma . . . . . . . . . . . . . . . . . . . . . . 11
4 Some results about projective varieties with numerically effective
tangent bundles and nef anticanonical bundles . . . . . . . . . . . 17
4.1 Some background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.2 Some results about numerically effective vector bundles . . . . . . . . 19
4.3 Some general results . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.4 Projective surfaces with nef tangent bundles . . . . . . . . . . . . . . 31
4.5 Projective 3-folds with nef tangent bundles . . . . . . . . . . . . . . . 33
4.6 Structure of the Albanese map . . . . . . . . . . . . . . . . . . . . . . 34
4.7 Projective surfaces with nef anticanonical bundles . . . . . . . . . . . 37
5 Weak positivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.1 Introduction and the roots out of section . . . . . . . . . . . . . . . . 41
5.2 The main lemma of weak positivity . . . . . . . . . . . . . . . . . . . 44
iii
5.3 The modification of weak positivity . . . . . . . . . . . . . . . . . . . 47
6 A new method to prove image theorem of varieties with nef anti-
canonical divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
6.1 Background and introduction . . . . . . . . . . . . . . . . . . . . . . 49
6.2 The main part of this new proof . . . . . . . . . . . . . . . . . . . . . 51
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
VITA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
iv
ABSTRACT
We shall study and discuss some important properties of the projective varieties
with nef anticanonical bundles and nef tangent bundles. And we shall review some
backgroundandhistoryaboutthesubject. Thenweshalluseweakpositivitytheorem
to give a new proof of a theorem of Olivier Debarre without using mod p reduction,
which gives an affirmative answer to a question raised by Fujino and Gongyo.
v
Chapter 1
Introduction
In the research of algebraic geometry, specially in the classification theory of higher
dimensionalvarieties, itisinevitablethatwehavetodealwithsomespecialprojective
varieties such as varieties with nef anticanonical and nef tangent bundles. For many
years, mathematician have searched for the geometrical structure of these varieties.
By the intensive works of many mathematicians, today we have plenty of knowledge
of those fascinating varieties.
A compact Riemann surface always has a hermitian metric with constant cur-
vature. The negative sign corresponds to curves of general type (genus ≥ 2), and
the case of zero curvature corresponds to elliptic curves (genus 1), and the positive
curvature corresponds to rational curves (genus 0). In higher dimensional cases, this
situation becomes much more complicated. For example, Frankel’s famous conjec-
ture on the characterization of Pn(C) on the compact Kahler manifold of dimension
n with positive sectional curvature. Later, Hartshorne gave a stronger conjecture by
strengthening Frankel’s conjecture and claiming that Pn(k) is the only nonsingular
projective algebraic variety with ample tangent vector bundle defined over an alge-
braically closed field k. In the coming years, Mori solved Harthsorne’s conjecture by
using his amazing characteristic p methods.
1
As a natural extension, by combining algebraic and analytic tools, several authors
have investigated the geometrical and topological properties of varieties with nef
anticanonical and nef tangent bundles. It was well-known that the classification of
those varieties in low dimensions. For example, varieties with ample anticanonical
bundle are Fano varieties. In dimension 2, these are the famous surfaces: Del-Pezzo
surface. Weshalldiscussmoreaboutthesevarietiesindimensiontwoinlatersections.
We shall introduce the weak positivity theorem, and use it to give a new proof
of the image theorem of varieties with nef anticanonical divisor of Olivier Debarre,
which answers a question in Fujino and Gongyo’s paper.
2
Chapter 2
Basic properties of ample and nef
divisors and some statements
2.1 Ample Divisors and Nef Divisors
Definition 2.1. (Cartier divisors) A Cartier divisor on X is a global section of
the quotient sheaf M∗ /O∗ . We denote by Div(X) the group of all such, so that
X X
Div(X) = Γ(X, M∗ /O∗ )
X X
More precisely, then, a divisor D ∈ Div(X) is represented by data {(U ,f )}
i i
consisting of an open covering {U } of X together with elements f ∈ Γ(U ,M∗),
i i i X
having the property that on U = U ∩U one can write
ij i j
f = g f for some g ∈ Γ(U ,O∗ )). (2.1)
i ij j ij ij X
thefunctionf iscalledalocal equation forD atanypointx ∈ U .Twosuchcollections
i i
3