The Noether inequality for algebraic threefolds (With an Appendix by J{a}nos Koll{a}r)

We establish the Noether inequality for projective $3$-folds. More precisely, we prove that the inequality $${rm vol}(X)geq tfrac{4}{3}p_g(X)-{tfrac{10}{3}}$$ holds for all projective $3$-folds $X$ of general type with either $p_g(X)leq 4$ or $p_g(X)geq 21$, where $p_g(X)$ is the geometric genus and ${rm vol}(X)$ is the canonical volume. This inequality is optimal due to known examples found by M. Kobayashi in 1992.

Varieties with vanishing holomorphic Euler characteristic II

We continue our study on smooth complex projective varieties $X$ of maximal Albanese dimension and of general type satisfying $chi(X, omega_X)=0$. We formulate a conjectural characterization of such varieties and prove this conjecture when the Albanese variety has only three simple factors.

The Noether inequality for Gorenstein minimal 3-folds

We prove the Conjecture of Catenese--Chen--Zhang: the inequality $K_X^3geq frac{4}{3}p_g(X)-frac{10}{3}$ holds for all projective Gorenstein minimal 3-folds $X$ of general type.

Explicit birational geometry of 3-folds and 4-folds of general type, III

Nonsingular projective 3-folds $V$ of general type can be naturally classified into 18 families according to the {it pluricanonical section index} $delta(V):=text{min}{m|P_mgeq 2}$ since $1leq delta(V)leq 18$ due to our previous series (I, II). Based on our further classification to 3-folds with $delta(V)geq 13$ and an intensive geometrical investigation to those with $delta(V)leq 12$, we prove that $text{Vol}(V) geq frac{1}{1680}$ and that the pluricanonical map $Phi_{m}$ is birational for all $m geq 61$, which greatly improves known results. An optimal birationality of $Phi_m$ for the case $delta(V)=2$ is obtained. As an effective application, we study projective 4-folds of general type with $p_ggeq 2$ in the last section.

Explicit birational geometry of 3-folds of general type, II

Let $V$ be a complex nonsingular projective 3-fold of general type. We shall give a detailed classification up to baskets of singularities on a minimal model of $V$. We show that the $m$-canonical map of $V$ is birational for all $mgeq 73$ and that the canonical volume $text{Vol}(V)geq {1/2660}$. When $chi(mathcal{O}_V)leq 1$, our result is $text{Vol}(V)geq {1/420}$, which is optimal. Other effective results are also included in the paper.

Explicit birational geometry of 3-folds of general type, I

Let $V$ be a complex nonsingular projective 3-fold of general type. We prove $P_{12}(V):=text{dim} H^0(V, 12K_V)>0$ and $P_{m_0}(V)>1$ for some positive integer $m_0leq 24$. A direct consequence is the birationality of the pluricanonical map $varphi_m$ for all $mgeq 126$. Besides, the canonical volume $text{Vol}(V)$ has a universal lower bound $ u(3)geq frac{1}{63cdot 126^2}$.

Factoring 3-fold flips and divisorial contractions to curves

We show that 3-fold terminal flips and divisorial contractions to a curve may be factored by a sequence of weighted blow-ups, flops, blow-downs to a locally complete intersection curve in a smooth 3-fold or divisorial contractions to a point.

The canonical volume of threefolds of general type with $chi<1$

We prove that the canonical volume $K^3geq {1/30}$ for all projective 3-folds of general type with $chi(mathcal{O})leq 0$. This bound is sharp.

On Uenos Conjecture K

We show that if $X$ is a smooth complex projective variety with Kodaira dimension $0$ then the Kodaira dimension of a general fiber of its Albanese map is at most $h^0(Omega ^1 _X)$.

On the geography of threefolds of general type

Let $X$ be a complex nonsingular projective 3-fold of general type. We show that there are positive constants $c$, $c$ and $m_1$ such that $chi (omega _X)geq -cVol (X)$ and $P_m(X)geq cm^3Vol (X)$ for all $mgeq m_1$.