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Meng Chen

Publications and source records attributed to Meng Chen.

162 records · Page 9Linked to original sources
arXiv2007

Characterization of the 4-canonical birationality of algebraic threefolds

In this article we present a 3-dimensional analogue of a well-known theorem of E. Bombieri (in 1973) which characterizes the bi-canonical birationality of surfaces of general type. Let $X$ be a projective minimal 3-fold of general type with $\mathbb{Q}$-factorial terminal singularities and the geometric genus $p_g(X)\ge 5$. We show that the 4-canonical map $ϕ_4$ is {\it not} birational onto its image if and only if $X$ is birationally fibred by a family $\mathscr{C}$ of irreducible curves of geometric genus 2 with $K_X\cdot C_0=1$ where $C_0$ is a general irreducible member in $\mathscr{C}$.

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arXiv2006

On the geography of Gorenstein minimal 3-folds of general type

Let $X$ be a minimal projective Gorenstein 3-fold of general type. We give two applications of an inequality between $χ(ω_X)$ and $p_g(X)$: 1) Assume that the canonical map $Φ_{|K_X|}$ is of fiber type. Let $F$ be a smooth model of a generic irreducible component in the general fiber of $Φ_{|K_X|}$. Then the birational invariants of $F$ are bounded from above. 2) If $X$ is nonsingular, then $c_1^3\leq {1/27} c_1c_2+{10/3}$ where $c_1$, $c_2$ are Chern invariants of $X$.

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arXiv2006

Complex multiplication, rationality and mirror symmetry for abelian varieties

We show that complex multiplication on abelian varieties is equivalent to the existence of a constant rational Kähler metric. We give a sufficient condition for a mirror of an abelian variety of CM-type to be of CM-type as well. We also study the relationship between complex multiplication and rationality of a toroidal lattice vertex algebra.

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arXiv2005

A sharp lower bound for the canonical volume of 3-folds of general type

Let V be a smooth projective 3-fold of general type. Denote by $K^3$, a rational number, the self-intersection of the canonical sheaf of any minimal model of V. One defines $K^3$ as the canonical volume of $V$. Assume $p_g\ge 2$. We show that $K^3\ge {1/3}$, which is a sharp lower bound. Then we classify those V with small $K^3$ up to explicit tructure. We also give some new examples with $p_g=2$ which have maximal canonical stability index. Finally we give an application to certain algebraic 4-folds.

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arXiv2005

The Noether inequality for smooth minimal 3-folds

Let X be a smooth projective minimal 3-fold of general type. We prove the sharp inequality K^3_X >= (2 /3)(2p_g(X) - 5), an analogue of the classical Noether inequality for algebraic surfaces of general type

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arXiv2004

Weak boundedness theorems for canonically fibered Gorenstein minimal threefolds

Let $X$ be a Gorenstein minimal projective 3-fold with at worst locally factorial terminal singularities. Suppose the canonical map is of fiber type. Denote by $F$ a smooth model of a generic irreducible component in fibers of the canonical map of $X$ and so $F$ is a smooth curve or a smooth surface. The main result of the paper is that there is a computable constant $K$ (independent of $X$) such that $g(F)\leq 647$ or $p_g(F)\leq 38$ whenever $p_g(X)\geq K$. The method heavily relies on both a Noether type of inequality and a Miyaoka-Yau inequality and that is the reason we only treat a Gorenstein object here. It is open whether the degree of the canonical map is universally bounded when the canonical map is generically finite.

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arXiv2003

On the Q-divisor method and its application

For smooth projective 3-folds of general type, we prove that the relative canonical stability $μ_s(3)\leq 8$. This is induced from our improved result of Kollár: the m-canonical map of a smooth projective 3-fold of general type is birational whenever $m\geq 5k+6$, provided $P_k(X)\geq 2$. The Q-divisor method is intensively developed to prove our results.

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arXiv2003

Inequalities of Noether type for 3-folds of general type

If $X$ is a smooth complex projective 3-fold with ample canonical divisor $K$, then the inequality $K^3\ge {2/3}(2p_g-7)$ holds, where $p_g$ denotes the geometric genus. This inequality is nearly sharp. We also give similar, but more complicated, inequalities for general minimal 3-folds of general type. (A noether type of inequality lies, by all means, on the other side of the Miyaoka-Yau inequality from the geographical point of view.)

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arXiv2003

Canonical stability of 3-folds of general type with $p_g\geq 3$

We study the canonical stability of a smooth projective 3-fold $V$ of general type. We prove that (1) $|5K_V|$ gives a birational map onto its image provided the geometric genus $p_g\geq 4$; (2) $|6K_V|$ gives a birational map provided $p_g=3$. Known examples show that both are optimal. This fact can be viewed as parallel to surface case, though people know very little on 3-folds of general type with $p_g\leq 1$.

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arXiv2001

On canonically derived families of surfaces of general type over curves

Suppose X is a smooth projective 3-fold of general type and |mK_X| is composed of a pencil of surfaces with m>1. This pencil naturally induces a fibration f:X->C onto a smooth curve C after the Stein-factorization, which is the main objects of this article. Based on Koll'ar's earlier works, we improve on it and try to understand the family in terms of discrete birational invariants of the total space as well as those of the general fiber and the base curve. The aim of this note is to build a little bit basic facts.

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arXiv2000

Canonical stability in terms of singularity index for algebraic threefolds

Let X be a projective 3-fold with at most Q-factorial terminal singularities on which K_X is nef and big. Suppose the canonical index r(X)>1. For any positive integer m, it is interesting to consider the base point freeness and birationality of the divisor mK_X. For example, we know the following results: (1) the system |5rK_X| is base point free (Ein-Lazarsfeld-Lee); (2) |mK_X| gives a birational map for all m>4r+2 (M. Hanamura). This article aims to present a better result in direction (2). As far as our method can tell here, |mK_X| gives a birational map for all m>2r+5. (Q-divisor method + patient calculation)

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arXiv1999

The relative pluricanonical stability for 3-folds of general type

This paper aims to improve a theorem of Janos Kollar. For a given Complex projective threefold X of general type, suppose the plurigenus p_k(X)\ge 2, Kollar proved that the (11k+5)-canonical map is birational. Here we show that either the (7k+3)-canonical map or the (7k+5)-canonical map is birational and that the m-canonical map is stably birational for m\ge 13k+6. If P_k(X)\ge 3, then the m-canonical map is stably birational for m\ge 10k+8. In particular, the 12-canonical map is birational when p_g(X)\ge 2 and the 11-canonical map is birational when p_g(X)\ge 3.

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arXiv1999

Kawamata-Viehweg vanishing and quint-canonical map of a Complex threefold

This paper proves that the 5-canonical map of a smooth minimal 3-fold is birational when the geometric genus is bigger than 2. A combination of the results in this paper and that of Ein-Lazarsfeld-Lee, the possible exceptional cases are very limited. Unfortunately, no counter examples have been found up to now.

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