[Paper Review] Pull-back of currents by meromorphic maps

Tuyen Trung Truong|arXiv (Cornell University)|Jul 8, 2011

Geometry and complex manifolds12 references3 citations

TL;DR

This paper introduces a well-defined pullback operator $f^{ atural}$ for $(p,p)$-currents of finite order on compact Kähler manifolds under dominant meromorphic maps, using regularization of DSH currents and weak limits via smooth approximations. The key contribution is a consistent, cohomologically compatible pullback that extends prior definitions and allows for the study of invariant currents, even when positivity is lost.

ABSTRACT

Let $X$ and $Y$ be compact Kähler manifolds, and let $f:X ightarrow Y$ be a dominant meromorphic map. Base upon a regularization theorem of Dinh and Sibony for DSH currents, we define a pullback operator $f^{\sharp}$ for currents of bidegrees $(p,p)$ of finite order on $Y$ (and thus for {\it any} current, since $Y$ is compact). This operator has good properties as may be expected. Our definition and results are compatible to those of various previous works of Meo, Russakovskii and Shiffman, Alessandrini and Bassanelli, Dinh and Sibony, and can be readily extended to the case of meromorphic correspondences. We give an example of a meromorphic map $f$ and two nonzero positive closed currents $T_1,T_2$ for which $f^{\sharp}(T_1)=-T_2$. We use Siu's decomposition to help further study on pulling back positive closed currents. Many applications on finding invariant currents are given.

Motivation & Objective

• To define a consistent pullback operator for $(p,p)$-currents of finite order on compact Kähler manifolds under dominant meromorphic maps.
• To ensure compatibility with existing pullback definitions for smooth, positive closed, and DSH currents.
• To extend the pullback to meromorphic correspondences and study invariance of currents under iteration.
• To investigate cases where the pullback of a positive current may not remain positive, particularly in higher bidegrees.
• To apply the theory to construct invariant currents using Siu’s decomposition and dynamical systems techniques.

Proposed method

• Use the duality identity $\int_X f^{ atural}(T) \wedge \alpha = \lim_{n\to\infty} \int_Y T \wedge K_n(f_*(\alpha))$ for smooth test forms $\alpha$, where $K_n$ are regularizations of $f_*(\alpha)$.
• Apply Dinh and Sibony’s regularization theorem to approximate $f_*(\alpha)$, which is a DSH current, by $C^s$ forms $K_n(f_*(\alpha))$.
• Define $f^{ atural}(T)$ via weak limit of $f^*(T_n)$ where $T_n$ are smooth approximations of $T$, ensuring consistency across sequences.
• Leverage compactness of positive currents and the $DSH$-norm to control convergence and define the pullback on general currents.
• Use Siu’s decomposition to analyze the pullback of positive closed currents by separating their singular and analytic parts.
• Apply the pullback to dynamical systems by constructing invariant currents via weak limits of averaged pullbacks of probability measures.

Experimental results

Research questions

• RQ1 Can a consistent pullback operator for $(p,p)$-currents be defined under dominant meromorphic maps, even when positivity is not preserved?
• RQ2 Does the proposed pullback operator agree with existing definitions for smooth, positive closed, and DSH currents?
• RQ3 Under what conditions is the pullback of a positive closed current under a meromorphic map still positive?
• RQ4 Can the pullback be extended to meromorphic correspondences and used to construct invariant currents?
• RQ5 Does the pullback satisfy the algebraic stability property $(f^n)^{ atural} = (f^{ atural})^n$ for general Kähler manifolds?

Key findings

• The pullback operator $f^{ atural}$ is well-defined for all $(p,p)$-currents of finite order on compact Kähler manifolds via regularization and weak limits.
• The operator is compatible with standard pullbacks on smooth forms and with prior constructions for positive closed and DSH currents.
• An example is constructed where $f^{ atural}(T_1) = -T_2$ for nonzero positive closed currents $T_1, T_2$, showing that positivity is not preserved in general.
• Using Siu’s decomposition, the paper provides a criterion to analyze the pullback of general positive closed currents by isolating their analytic and singular parts.
• The pullback preserves the cohomology class of currents, ensuring compatibility with cohomological dynamics.
• For the involution $J_X$, the pullback satisfies $(J_X^{ atural})^2 = \mathrm{id}$ on $(2,2)$-currents, and $J_X$ is $2$-algebraically stable.