Metric method in Birational Geometry

In this series of notes, we summarize the metric methods that have become standard tools in birational geometry.

Part I. Singular Metrics, Currents, and Multiplier Ideals

Positivity in this theory is carried by singular Hermitian metrics and the closed positive currents they define. The order here is: define the objects, prove the vanishing theorem, then establish the two approximation results that let one replace an arbitrary metric by a manageable one.

Note-I.1 Singular Hermitian Metrics, Closed Positive Currents, and Lelong Numbers (in preparation)

Note-I.2 Multiplier Ideal Sheaves and Nadel Vanishing (in preparation)

Note-I.3 The Dictionary with Divisorial Singularities: klt, lc, and Log Canonical Thresholds (in preparation)

Note-I.4 Demailly Regularization and Approximation by Analytic Singularities (in preparation)

Note-I.5 Metrics with Minimal Singularities and Equisingular Approximations (in preparation)

Part II. The Ohsawa–Takegoshi Extension Theorem

The Ohsawa–Takegoshi theorem is one of the key technical tools used throughout this series of notes.

Note-II.1 The Ohsawa–Takegoshi Extension Theorem [upd 10.8]

Note-II.2 The Optimal Constant: Błocki, Guan–Zhou, and Berndtsson–Lempert (in preparation)

Part III. Bergman Kernel, Narasimhan–Simha, and Hodge Metrics

Part II produces sections. To produce a metric one needs a construction that is canonical in the family, and there are three standard ones. The first two are $L^2$ constructions, on the relative canonical bundle and then on its higher powers; the third comes from Hodge theory and is the one whose positivity was understood first.

Note-III.1 The Bergman Kernel Metric and Berndtsson’s Positivity Theorem (in preparation)

Note-III.2 The Narasimhan–Simha Metric on Relative Pluricanonical Bundles (in preparation)

Note-III.3 Hodge Metrics, Griffiths Positivity, and Variation of Hodge Structure (in preparation)

Note-III.4 Comparison of the Three Constructions and Their Curvature Properties (in preparation)

Part IV. Extension of Pluricanonical Forms and Invariance of Plurigenera

The first genuine use of Ohsawa–Takegoshi: sections of $mK_X$ on a fibre or a divisor extend to the ambient space.

Note-IV.1 Păun’s Analytic Proof of Invariance of Plurigenera [upd 5.11]

Note-IV.2 Berndtsson–Păun’s Extension Theorem for Metrics with Minimal Singularities (in preparation)

Note-IV.3 Demailly–Hacon–Păun’s Proof of the DLT Extension Theorem (in preparation)

Part V. Positivity of Direct Images

The same extension theorem, applied fibrewise, puts a singular Hermitian metric on $f_* \omega_{X/Y}^{\otimes m}$ with semipositive curvature. We fix the vocabulary for what that means first, then give the two constructions, then the application to the Iitaka conjecture.

Note-V.1 Positivity Notions for Torsion-Free Sheaves and Singular Hermitian Vector Bundles (in preparation)

Note-V.2 Păun–Takayama’s Construction of Singular Hermitian Metrics on Direct Images of Relative (Pluri)canonical Sheaves (in preparation)

Note-V.3 Hacon–Popa–Schnell’s Construction of Singular Hermitian Metrics on Direct Images of Relative (Pluri)canonical Sheaves [upd 10.24]

Note-V.4 The Iitaka Conjecture: Subadditivity of Kodaira Dimension (in preparation)

Part VI. Metric Methods in the Canonical Bundle Formula

The canonical bundle formula splits $K_X$ along a fibration into a discriminant part and a moduli part, and the moduli part is the hard one. Positivity of direct images from Part V gives it directly, without variation of Hodge structure.

Note-VI.1 Hacon–Păun’s Metric Method for the Canonical Bundle Formula (in preparation)

Note-VI.2 Positivity of the Moduli Part: the Metric Proof beside the Hodge-Theoretic One (in preparation)

Note-VI.3 The Semi-ampleness Conjecture for the Moduli Part (in preparation)

The algebraic development of the canonical bundle formula, subadjunction, and inversion of adjunction is in the adjunction notes.

Part VII. Effective Birationality and Boundedness

Now the estimates are used quantitatively. The constants in Parts I and II become explicit bounds on the multiple of $K_X$ that gives a birational map, and boundedness of the family follows from that bound together with a lower bound on volume.

Note-VII.1 Cutting Log Canonical Centers and the Theorem of Angehrn and Siu (in preparation)

Note-VII.2 Effective Base Point Freeness and Separation of Jets (in preparation)

Note-VII.3 Volumes, the DCC Conjecture, and the Statement of Birational Boundedness (in preparation)

Note-VII.4 Effective Birationality of Pluricanonical Maps: the Analytic Argument (in preparation)

Note-VII.5 Birational Boundedness for Varieties of General Type (in preparation)

Part VIII. Non-vanishing, Abundance, and Finite Generation

The deepest applications, and the ones still open. The difficulty is uniform across them: one needs a section of $mK_X$ with no divisor available to extend from, so the metric has to be produced first and the section extracted from it afterwards. We set up the numerical framework, then take non-vanishing, abundance, and finite generation in that order, since each is used in the next.

Note-VIII.1 Numerical Dimension, Nakayama’s $\sigma$-Decomposition, and the Shape of the Abundance Conjecture (in preparation)

Note-VIII.2 Siu and Păun’s Analytic Approach to Shokurov’s Non-vanishing (in preparation)

Note-VIII.3 Supercanonical Metrics and the Abundance Conjecture (in preparation)

Note-VIII.4 The Analytic Proof of Finite Generation of the Canonical Ring (in preparation)