This book collects the proceedings of a series of conferences dedicated to birational geometry of Fano varieties held in Moscow, Shanghai and Pohang
The conferences were focused on the following two related problems:
existence of KählerEinstein metrics on Fano varieties
degenerations of Fano varieties
on which two famous conjectures were recently proved. The first is the famous BorisovAlexeevBorisov Conjecture on the boundedness of Fano varieties, proved by Caucher Birkar (for which he was awarded the Fields medal in 2018), and the second one is the (arguably even more famous) TianYauDonaldson Conjecture on the existence of KählerEinstein metrics on (smooth) Fano varieties and K-stability, which was proved by Xiuxiong Chen, Sir Simon Donaldson and Song Sun. The solutions for these longstanding conjectures have opened new directions in birational and Kähler geometries. These research directions generated new interesting mathematical problems, attracting the attention of mathematicians worldwide.
These conferences brought together top researchers in both fields (birational geometry and complex geometry) to solve some of these problems and understand the relations between them. The result of this activity is collected in this book, which contains contributions by sixty nine mathematicians, who contributed forty three research and survey papers to this volume. Many of them were participants of the MoscowShanghaiPohang conferences, while the others helped to expand the research breadth of the volumethe diversity of their contributions reflects the vitality of modern Algebraic Geometry.
T. Abe, Classification of exceptional complements: elliptic curve case.-
E. Ballico, E. Gasparim, F. Rubilar, B. Suzuki, LAGRANGIAN SKELETA, COLLARS
AND DUALITY.- G. Belousov, CYLINDERS IN DEL PEZZO SURFACES OF DEGREE TWO.- M.
Benzerga, FINITENESS OF REAL STRUCTURES ON KLT CALABI-YAU REGULAR SMOOTH
PAIRS OF DIMENSION 2.- C. Birkar, ANTICANONICAL VOLUME OF FANO 4-FOLDS.- C.
Boyer Christina Tonnesen-Friedman, CONSTANT SCALAR CURVATURE SASAKI METRICS
AND PROJECTIVE BUNDLES.- G. Brown, J. Buczynski, A. Kasprzyk, TORIC SARKISOV
LINKS.- I. Burban, DU VAL SINGULARITIES.- I. Cheltsov, H. Suess,
K-POLYSTABILITY OF TWO SMOOTH FANO THREEFOLDS.- G. Codogni, Z. Patakfalvi, A
NOTE ON FAMILIES OF K-SEMISTABLE LOG-FANO PAIRS.- T. Delcroix, THE
YAU-TIAN-DONALDSON CONJECTURE FOR COHOMOGENEITY ONE MANIFOLDS.- A. Dubouloz,
FIBRATIONS BY AFFINE LINES ON RATIONAL AFFINE SURFACES WITH IRREDUCIBLE
BOUNDARIES.- K. Fujita, ON FANO THREEFOLDS OF DEGREE 22 AFTER CHELTSOV AND
SHRAMOV.- K. Fujita, Y. Liu, H. Suess,K. Zhang, Z. Zhuang, ON THE
CHELTSOV-RUBINSTEIN CONJECTURE.- S. Grishin, Ilya Karzhemanov, Ming-Chang
Kang, RATIONALITY OF QUOTIENTS BY FINITE HEISENBERG GROUPS.- Y. Hashimoto.-
J. Keller, QUOT-SCHEME LIMIT OF FUBINISTUDY METRICS AND ITS APPLICATIONS TO
BALANCED METRICS.- Z. Hu, EXISTENCE OF CANONICAL MODELS FOR KAWAMATA LOG
TERMINAL PAIRS.- Y. Imagi, GENERALIZED THOMASYAU UNIQUENESS THEOREMS.- K.
Jamieson, BIRATIONALLY RIGID COMPLETE INTERSECTIONS OF CODIMENSION 3.- D.
Jeong.- J. Park, SIMPLY CONNECTED SASAKI-EINSTEIN 5-MANIFOLDS: OLD AND NEW.-
C. Jiang, CHARACTERIZING Q-FANO THREEFOLDS WITH THE SMALLEST ANTI-CANONICAL
VOLUME.- L. Katzarkov, Kyoung-Seog Lee, J. Svoboda, A. Petkov,
INTERPRETATIONS OF SPECTRA.- Young-Hoon Kiem, Kyoung-Seog Lee, FANO VISITORS,
FANO DIMENSION AND FANO ORBIFOLDS.- In-kyun Kim, N. Viswanathan, J. Won, ON
SINGULAR DEL PEZZO HYPERSURFACES OF INDEX 3.- S. Kudryavtsev, Blow-ups of
three-dimensional toric singularities.- N. Kurnosov, E. Yasinsky,
AUTOMORPHISMS OF HYPERKAHLER MANIFOLDS AND GROUPS ACTING ON CAT(0) SPACES.-
A. Laface, R. Quezada, ON GENERALIZED BUCHI SURFACES.- Chi Li, K-STABILITY
AND FUJITA APPROXIMATION.- Y. Li, Zhenye Li, ON A CONJECTURE OF FULTON ON
ISOTROPIC GRASSMANNIANS.- Y. Maeda, Y. Odaka, FANO SHIMURA VARIETIES WITH
MOSTLY BRANCHED CUSP.- L. Makar-Limanov, ON LOCALLY NILPOTENT DERIVATIONS OF
DANIELEWSKI DOMAINS.- D. Markouchevitch, A. Moreau, ACTION OF THE
AUTOMORPHISM GROUP ON THE JACOBIAN OF KLEIN'S QUARTIC CURVE.- J.
Martinez-Garcia, C. Spotti, SOME OBSERVATIONS ON THE DIMENSION OF FANO
K-MODULI.- D. Witt Nystrom, OKOUNKOV BODIES AND THE KAHLER GEOMETRY OF
PROJECTIVE MANIFOLDS.- J. Park, SINGULARITIES OF PLURI-FUNDAMENTAL DIVISORS
ON GORENSTEIN FANO VARIETIES OF COINDEX.- J. Paulhus, A DATABASE OF GROUP
ACTIONS ON RIEMANN SURFACES.- A. Petracci, A 1-DIMENSIONAL COMPONENT OF
K-MODULI OF DEL PEZZO SURFACES.- T. De Piro, A NON-STANDARD BEZOUT THEOREM
FOR CURVES.- Y. Prokhorov, EMBEDDINGS OF THE SYMMETRIC GROUPS TO THE SPACE
CREMONA GROUP.- J. Ross, M. Toma, ON HODGE-RIEMANN COHOMOLOGY CLASSES.- Y.
Rubinstein, ON LARGE DEVIATION PRINCIPLES AND THE MONGEAMPERE EQUATION
(FOLLOWING BERMAN, HULTGREN).- T. Sano, ON BIRATIONAL BOUNDEDNESS OF SOME
CALABI-YAU HYPERSURFACES.- Y. Zarhin, ABELIAN VARIETIES, QUATERNION TRICK AND
ENDOMORPHISMS.
