Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn the Boundedness of Canonical Models
89
0
0.0
(
0
)
Authors
Junpeng Jiao
Ask ChatGPT about the research
No Arabic abstract
It is conjectured that the canonical models of varieties (not of general type) are bounded when the Iitaka volume is fixed. We confirm this conjecture when the general fibers of the corresponding Iitaka fibration are in a fixed bounded family of polarized Calabi-Yau pairs. As a consequence, we prove that in this case, the fibration is birationally bounded, and when it has terminal singularities, the corresponding minimal model is bounded in codimension 1.
rate research
Read More
We give a new simple proof of boundedness of the family of semistable sheaves with fixed numerical invariants on a fixed smooth projective variety. In characteristic zero our method gives a quick proof of Bogomolovs inequality for semistable sheaves on a smooth projective variety of any dimension $ge 2$ without using any restriction theorems.
Let $Gamma$ be a finite set, and $X i x$ a fixed klt germ. For any lc germ $(X i x,B:=sum_{i} b_iB_i)$ such that $b_iin Gamma$, Nakamuras conjecture, which is equivalent to the ACC conjecture for minimal log discrepancies for fixed germs, predicts that there always exists a prime divisor $E$ over $X i x$, such that $a(E,X,B)={rm{mld}}(X i x,B)$, and $a(E,X,0)$ is bounded from above. We extend Nakamuras conjecture to the setting that $X i x$ is not necessarily fixed and $Gamma$ satisfies the DCC, and show it holds for surfaces. We also find some sufficient conditions for the boundedness of $a(E,X,0)$ for any such $E$.
An important classification problem in Algebraic Geometry deals with pairs $(E,phi)$, consisting of a torsion free sheaf $E$ and a non-trivial homomorphism $phicolon (E^{otimes a})^{oplus b}lradet(E)^{otimes c}otimes L$ on a polarized complex projective manifold $(X,O_X(1))$, the input data $a$, $b$, $c$, $L$ as well as the Hilbert polynomial of $E$ being fixed. The solution to the classification problem consists of a family of moduli spaces ${cal M}^delta:={cal M}^{delta-rm ss}_{a/b/c/L/P}$ for the $delta$-semistable objects, where $deltainQ[x]$ can be any positive polynomial of degree at most $dim X-1$. In this note we show that there are only finitely many distinct moduli spaces among the ${cal M}^delta$ and that they sit in a chain of GIT-flips. This property has been known and proved by ad hoc arguments in several special cases. In our paper, we apply refined information on the instability flag to solve this problem. This strategy is inspired by the fundamental paper of Ramanan and Ramanathan on the instability flag.
We show the existence of $(epsilon,n)$-complements for $(epsilon,mathbb{R})$-complementary surface pairs when the coefficients of boundaries belong to a DCC set.
We prove that the log canonical ring of a projective log canonical pair in Kodaira dimension two is finitely generated.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
