No Arabic abstract
Let $X$ be a smooth projective surface and $Delta$ is a normal crossing curve on $X$ such that $K_X+Delta$ is big. We show that the minimal possible volume of the pair $(X, Delta)$ is $frac{1}{143}$ if its (log) geometric genus is positive. Based on this, we establish a Noether type inequality for stable log surfaces, be they normal or non-normal. In the other direction, we show that, if the volume of $(X, Delta)$ is less than $frac{1}{143}$ then $X$ must be a rational surface and the connected components of $Delta$ are trees of smooth rational curves.
We introduce the notion of generalized MR log canonical surfaces and establish the minimal model theory for generalized MR log canonical surfaces in full generality.
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 $E$.
We prove that the canonical volume $K^3geq {1/30}$ for all projective 3-folds of general type with $chi(mathcal{O})leq 0$. This bound is sharp.
This paper shows that Mustata-Nakamuras conjecture holds for pairs consisting of a smooth surface and a multiideal with a real exponent over the base field of positive characteristic. As corollaries, we obtain the ascending chain condition of the minimal log discrepancies and of the log canonical thresholds for those pairs. We also obtain finiteness of the set of the minimal log discrepancies of those pairs for a fixed real exponent.
Projective varieties with ample cotangent bundle satisfy many notions of hyperbolicity, and one goal of this paper is to discuss generalizations to quasi-projective varieties. A major hurdle is that the naive generalization fails, i.e. the log cotangent bundle is never ample. Instead, we define a notion called almost ample which roughly asks that the log cotangent is as positive as possible. We show that all subvarieties of a quasi-projective variety with almost ample log cotangent bundle are of log general type. In addition, if one assumes globally generated then we obtain that such varieties contain finitely many integral points. In another direction, we show that the Lang-Vojta conjecture implies the number of stably integral points on curves of log general type, and surfaces of log general type with almost ample log cotangent sheaf are uniformly bounded.