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Register a new userOn numerically trivial automorphisms of threefolds of general type
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In this paper, we prove that the group $mathrm{Aut}_mathbb{Q}(X)$ of numerically trivial automorphisms are uniformly bounded for smooth projective threefolds $X$ of general type which either satisfy $q(X)geq 3$ or have a Gorenstein minimal model. If $X$ is furthermore of maximal Albanese dimension, then $|mathrm{Aut}_mathbb{Q}(X)|leq 4$, and equality can be achieved by an unbounded family of threefolds previously constructed by the third author. Along the way we prove a Noether type inequality for log canonical pairs of general type with the coefficients of the boundary divisor from a given subset $mathcal{C}subset (0,1]$ such that $mathcal{C}cup{1}$ attains the minimum.
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Let $X$ be a complex nonsingular projective 3-fold of general type. We show that there are positive constants $c$, $c$ and $m_1$ such that $chi (omega _X)geq -cVol (X)$ and $P_m(X)geq cm^3Vol (X)$ for all $mgeq m_1$.
Let $V$ be a complex nonsingular projective 3-fold of general type. We prove $P_{12}(V)>0$ and $P_{24}(V)>1$ (which answers an open problem of J. Kollar and S. Mori). We also prove that the canonical volume has an universal lower bound $text{Vol}(V) geq 1/2660$ and that the pluri-canonical map $Phi_m$ is birational onto its image for all $mgeq 77$. As an application of our method, we prove Fletchers conjecture on weighted hyper-surface 3-folds with terminal quotient singularities. Another featured result is the optimal lower bound $text{Vol}(V)geq {1/420}$ among all those 3-folds $V$ with $chi({mathcal O}_V)leq 1$.
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.
In this paper, we investigate automorphisms of compact Kahler manifolds with different levels of topological triviality. In particular, we provide several examples of smooth complex projective surfaces X whose groups of $C^infty$-isotopically trivial automorphisms, resp. cohomologically trivial automorphisms, have a number of connected components which can be arbitrarily large.
We consider the Noether-Lefschetz problem for surfaces in Q-factorial normal 3-folds with rational singularities. We show the existence of components of the Noether-Lefschetz locus of maximal codimension, and that there are indeed infinitely many of them. Moreover, we show that their union is dense in the natural topology.
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