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Register a new userFujita-type freeness for quasi-log canonical threefolds
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Authors
Haidong Liu
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In this paper, we show that Fujitas basepoint-freeness conjecture for projective quasi-log canonical singularities holds true in dimension three. Immediately, we prove Fujita-type basepoint-freeness for projective semi-log canonical threefolds.
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We establish a relative spannedness for log canonical pairs, which is a generalization of the basepoint-freeness for varieties with log-terminal singularities by Andreatta--Wisniewski. Moreover, we establish a generalization for quasi-log canonical pairs.
We establish a kind of subadjunction formula for quasi-log canonical pairs. As an application, we prove that a connected projective quasi-log canonical pair whose quasi-log canonical class is anti-ample is simply connected and rationally chain connected. We also supplement the cone theorem for quasi-log canonical pairs. More precisely, we prove that every negative extremal ray is spanned by a rational curve. Finally, we treat the notion of Mori hyperbolicity for quasi-log canonical pairs.
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.
Using inversion of adjunction, we deduce from Nadels theorem a vanishing property for ideals sheaves on projective varieties, a special case of which recovers a result due to Bertram--Ein--Lazarsfeld. This enables us to generalize to a large class of projective schemes certain bounds on Castelnuovo--Mumford regularity previously obtained by Bertram--Ein--Lazarsfeld in the smooth case and by Chardin--Ulrich for locally complete intersection varieties with rational singularities. Our results are tested on several examples.
Let T_n denote the set of log canonical thresholds of pairs (X,Y), with X a nonsingular variety of dimension n, and Y a nonempty closed subscheme of X. Using non-standard methods, we show that every limit of a decreasing sequence in T_n lies in T_{n-1}, proving in this setting a conjecture of Koll{a}r. We also show that T_n is a closed subset in the set of real numbers; in particular, every limit of log canonical thresholds on smooth varieties of fixed dimension is a rational number. As a consequence of this property, we see that in order to check Shokurovs ACC Conjecture for all T_n, it is enough to show that 1 is not a point of accumulation from below of any T_n. In a different direction, we interpret the ACC Conjecture as a semi-continuity property for log canonical thresholds of formal power series.
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