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This version corrects a wrong proof of Proposition 6.3.2 and simplifies the exposition in Section 6.
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We introduce new invariants in equivariant birational geometry and study their relation to modular symbols and cohomology of arithmetic groups.
In this paper we study smooth projective varieties and polarized pairs with an action of a one dimensional complex torus. As a main tool, we define birational geometric counterparts of these actions, that, under certain assumptions, encode the information necessary to reconstruct them. In particular, we consider some cases of actions of low complexity -- measured in terms of two invariants of the action, called bandwidth and bordism rank -- and discuss how they are determined by well known birational transformations, namely Atiyah flips and Cremona transformations.
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 $chi({mathcal O}_V)leq 1$.
Let $V$ be a complex nonsingular projective 3-fold of general type. We prove $P_{12}(V):=text{dim} H^0(V, 12K_V)>0$ and $P_{m_0}(V)>1$ for some positive integer $m_0leq 24$. A direct consequence is the birationality of the pluricanonical map $varphi_m$ for all $mgeq 126$. Besides, the canonical volume $text{Vol}(V)$ has a universal lower bound $ u(3)geq frac{1}{63cdot 126^2}$.
Let $V$ be a complex nonsingular projective 3-fold of general type. We shall give a detailed classification up to baskets of singularities on a minimal model of $V$. We show that the $m$-canonical map of $V$ is birational for all $mgeq 73$ and that the canonical volume $text{Vol}(V)geq {1/2660}$. When $chi(mathcal{O}_V)leq 1$, our result is $text{Vol}(V)geq {1/420}$, which is optimal. Other effective results are also included in the paper.
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