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Register a new userA new geometric approach to problems in birational geometry
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A classical set of birational invariants of a variety are its spaces of pluricanonical forms and some of their canonically defined subspaces. Each of these vector spaces admits a typical metric structure which is also birationally invariant. These vector spaces so metrized will be referred to as the pseudonormed spaces of the original varieties. A fundamental question is the following: given two mildly singular projective varieties with some of the first varietys pseudonormed spaces being isometric to the corresponding ones of the second varietys, can one construct a birational map between them which induces these isometries? In this work a positive answer to this question is given for varieties of general type. This can be thought of as a theorem of Torelli type for birational equivalence.
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This work is dedicated to a new completely algebraic approach to Arakelov geometry, which doesnt require the variety under consideration to be generically smooth or projective. In order to construct such an approach we develop a theory of generalized rings and schemes, which include classical rings and schemes together with exotic objects such as F_1 (field with one element), Z_infty (real integers), T (tropical numbers) etc., thus providing a systematic way of studying such objects. This theory of generalized rings and schemes is developed up to construction of algebraic K-theory, intersection theory and Chern classes. Then existence of Arakelov models of algebraic varieties over Q is shown, and our general results are applied to such models.
This version corrects a wrong proof of Proposition 6.3.2 and simplifies the exposition in Section 6.
We introduce new invariants in equivariant birational geometry and study their relation to modular symbols and cohomology of arithmetic groups.
Bershadsky, Cecotti, Ooguri and Vafa constructed a real valued invariant for Calabi-Yau manifolds, which is now called the BCOV invariant. The BCOV invariant is conjecturally related to the Gromov-Witten theory via mirror symmetry. Based upon previous work of the second author, we prove the conjecture that birational Calabi-Yau manifolds have the same BCOV invariant. We also extend the construction of the BCOV invariant to Calabi-Yau varieties with Kawamata log terminal singularities, and prove its birational invariance for Calabi-Yau varieties with canonical singularities. We provide an interpretation of our construction using the theory of motivic integration.
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 $chi({mathcal O}_V)leq 1$.
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