2015年3期 (共 篇) 引用文章 全选
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A fundamental result in the theory of minimal rational curves on projective manifolds is Cartan- Fubini extension theorem proved by Hwang and Mok, which describes the extensibility of biholomorphisms between connected open subsets of two Fano manifolds of Picard number 1 which preserve varieties of minimal rational tangents (VMRT), under a mild geometric assumption on the second fundamental forms of VMRT's. Hong and Mok have developed Cartan-Fubini extension for non-equidimensional holomorphic immersions from a connected open subset of a Fano manifold of Picard number 1 into a uniruled projective manifold, under the assumptions that the map sends VMRT's onto linear sections of VMRT's and it satisfies a mild geometric condition formulated in terms of second fundamental forms on VMRT's. In the current paper, we give a generalization of Hong and Mok's result, under the same condition on second fundamental forms, assuming only that the holomorphic immersions send VMRT's to VMRT's. Our argument is different from Hong and Mok's and is based on the study of natural foliations on the total family of VMRT's. This gives a substantially simpler proof than Hong and Mok's argument.
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Let X ⊄ PNC be an n-dimensional nondegenerate smooth projective variety containing an mdimensional subvariety Y. Assume that either m > n/2 and X is a complete intersection or that m > N/2. We show deg(X)|deg(Y) and codim Y ≥ codimPNX, where is the linear span of Y. These bounds are sharp. As an application, we classify smooth projective n-dimensional quadratic varieties swept out by m ≥ [n/2] + 1 dimensional quadrics passing through one point.
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We show that every connected commutative algebraic group over an algebraically closed field of characteristic 0 is the Picard variety of some projective variety having only finitely many non-normal points. In contrast, no Witt group of dimension at least 3 over a perfect field of prime characteristic is isogenous to a Picard variety obtained by this construction.
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Let Md be the moduli space of stable sheaves on P2 with Hilbert polynomial dm+1. In this paper, we determine the effective and the nef cone of the space Md by natural geometric divisors. Main idea is to use the wall-crossing on the space of Bridgeland stability conditions and to compute the intersection numbers of divisors with curves by using the Grothendieck-Riemann-Roch theorem. We also present the stable base locus decomposition of the space M6. As a byproduct, we obtain the Betti numbers of the moduli spaces, which confirm the prediction in physics.
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We classify normal supersingular K3 surfaces Y with total Milnor number 20 in characteristic p, where p is an odd prime that does not divide the discriminant of the Dynkin type of the rational double points on Y. This paper appeared in preprint form in the home page of the first author in the year 2005.
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Let S be a complete intersection of a smooth quadric 3-fold Q and a hypersurface of degree d in P4. We analyze GIT stability of S with respect to the natural G = SO(5, C)-action. We prove that if d > 4 and S has at worst semi-log canonical singularities then S is G-stable. Also, we prove that if d > 3 and S has at worst semi-log canonical singularities then S is G-semistable.
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Let R be a Noetherian unique factorization domain such that 2 and 3 are units, and let A = R[α] be a quartic extension over R by adding a root α of an irreducible quartic polynomial p(z) = z4 + az2 + bz + c over R. We will compute explicitly the integral closure of A in its fraction field, which is based on a proper factorization of the coefficients and the algebraic invariants of p(z). In fact, we get the factorization by resolving the singularities of a plane curve defined by z4+a(x)z2+b(x)z+c(x) = 0. The integral closure is expressed as a syzygy module and the syzygy equations are given explicitly. We compute also the ramifications of the integral closure over R.
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We consider base spaces of Lagrangian fibrations from singular symplectic varieties. After defining cohomologically irreducible symplectic varieties, we construct an example of Lagrangian fibration whose base space is isomorphic to a quotient of the projective space. We also prove that the base space of Lagrangian fibration from a cohomologically symplectic variety is isomorphic to the projective space provided that the base space is smooth.
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Strongly pseudoconvex CR manifolds are boundaries of Stein varieties with isolated normal singularities. We introduce a series of new invariant plurigenera δm,m ∈ Z+ for a strongly pseudoconvex CR manifold. The main purpose of this paper is to present the following result: Let X1 and X2 be two compact strongly pseudoconvex embeddable CR manifolds of dimension 2n-1 > 3. If there is a non-constant CR morphism from X1 to X2, then δm(X2) 6 δm(X1) where δm(Xi) is the plurigeneus of Xi (see Definition 2.4).
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We give the classification of globally generated vector bundles of rank 2 on a smooth quadric surface with c1 6 (2, 2) in terms of the indices of the bundles, and extend the result to arbitrary higher rank case. We also investigate their indecomposability and give the sufficient and necessary condition on numeric data of vector bundles for indecomposability.
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Laszlo and Olsson constructed Grothendieck's six operations for constructible complexes on Artin stacks in étale cohomology under an assumption of finite cohomological dimension, with base change established on the level of sheaves. We give a more direct construction of the six operations for complexes on Deligne- Mumford stacks without the finiteness assumption and establish base change theorems in derived categories. One key tool in our construction is the theory of gluing finitely many pseudofunctors developed by Zheng (2014). As an application, we prove a Lefschetz-Verdier formula for Deligne-Mumford stacks. We include both torsion and ℓ-adic coefficients.
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We present the complete list of all singularity types on Gorenstein Q-homology projective planes, i.e., normal projective surfaces of second Betti number one with at worst rational double points. The list consists of 58 possible singularity types, each except two types supported by an example.
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This special issue on algebraic geometry contains 12 articles, most of which are invited from the participants of the International Conference “Algebraic Geometry in East Asia”, which was held at Morningside Center of Mathematics, Academy of Mathematics and Systems Science (AMSS), Chinese Academy of Sciences (CAS) in Beijing from October 14 to October 18, 2013. The conference “Algebraic Geometry in East Asia” (AGEA) started with its first meeting held at the International Institute for Advanced Studies in Kyoto in 2001. A primary goal of the conference has been to provide opportunities to meet and exchange ideas among East Asian algebraic geometers. This is the fifth meeting following the second at Hanoi in 2005, the third at Seoul in 2008 and the fourth at Taipei in 2011. The scientific committee consisted of Jungkai Alfred Chen, Yujiro Kawamata, JongHae Keum, Xiaotao Sun and Stephen Yau. The local organizing committee were served by Yifei Chen, Baohua Fu and Xiaotao Sun. The conference featured 21 invited talks with about 60 participants from East Asia and Europe. It was financially supported by National Natural Science Foundation of China and by AMSS of Chinese Academy of Sciences. We are grateful to these organizations for their generous support. We would like to thank all the speakers and participants for the wonderful talks and stimulating discussions, which made the conference successful. Most of the articles in this special issue were invited by the organizers as guest editors of the journal SCIENCE CHINA Mathematics. All of the articles went through the regular refereeing procedure of the journal. We would like to thank all the authors and referees for the cooperation.