Stable reflexive sheaves
01 Jun 1980-Mathematische Annalen (Springer Science and Business Media LLC)-Vol. 254, Iss: 2, pp 121-176
About: This article is published in Mathematische Annalen.The article was published on 1980-06-01. It has received 717 citations till now. The article focuses on the topics: Reflexive sheaf & Chern class.
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08 Oct 2012TL;DR: In this paper, the authors present a survey of the geometry of lines and cubic surfaces, including determinantal equations, theta characteristics, and the Cremona transformations.
Abstract: Preface 1. Polarity 2. Conics and quadrics 3. Plane cubics 4. Determinantal equations 5. Theta characteristics 6. Plane quartics 7. Cremona transformations 8. Del Pezzo surfaces 9. Cubic surfaces 10. Geometry of lines Bibliography Index.
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Cites background from "Stable reflexive sheaves"
...They are locally free outside of a closed subset of codimension ≥ 3 (see [284])....
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TL;DR: In this paper, the deformation theory necessary to obtain virtual moduli cycles of stable sheaves whose higher obstruction groups vanish has been developed, and the moduli spaces of sheaves on a general $K3$ fibration have been computed.
Abstract: We briefly review the formal picture in which a Calabi-Yau $n$-fold is the complex analogue of an oriented real $n$-manifold, and a Fano with a fixed smooth anticanonical divisor is the analogue of a manifold with boundary, motivating a holomorphic Casson invariant counting bundles on a Calabi-Yau 3-fold. We develop the deformation theory necessary to obtain the virtual moduli cycles of \cite{LT}, \cite{BF} in moduli spaces of stable sheaves whose higher obstruction groups vanish. This gives, for instance, virtual moduli cycles in Hilbert schemes of curves in $\Pee^3$, and Donaldson-- and Gromov-Witten-- like invariants of Fano 3-folds. It also allows us to define the holomorphic Casson invariant of a Calabi-Yau 3-fold $X$, prove it is deformation invariant, and compute it explicitly in some examples. Then we calculate moduli spaces of sheaves on a general $K3$ fibration $X$, enabling us to compute the invariant for some ranks and Chern classes, and equate it to Gromov-Witten invariants of the ``Mukai-dual'' 3-fold for others. As an example the invariant is shown to distinguish Gross' diffeomorphic 3-folds. Finally the Mukai-dual 3-fold is shown to be Calabi-Yau and its cohomology is related to that of $X$.
464 citations
TL;DR: The existence of Secant Lines has been studied extensively in the literature, see as discussed by the authors for a survey. But the main focus of this paper is on the relation between Secant lines and regularity and rationality.
Abstract: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491 w Notat ion and Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493 w A Regularity Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494 w A Rationality Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498 w The Existence of Secant Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 w Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506
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01 Dec 1988
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Cites background from "Stable reflexive sheaves"
...We refer to [219, Section 2], as well as to [275] for details, proofs, and further results....
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...The following extension result is an important characterization of reflexive sheaves: if X is also assumed to be normal, then a coherent OX -module F is reflexive if and only if for every open subset U ⊆ X and every closed subset Z ⊆ U of codimension ≥ 2 the natural restriction map H(U,F|U ) → H(U\Z,F|U\Z) is an isomorphism, see [275], Proposition 1....
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TL;DR: In this paper, the degenerate complex Monge-Ampere equations of the form $(\omega+dd^c\f)^n = e^{t \f}\mu$ were studied.
Abstract: We study degenerate complex Monge-Ampere equations of the form $(\omega+dd^c\f)^n = e^{t \f}\mu$ where $\omega$ is a big semi-Kahler form on a compact Kahler manifold $X$ of dimension $n$, $t \in \R^+$, and $\mu=f\omega^n$ is a positive measure with density $f\in L^p(X,\omega^n)$, $p>1$. We prove the existence and unicity of continuous $\o$-plurisubharmonic solutions. In case $X$ is projective and $\omega=\psi^*\omega'$, where $\psi:X\to V$ is a proper birational morphism to a normal projective variety, $[\omega']\in NS_{\R} (V)$ is an ample class and $\mu$ has only algebraic singularities, we prove that the solution is smooth in the regular locus of the equation. We use these results to construct singular Kahler-Einstein metrics of non-positive curvature on projective klt pairs, in particular on canonical models of algebraic varieties of general type.
289 citations
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01 Jan 1966
TL;DR: In this paper, Mumford's lecture on algebraic geometry is devoted to a study of properties of families of algebraic curves, on a non-singular projective algebraic curve defined over an algebraically closed field of arbitrary characteristic.
Abstract: These lectures, delivered by Professor Mumford at Harvard in 1963-1964, are devoted to a study of properties of families of algebraic curves, on a non-singular projective algebraic curve defined over an algebraically closed field of arbitrary characteristic. The methods and techniques of Grothendieck, which have so changed the character of algebraic geometry in recent years, are used systematically throughout. Thus the classical material is presented from a new viewpoint.
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