Title    name Lastest   |   Most hits   년도선택 최근 50건2012201120102009200820072006200520042003200220012000     KIAS Seminar Around $p_g=0$ stable surfaces Giancarlo Urzua (Pontificia Universidad Catolica de Chile) 2012-07-02   We now apply the previous talk to the $p_g=0$ surfaces of general type constructed by Y. Lee and J. Park, and H. Park, J. Park, and D. Shin. Given such a projective surface $X$ with only $T$-singularities, we identify the minimal model of the minimal resolutions of all partial $mathbb{Q}$-Gorenstein smoothings of $X$. This is about mapping the Koll'ar--Shepherd-Barron and Alexeev moduli space of stable surfaces around the point corresponding to $X$. Here we find other special rational surfaces (e.g. coming from $2$-Halphen pencils), Dolgachev surfaces with certain embedded curves, surfaces of general type, etc. This is also joint with P. Hacking and J. Tevelev.   KIAS Seminar Flipping surfaces Giancarlo Urzua (Pontificia Universidad Catolica de Chile) 2012-07-02   Let $X$ be a projective surface with only $T$-singularities. Assume it has a $mathbb{Q}$-Gorenstein smoothing $X subset mathcal{X} to 0 in mathbb{D}$ over a smooth curve germ, whose smooth fibers are of general type. What is the Koll'ar--Shepherd-Barron stable family of $mathcal{X} to mathbb{D}$? In this talk we will show how to find explicitly its minimal model. The main point is to relate Mori flips of type $k1A$ and $k2A$ via deformations and continued fractions. We will discuss certain properties such as the existence of a universal family for them, and the amount of these families over a fixed cyclic quotient singularity (this allows fast calculations). This is a joint work with P. Hacking and J. Tevelev.   KIAS Seminar Introduction to Noncommutative Geometry 6 Dmitry Kaledin (University of Moscow) 2009-10-23     KIAS Seminar Introduction to Noncommutative Geometry 5 Dmitry Kaledin (University of Moscow) 2009-10-23     KIAS Seminar Introduction to Noncommutative Geometry 4 Dmitry Kaledin (University of Moscow) 2009-10-15     KIAS Seminar Introduction to Noncommutative Geometry 3 Dmitry Kaledin (University of Moscow) 2009-10-15     KIAS Seminar Introduction to spherical varieties and description of some particular classes of spherical varieties 8 Boris Pasquier (Hausdorff Center) 2009-07-02     KIAS Seminar Introduction to spherical varieties and description of some particular classes of spherical varieties 7 Boris Pasquier (Hausdorff Center) 2009-07-02     KIAS Seminar Introduction to spherical varieties and description of some particular classes of spherical varieties 6 Boris Pasquier (Hausdorff Center) 2009-07-01     KIAS Seminar Introduction to spherical varieties and description of some particular classes of spherical varieties 5 Boris Pasquier (Hausdorff Center) 2009-07-01     KIAS Seminar Introduction to spherical varieties and description of some particular classes of spherical varieties 4 Boris Pasquier (Hausdorff Center) 2009-06-30     KIAS Seminar Introduction to spherical varieties and description of some particular classes of spherical varieties 3 Boris Pasquier (Hausdorff Center) 2009-06-30     KIAS Seminar Introduction to spherical varieties and description of some particular classes of spherical varieties 2 Boris Pasquier (Hausdorff Center) 2009-06-29     KIAS Seminar Introduction to spherical varieties and description of some particular classes of spherical varieties 1 Boris Pasquier (Hausdorff Center) 2009-06-29     KIAS Seminar Lectures on abelian varieties and integrable systems 8 Sam Grushevsky (Princeton University) 2009-06-25     KIAS Seminar Lectures on abelian varieties and integrable systems 7 Sam Grushevsky (Princeton University) 2009-06-25     KIAS Seminar Lectures on abelian varieties and integrable systems 6 Sam Grushevsky (Princeton University) 2009-06-24     KIAS Seminar Lectures on abelian varieties and integrable systems 5 Sam Grushevsky (Princeton University) 2009-06-24     KIAS Seminar Lectures on abelian varieties and integrable systems 4 Sam Grushevsky (Princeton University) 2009-06-23     KIAS Seminar Lectures on abelian varieties and integrable systems 3 Sam Grushevsky (Princeton University) 2009-06-23     KIAS Seminar Lectures on abelian varieties and integrable systems 2 Sam Grushevsky (Princeton University) 2009-06-22     KIAS Seminar Lectures on abelian varieties and integrable systems 1 Sam Grushevsky (Princeton University) 2009-06-22     KIAS Seminar Sheaves on toric Calabi-Yau hypersurfaces Vladimir Baranovsky (UC Irvine) 2009-04-07     We are planning to discuss a re-interpretation and generalization of the following result due to D. Orlov. Let $mathbb{P} = (Vsetminus 0)/mathbb{C}^*$ be a (weighted) projective space and assume that $Y subset mathbb{P}$ is a quasi-smooth Calabi-Yau hypersurface given by vanishing of a (weighted) homogeneous polynomial $W: V to mathbb{C}$. Let $D^b(Y)$ be the derived category of coherent sheaves (or algebraic vector bundles) on $Y$ and $D^b_{sing}(W)$ be the singular category of $W$, i.e. the quotient of finite complexes of coherent sheaves on $Z(W) subset V$, by finite complexes of projective modules. A general result proved by Orlov implies an equivalence $$ D^b(Y) = D^b_{sing}(W) $$ The original proof of this result is mostly algebraic. This talk will interpret it geometrically (as a particular case of a generalized ``McKay correspondence') and extend to the case when $mathbb{P}$ is a projective smooth toric Fano variety.   KIAS Seminar Weak Landau-Ginzburg models and rationality problems Victor Przyjalkowski (Steklov Mathematical Institute) 2009-04-07     We will talk about new (mostly conjectural yet) approach to rationality problems of Fano varieties. Mirror Symmetry enables one to define some birational invariants of Fano varieties. In particular, the monodromy of perverse sheaf of vanishing cycles associated to Landau-Ginzburg model dual to a Fano variety is invariant with respect to blow-ups and blow-downs and trivial for projective spaces. We use the simplified version of Mirror Symmetry, namely, Mirror Symmetry of Hodge structures variations for predicting this monodromy. This version reduces the mirror correspondence to finding Gromov-Witten invariants of Fano varieties and some combinatorial problems. We demonstrate this approach on some examples, such as Fano threefolds or four-dimensional cubic.   KIAS Seminar Lectures on Toric Degenerations, Potential Functions, and Gelfand-Cetlin Systems Yuichi Nohara (Tohoku University) 2009-03-17    It is well known that polarized toric varieties are related to the moment polytopes in two different ways, the moment map and monomial basis. Also in the case of flag manifolds, certain polytopes, called the Gelfand-Cetlin polytopes, appear in different stories: the Gelfand-Cetlin system, a completely integrable system; and the Gelfand-Cetlin basis, a basis of an irreducible representation. Furthermore the flag manifold admits a degeneration into the toric variety corresponding to the Gelfand-Cetlin polytope. Kogan and Miller proved that the Gelfand-Cetlin basis can be deformed into monomial basis on the toric variety under the degeneration. We show that the Gelfand-Cetlin system can be deformed into a moment map on the toric variety. We will also explain that the idea of toric degeneration can be applied to the study of modern symplectic geometry of flag manifolds. In particular, we will calculate the so-called potential functions (or obstructions) in Floer cohomology of flag manifolds, which imply some geometric consequences for the Lagrangian fibers of Gelfand-Cetlin systems. The calculation of potential functions is based on Cho-Oh's study of holomorphic disks in toric manifolds.   KIAS Seminar Lectures on Toric Degenerations, Potential Functions, and Gelfand-Cetlin Systems Takeo Nishinou (Tohoku University) 2009-03-17    It is well known that polarized toric varieties are related to the moment polytopes in two different ways, the moment map and monomial basis. Also in the case of flag manifolds, certain polytopes, called the Gelfand-Cetlin polytopes, appear in different stories: the Gelfand-Cetlin system, a completely integrable system; and the Gelfand-Cetlin basis, a basis of an irreducible representation. Furthermore the flag manifold admits a degeneration into the toric variety corresponding to the Gelfand-Cetlin polytope. Kogan and Miller proved that the Gelfand-Cetlin basis can be deformed into monomial basis on the toric variety under the degeneration. We show that the Gelfand-Cetlin system can be deformed into a moment map on the toric variety. We will also explain that the idea of toric degeneration can be applied to the study of modern symplectic geometry of flag manifolds. In particular, we will calculate the so-called potential functions (or obstructions) in Floer cohomology of flag manifolds, which imply some geometric consequences for the Lagrangian fibers of Gelfand-Cetlin systems. The calculation of potential functions is based on Cho-Oh's study of holomorphic disks in toric manifolds.

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