Algebraic Geometry Seminar
Spring 2013
Regular Day: Tuesday
Regular Time: 3:30PM - 4:30PM
Regular Location: LCB 215
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January 8 |
Yongbin Ruan
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Traditionally, we use mirror symmetry to map a difficult problem (A-model) to an easier problem (B-model). Recently, there is a great deal of activities in mathematics to understand the modularity properties of Gromov-Witten theory, a phenomenon suggested by BCOV almost twenty years ago. Mirror symmetry is again used in a crucial way. However, the new usage of mirror does not map a difficult problem to easy problem. Instead, we make both side of mirror symmetry to work together in a deep way. I will explain this interesting phenomenon in the talk.
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January 15 |
Sofia Tirabassi
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We present some recent results concerning the birational geometry of varieties of maximal Albanese dimension obtained by the means of Pareschi--Popa M-regularity theory. Many of these result had been obtained in collaboration with Z. Jiang and M. Lahoz.
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January 22 |
Zachary Maddock
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Over perfect fields, the geometry of regular del Pezzo surfaces has been classified, but over imperfect fields, the problem remains largely open. We construct the first examples of regular del Pezzo surfaces X that have positive irregularity h^1(X, O_X ) > 0. Our construction is by quotienting a regular, quasi-linear surface (i.e. a regular variety that is geometrically a non-reduced first-order neighborhood of a plane) by explicit rank 1 foliations. We also find a restriction on the integer pairs that are possible as the anti-canonical degree and irregularity of such surfaces.
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January 29 |
Zsolt Patakfalvi
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This talk is a continuation of a talk given by Karl Schwede during the Fall semester. It is based on a joint project with Karl Schwede and Wenliang Zhang aiming to interpret many notions of the F-singularity theory in the relative setting. One of the surprising results is the definition of a relative test ideal that restricts to the test ideal on each fiber. The focus of the current talk will be applications to global geometry, such as: behavior of the canonical adjoint linear sub-system S^0 in flat families, definition of canonical subsheaves of pushforards of adjoint line bundles, global generation result for the latest, etc.
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February 5 |
Tommaso de Fernex
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Hironaka's theorem on resolution of singularities allows to study the geometry of a singular variety by associating to it a smooth birational model. A more intrinsic approach to study singularities was proposed by Nash. The idea is to look at the space of arcs (i.e. analytic germs of curves) passing through the singular points. This space decomposes into finitely many irreducible families, and carries much of the information encoded in a resolution. The Nash problem gives a precise formulation of how such families of arcs should relate to resolutions of singularities. In this talk I will give an overview of the history and solution of the problem.
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February 12 |
Aaron Bertram
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February 19 |
Yu-Han Liu
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In this talk I will explain how some familiar constructions in birational geometry can be studied using tensor triangular geometry.
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February 26 |
Mihnea Popa
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I will report on recent work with C. Schnell, in which we prove that every holomorphic one-form on a variety of general type must vanish at some point (together with a suitable generalization to arbitrary Kodaira dimension). The proof makes use of generic vanishing theory for Hodge D-modules on abelian varieties.
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March 5 |
Domingo Toledo
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The title refers to a class of homogeneous complex manifolds which includes the period domains for Hodge structures of weight at least two. The absence of automorphic forms for higher weight variations of Hodge structure suggests the theorem in the title. This talk will present a proof of this theorem, which requires some interesting geometry of non-classical domains. This is joint work with Phillip Griffiths and Colleen Robles.
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March 12 |
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March 19 |
Bumsig Kim
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The moduli spaces of stable quasimaps unify various moduli appearing in the study of Gromov-Witten Theory. We introduce big I-functions as the quasimap version of J-functions, generalizing Givental's small I-functions of smooth toric complete intersections. The J-functions are the GW counterparts of periods of mirror families. We discuss some advantages of I-functions, in particular an explanation of mirror maps. This is joint work with I. Ciocan-Fontanine. If time permitted, I will also report on the stable quasimaps for orbifold targets and I-functions of toric orbifolds. This is joint work with I. Ciocan-Fontanine and D. Cheong.
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March 26 |
Yifan Chen
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This talk is aimed to give a classification of the pair (S, G), where S is a smooth minimal surface of general type with p_g=0 and K^2=7, and G is a subgroup of Aut(S), and G is isomorphic to (Z/2Z)^2. From the classification, a new family of surfaces is constructed. These new surfaces are the first known surfaces with p_g=0, K^2=7 and with birational bicanonical maps.
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April 2 |
Samouil Molcho
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The space of stable maps to a smooth variety relative to a smooth divisor, defined by Jun Li, is an important moduli space in Gromov-Witten theory, which arises naturally when one tries to study Gromov-Witten invariants by degeneration. It's deformation theory however is complicated. A related space is the space of logarithmic stable maps, defined by B.Kim. This space has the same Gromov-Witten invariants as the space of relative stable maps, but its deformation theory has better formal properties. In this talk I will briefly describe these moduli spaces and explain how to do localization in the space of logarithmic stable maps when the targets admit a torus action. This is joint work with E. Routis.
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April 10 (Wednesday) Time/Location: 2pm - 3pm JTB 320 |
Mark De Cataldo
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I report on joint work with Luca Migliorini at Bologna. If you have a map of complex projective manifolds, then the rational cohomology of the domain splits into a direct sum of pieces in a way dictated by the singularities of the map. By Poincare' duality, the corresponding projections can be viewed as cohomology classes (projectors) on the self-product of the domain. These projectors are Hodge classes, i.e. rational and of type (p,p) for the Hodge decomposition. Take the same situation after application of an automorphism of the ground field of complex numbers. The new projectors are of course Hodge classes. On the other hand, you can also transplant, using the field automorphism, the old projectors into the new situation and it is not clear that the new projectors and the transplants of the old projectors coincide. We prove they do, thus proving that the projectors are absolute Hodge classes, i.e. their being of Hodge type survives the totally discontinuous process of a field automorphism. We also prove that these projectors are motivated in the sense of Andre.
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April 16 |
Christopher Hacon
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April 23 |
Renzo Cavalieri
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The question that the Crepant Resolution Conjecture (CRC) wants to address is: given an orbifold X that admits a crepant resolution Y, can we systematically compare the Gromov-Witten theories of the two spaces? That this should happen was first observed by physicists and the question was imported into mathematics by Y.Ruan, who posited as the search for an isomorphism in the quantum cohomologies of the two spaces. In the last fifteen years this question has evolved and found different formulations which various degree of generality and validity. Perhaps the most powerful approach to the CRC is through Givental's formalism. In this case, Coates, Corti, Iritani and Tseng propose that the CRC should consist of the natural comparison of geometric objects constructed from the GW potential fo the space. We explore this approach in the setting of open GW invariants. We formulate an open version of the CRC using this formalism, and verify it for the family of A_n singularities. Our approach is well tuned with Iritani's approach to the CRC via integral structures
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