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Quantum Cohomology and Mirror Symmetry / Cohomologie quantique et symétrie miroir
Org: Kai Behrend (UBC)
- IONUT CIOCAN-FONTANINE, University of Minnesota, School of Mathematics, Minneapolis,
Minnesota
Gromov-Witten invariants of GIT quotients
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I will discuss some conjectures about the relationship between the
genus zero Gromov-Witten invariants of a non-abelian GIT quotient
X//G, and those of the associated abelian quotient X//T, for T a
maximal torus in G. The conjectures were inspired by explicit
computations of Gromov-Witten invariants on Grassmannians, and may be
viewed as "quantum" generalizations of results (due to
Ellingsrud-Strømme, and Martin) relating the cohomology rings of
X//G and X//T. I will then sketch proofs for some of the
conjectures in the cases when localization methods apply, in particular
for all type A flag manifolds. This is joint work with Aaron Bertram
and Bumsig Kim.
- ALASTAIR CRAW, SUNY at Stony Brook
Towards the McKay correspondence in higher dimensions
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For a finite subgroup G of SL(n,C), if the quotient
singularity Cn/G admits a crepant resolution Y then the
McKay correspondence principle asserts that the geometry of Y
should be equivalent to the G-equivariant geometry of
Cn. This is always true for the euler number of Y, a
weak geometric invariant.
With additional assumptions (which hold in dimension two or three),
Bridgeland, King and Reid established the McKay correspondence as an
equivalence of derived categories for a special choice of a crepant
resolution. This talk will discuss a programme to generalise this
result to higher dimensions.
- CHUCK DORAN, Columbia University, New York, New York, USA
Integral structures, toric geometry, and homological
mirror symmetry
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We establish the isomorphisms over Z of
cohomology/K-theory, global monodromy, and invariant symplectic forms
predicted by Kontsevich's Homological Mirror Symmetry Conjecture for
certain one dimensional families of Calabi-Yau threefolds with h2,1 = 1. These families arise as hypersurfaces or complete intersections
in Gorenstein toric Fano varieties, and their mirrors are described by
the Batyrev-Borisov construction. Our method involves (1) classifying
all rank four integral variations of Hodge structure over P1\{0,1,¥} with maximal unipotent local monodromy about
0 and local monodromy about 1 unipotent of rank 1, and
(2) checking, using properties of nef partitions of reflexive
polytopes, that the Z-VHS of our Calabi-Yau families match
those picked out by the K-theory of their mirrors via the HMS
Conjecture. This is joint work with John Morgan.
- BARBARA FANTECHI, SISSA, Via Beirut 4, 34014 Trieste, Italy
Quantum cohomology for orbifiolds
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In this mostly expository talk we will recall the definition of quantum
cohomology (and Gromov Witten invariants) for orbifolds, i.e.
smooth proper complex Deligne Mumford stacks with projective coarse
moduli space. We will stress the main differences with the quantum
cohomology for smooth varieties: the need to allow for stacky points in
the nodal curves which are the domain of stable maps, and as a
consequence the fact that the evaluation maps have the inertia stack as
target (the definition of inertia stack will also be recalled). Special
attention will be devoted to the degree zero case, defining what is
called orbifold or Chen Ruan cohomology. We will then present an
overview of the (not very many) computational results available so far:
indeed, perhaps the most striking difference between quantum cohomology
for varities and for orbifolds is that for varieties many computations
were carried out even before a general definition was complete, while
for orbifolds the definition (both in the symplectic and in the
algebraic language) is already a few years old but explicit results are
still few and far apart. The talk should be understandable to
mathematicians working in algebraic or symplectic geometry: previous
familiarity with either quantum cohomology or orbifolds/stacks (or
both) will be useful but not necessary.
- HOLGER KLEY, Colorado
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- SÁNDOR KOVÁCS, University of Washington, Department of Mathematics, Seattle,
Washington 98195, USA
Zero sets of holomorphic one forms on varieties of general
type
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This is joint work with Christopher Hacon (Utah).
It has been conjectured that a global holomorphic one form on a variety
of general type always has a non-empty sero set. One of several
applications of this statement is that a variety with this condition
does not admit a smooth morphism to an abelian variety.
This conjecture has been confirmed by Zhang for canonically polarized
varieties and by Luo and Zhang for threefolds. We confirm the
conjecture for several other cases, including all smooth minimal models
of general type of arbitrary dimension.
- RAVI VAKIL, Stanford University, Stanford, California, USA
Schubert induction
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I'll describe a Schubert induction theorem, a tool for analyzing
intersections on a Grassmannian over an arbitrary base ring. The key
ingredient in the proof is the geometric Littlewood-Richardson rule.
Schubert problems are among the most classical problems in enumerative
geometry of continuing interest. As an application of Schubert
induction, I will address several natural questions related to Schubert
problems, including: the "reality" of solutions; effective numerical
methods; solutions over algebraically closed fields of positive
characteristic; solutions over finite fields; a generic smoothness
(Kleiman-Bertini) theorem; and monodromy groups of Schubert problems.
I may spend some time at the end discussing the geometry behind the
geometric Littlewood-Richardson rule.
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