Blow Up Formulae and Invariants I - Blow-up Formula for Quantum Cohomology
Dr. Maxim Kontsevich
IHES
University of Miami & IMSA
9:00am - 10:00am
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Abstract: I will explain heuristics behind the conjectural formula expressing genus zero Gromov-Witten invariants of a blown up complex algebraic variety in terms of GW-invariants for the initial variety, and for the center. I'll also propose an approach to the proof via an equivariant localization.
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Blow Up Formulae and Invariants II - Dimension Theory
Dr. Maxim Kontsevich
IHES
University of Miami & IMSA
10:30am - 11:30am
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Abstract: I will introduce an numerical invariant ("dimension") for a meromorphic connection over formal disc, with the second order pole singularity. In the case of Landau-Ginzburg models associated with isolated singularity, it coincides with an invariant considered originally by V.Arnold. In analogy with singularity theory, one conjectures certain semi-continituity of this invariant. Combination of the semi-continuity with the blow-up formula leads to a new strong criterion for non-rationality of higher-dimensional Fano varieties.
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Secondary Fan, Theta Functions, and Moduli of Calabi-Yau Pairs
Dr. Tony Yue Yu
Laboratoire de Mathématiques d'Orsay
1:00pm - 2:00pm
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Abstract: We conjecture that any connected component $Q$ of the moduli space of triples $(X,E=E_1+\dots+E_n,\Theta)$ where $X$ is a smooth projective variety, $E$ is a normal crossing anti-canonical divisor with a 0-stratum, every $E_i$ is smooth, and $\Theta$ is an ample divisor not containing any 0-stratum of $E$, is unirational. More precisely, we conjecture that the compactification of $Q$ inside the moduli space of Kollár-Shepherd-Barron-Alexeev stable pairs admits a finite cover by a complete toric variety. We construct the associated complete toric fan, generalizing the Gelfand-Kapranov-Zelevinski secondary fan for reflexive polytopes. Inspired by mirror symmetry, we speculate a synthetic construction of the universal family over this toric variety, as the Proj of a sheaf of graded algebras with a canonical basis, whose structure constants are given by counts of non-archimedean analytic disks. In the Fano case and under the assumption that the mirror contains a Zariski open torus, we construct the conjectural universal family, generalizing the families of Kapranov-Sturmfels-Zelevinski and Alexeev in the toric case. In the case of del Pezzo surfaces with an anti-canonical cycle of $(-1)$-curves, we prove the full conjecture. The reference is arXiv:2008.02299 joint with Hacking and Keel.
NC Spectra
Dr. Ludmil Katzarkov
University of Miami & IMSA
2:30pm - 3:30pm
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Abstract: This talk is continuation of the short course by M. Kontsevich. We will offer some examples and applications to nonrationality questions. Generalizations will be considered. The talk is based on joint works with Kontsevich and Pantev.
Surfaces on Cubic Fourfolds via Stability Conditions
Dr. Alex Perry
University of Michigan
4:00pm - 5:00pm
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Abstract: Any cubic fourfold has an associated K3 category, defined by Kuznetsov as a semiorthogonal component of the derived category. I will explain some geometric applications of stability conditions on this K3 category, in particular the construction of special surfaces on cubic fourfolds and relations to the rationality problem. This is based on joint work with Arend Bayer, Aaron Bertram, and Emanuele Macri.
