Paolo Cascini, Ivan Cheltsov, James McKernan and Chenyang Xu
Meng Chen
Antonella Grassi
Chen Jiang
Yongnam Lee
John Lesieutre
Emanuele Macri
Michael McQuillan
Alexander Perry
Nick Shepherd-Barron
Bernd Siebert
Chengxi Wang
De-Qi Zhang
Wednesday, October 25th
9:30 - 10:30 Nick Shepherd-Barron
11:00 - 12:00 Emanuele Macri
12:00 - 1:00 Lunch
1:30 - 2:30 De-Qi Zhang
3:00 - 4:00 Yongnam Lee
Thursday, October 26th
9:30 - 10:30 Antonella Grassi
11:00 - 12:00 Alexander Perry
12:00 - 1:00 Lunch
1:30 - 2:30 Bernd Siebert
3:00 - 4:00 Michael McQuillan
4:00 - 5:00 Chen Jiang
Friday, October 27th
9:30 - 10:30 Chengxi Wang
11:00 - 12:00 Meng Chen
12:00 - 1:00 Lunch
1:30 - 2:30 John Lesieutre
Meng Chen - A lifting principle for canonical stability indices of varieties of general type
Abstract. For any integer $n>0$, the $n$-th canonical stability index $r_n$ is defined to be the smallest positive integer so that the $r_n$-canonical map $\Phi_{r_n}$ is stably birational onto its image for all nonsingular projective $n$-folds of general type. In this talk, I will explain the main steps towards proving the following conjecture(="lifting principle”): $r_n$ equals to the maximum of the set of those canonical stability indices of smooth projective $(n+1)$-folds with sufficiently large canonical volumes. This is a joint work with Hexu Liu.
Chen Jiang - Numerically trivial automorphisms of hyperkähler 4-folds
Abstract: An automorphism on a smooth projective variety is said to be numerically trivial if it induces a trivial action on the cohomology ring. A well-known result says that there is no numerically trivial automorphism for a K3 surface. I will report our recent progress on the absence of numerically trivial automorphisms of hyperkähler 4-folds.
Antonella Grassi - Local, global, local-to-global “principles”, and elliptic Calabi-Yau threefolds with certain singularities.
Abstract: We present local, global and local-to-global properties of threefolds with certain singularities, in particular criteria for these threefolds to be rational homology manifolds and to satisfy rational Poincaré duality. The motivation comes from physics of string theory. We motivate and state a conjecture on the extension of Kodaira’s classification of singular fibers on relatively minimal elliptic surfaces to the class of birationally equivalent relatively minimal elliptically fibered varieties. We prove results when X is an elliptic Calabi-Yau.
Yongnam Lee - Smooth specialization of hypersurfaces in projective manifolds
Abstract: In this presentation, we give a structure theorem for projective manifolds $W_0$ with the property of admitting a one parameter deformation where a general fiber $W_t$ is a smooth hypersurface in a projective smooth manifold $Z_t$. Their structure is the one of special iterated univariate coverings, which we call normal type. We give an application to the case where $Z_t$ is a projective space, respectively an abelian variety. We also give a characterizaton of smooth ample hypersurfaces in abelian varieties and describe an irreducible connected component of their moduli space. This talk is mainly based on joint work with Fabrizio Catanese.
John Lesieutre - More pathologies of the volume function
Abstract: I will report on joint work with Valentino Tosatti and Simion Filip in which we show by example that for a pseudoeffective divisor D and ample A, the volume function vol(D+tA) for small values of t can exhibit various pathological behaviors.
Emanuele Macri - Vector bundles on Fano threefolds
Abstract: A celebrated part in the classification of Fano threefold is Mukai's vector bundle method. One of the main result is an existence (and rigidity) result for vector bundles with rank dividing the genus, for prime Fano threefolds of index 1. Unfortunately, in the literature, the proof has a gap. I will present joint work with Arend Bayer and Alexander Kuznetsov, where we fill this gap.
Michael McQuillan - Functorial resolution of singularities.
Abstract: The talk will give a general framework for resolution of singularities whether of varieties, resp. foliations, by way of a flow determined by an ideal, resp. a derivation, on the space of valuations of a complete regular local ring. In particular the convergence of this flow implies a wholly functorial solution of the resolution problem in question.
Alex Perry - Hyperkähler varieties from CY2 categories
Abstract: Hyperkähler varieties form one of the three building blocks for projective varieties with trivial canonical bundle. Their classification is widely open, but there is growing evidence that the problem is closely related to the classification of 2-dimensional Calabi-Yau categories, with the most optimistic hope being that any hyperkähler variety arises from a suitable moduli space of objects in such a category. I will discuss this circle of ideas, including work in progress with Arend Bayer, Laura Pertusi, and Xiaolei Zhao completing this program for hyperkähler varieties of Kummer type.
Nick Shepherd-Barron - Generic Torelli with denominators for elliptic surfaces.
Abstract: It is well known that a very general curve of genus at least 5 is determined by its polarised rational Hodge structure. We prove an analogous result for elliptic surfaces.
Bernd Siebert - Intrinsic Mirror Symmetry
Abstract: One of the fundamental questions of mirror symmetry asks "How broadly do mirrors exist"? In the talk I will survey joint work with Mark Gross giving canonical candidate mirror varieties in the two basic cases (1) maximally unipotent degenerations of Calabi-Yau manifolds and (2) pairs (X,D) with trivial logarithmic canonical bundle. The construction is known or conjectured to reproduce all known algebraic-geometric mirror pairs.The coordinate rings of the intrinsic mirror families have module generators given by contact orders of curves with the irreducible components of the central fibers and with D, respectively. The structure coefficients are given by a variant of logarithmic Gromov-Witten invariants admitting negative contact orders, recently defined in joined work with Dan Abramovich and Qile Chen. Many interesting questions arise concerning the properties of intrinsic mirror families, including applications to moduli problems and classification theory.
Chengxi Wang - Fano varieties with extreme behavior
Abstract: It is attractive to classify Fano varieties with various types of singularities that originated from the minimal model program. For a Fano variety, the Fano index is the largest integer m such that the anti-canonical divisor is Q-linearly equivalent to m times some Weil divisor. For Fano varieties of various singularities, I show the Fano indexes can grow double exponentially with respect to the dimension. Those examples are also conjecturally optimal and have a close connection with Calabi-Yau varieties of extreme behavior.
De-Qi Zhang - Structures theorems and applications of non-isomorphic surjective endomorphisms of smooth projective threefolds
Abstract: Let f: X to X be a non-isomorphic surjective endomorphism of a smooth projective threefold X. We prove that any birational minimal model program becomes f-equivariant after iteration, provided that f is delta-primitive. Here by ‘delta-primitive’ we mean that there is no f-equivariant dominant rational map X to Y to a positive lower-dimensional projective variety Y such that the first dynamical degree delta_f remains unchanged. We further determine the building blocks of f. As the first application, we prove the Kawaguchi-Silverman conjecture (about the equality of dynamic degree and arithmetic degree) for every non-isomorphic surjective endomorphism of a smooth projective threefold. As the second application, we reduce the Zariski dense orbit conjecture for f to a terminal threefold with only f-equivariant Fano contractions.