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Frühjahrssemester 2020

Datum / Zeit Referent:in Titel Ort
* 20. Januar 2020
10:45-12:00 		Dr. Junliang Shen
MIT
Details

Algebraic Geometry and Moduli Seminar

Titel Hitchin systems, compact hyper-Kaehler geometry, and the P=W conjecture I
Referent:in, Affiliation Dr. Junliang Shen, MIT
Datum, Zeit 20. Januar 2020, 10:45-12:00
Ort HG G 43
Abstract The topology of Hitchin systems play a central role in geometry, mathematical physics, and representation theory. In 2010 , de Cataldo, Hausel, and Migliorini predicts a surprising connection between the topology of Hitchin systems and the Hodge theory of character varieties, which is now referred to as the P=W conjecture. In the three lectures, we will discuss structures of the cohomology of the moduli of Higgs bundles, recent progress on the P=W conjecture, and compact analogs of the P=W conjecture. We will present a proof of the genus 2 case for arbitrary rank, and a proof of the P=W for the even tautological ring for arbitrary genus. Our strategy is to use hidden symmetries in the compact hyper-Kaehler geometry and a degeneration method connecting the compact geometry and Hitchin systems. If time permits, we will also discuss connections to curve counting invariants and some open questions.
Hitchin systems, compact hyper-Kaehler geometry, and the P=W conjecture Iread_more
HG G 43
* 20. Januar 2020
14:30-15:45 		Dr. Andrey Soldatenkov
Humboldt Universität zu Berlin
Details

Algebraic Geometry and Moduli Seminar

Titel Hodge structures of compact hyperkähler manifolds
Referent:in, Affiliation Dr. Andrey Soldatenkov, Humboldt Universität zu Berlin
Datum, Zeit 20. Januar 2020, 14:30-15:45
Ort HG G 43
Abstract Cohomology of a compact hyperkähler manifold carries a natural Lie algebra action introduced by Verbitsky and Looijenga-Lunts. In my talk, I will discuss some applications of this fact. First, one can use it to recover the Hodge structures on higher degree cohomology groups from the Hodge structure in degree two. Second, it controls the monodromy action on the cohomology in smooth families of hyperkähler manifolds. The latter observation can be used to study the behaviour of Hodge structures under degeneration.
Hodge structures of compact hyperkähler manifoldsread_more
HG G 43
* 20. Januar 2020
16:15-17:30 		Prof. Dr. Qizheng Yin
Peking University
Details

Algebraic Geometry and Moduli Seminar

Titel A compact analogue of P=W
Referent:in, Affiliation Prof. Dr. Qizheng Yin, Peking University
Datum, Zeit 20. Januar 2020, 16:15-17:30
Ort HG G 43
Abstract I will present the proof of a compact version of P=W, which relates the topology of holomorphic Lagrangian fibrations to the Hodge theory of degenerations of compact hyper-Kähler manifolds. Hopefully this reveals certain hyper-Kähler nature of the P=W phenomenon. I will also discuss possible ways to categorify the statement. Joint work with Andrew Harder, Zhiyuan Li, and Junliang Shen.
A compact analogue of P=Wread_more
HG G 43
* 21. Januar 2020
10:45-12:00 		Dr. Junliang Shen
MIT
Details

Algebraic Geometry and Moduli Seminar

Titel Hitchin systems, compact hyper-Kaehler geometry, and the P=W conjecture II
Referent:in, Affiliation Dr. Junliang Shen, MIT
Datum, Zeit 21. Januar 2020, 10:45-12:00
Ort HG G 43
Abstract The topology of Hitchin systems play a central role in geometry, mathematical physics, and representation theory. In 2010 , de Cataldo, Hausel, and Migliorini predicts a surprising connection between the topology of Hitchin systems and the Hodge theory of character varieties, which is now referred to as the P=W conjecture. In the three lectures, we will discuss structures of the cohomology of the moduli of Higgs bundles, recent progress on the P=W conjecture, and compact analogs of the P=W conjecture. We will present a proof of the genus 2 case for arbitrary rank, and a proof of the P=W for the even tautological ring for arbitrary genus. Our strategy is to use hidden symmetries in the compact hyper-Kaehler geometry and a degeneration method connecting the compact geometry and Hitchin systems. If time permits, we will also discuss connections to curve counting invariants and some open questions.
Hitchin systems, compact hyper-Kaehler geometry, and the P=W conjecture IIread_more
HG G 43
* 21. Januar 2020
14:30-15:45 		Dr. Laura Pertusi
Università degli Studi di Milano
Details

Algebraic Geometry and Moduli Seminar

Titel Gushel-Mukai varieties and stability conditions
Referent:in, Affiliation Dr. Laura Pertusi, Università degli Studi di Milano
Datum, Zeit 21. Januar 2020, 14:30-15:45
Ort HG G 43
Abstract A generic Gushel-Mukai variety X is a quadric section of a linear section of the Grassmannian Gr(2,5). Kuznetsov and Perry proved that the bounded derived category of X has a semiorthogonal decomposition with exceptional objects and a non-trivial subcategory Ku(X), known as the Kuznetsov component. In this talk we will discuss the construction of stability conditions on Ku(X) and, consequently, on the bounded derived category of X. As applications, for X of even-dimension, we will construct locally complete families of hyperkaehler manifolds from moduli spaces of stable objects in Ku(X) and we will determine when X has a homological associated K3 surface. This is a joint work with Alex Perry and Xiaolei Zhao.
Gushel-Mukai varieties and stability conditionsread_more
HG G 43
* 21. Januar 2020
16:15-17:30 		Dr. Johannes Schmitt
Universität Bonn
Details

Algebraic Geometry and Moduli Seminar

Titel Almost three definitions of double ramification cycles and why they are equivalent
Referent:in, Affiliation Dr. Johannes Schmitt, Universität Bonn
Datum, Zeit 21. Januar 2020, 16:15-17:30
Ort HG G 43
Abstract Given a smooth projective curve C and distinct points p1,...,pn in C, one can study the condition that there exists a meromorphic k-differential form on C with zeros and poles at the points p1, ..., pn of given orders. The double ramification cycles are cohomology classes on the moduli space of stable curves (C,p1,...,pn) associated to extending this condition to all stable curves. Different approaches have been proposed for defining these cycles. I will introduce some of them and discuss how work in progress with Bae, Holmes, Pandharipande and Schwarz finishes the proof that these give the same cohomology classes.
Almost three definitions of double ramification cycles and why they are equivalentread_more
HG G 43
* 22. Januar 2020
10:45-12:00 		Dr. Junliang Shen
MIT
Details

Algebraic Geometry and Moduli Seminar

Titel Hitchin systems, compact hyper-Kaehler geometry, and the P=W conjecture III
Referent:in, Affiliation Dr. Junliang Shen, MIT
Datum, Zeit 22. Januar 2020, 10:45-12:00
Ort HG G 43
Abstract The topology of Hitchin systems play a central role in geometry, mathematical physics, and representation theory.
In 2010 , de Cataldo, Hausel, and Migliorini predicts a surprising connection between the topology of Hitchin systems and the Hodge theory of character varieties, which is now referred to as the P=W conjecture. In the three lectures, we will discuss structures of the cohomology of the moduli of Higgs bundles, recent progress on the P=W conjecture, and compact analogs of the P=W conjecture. We will present a proof of the genus 2 case for arbitrary rank, and a proof of the P=W for the even tautological ring for arbitrary genus. Our strategy is to use hidden symmetries in the compact hyper-Kaehler geometry and a degeneration method connecting the compact geometry and Hitchin systems. If time permits, we will also discuss connections to curve counting invariants and some open questions.
Hitchin systems, compact hyper-Kaehler geometry, and the P=W conjecture IIIread_more
HG G 43
* 22. Januar 2020
14:30-15:45 		Prof. Dr. Anton Mellit
Universität Vienna
Details

Algebraic Geometry and Moduli Seminar

Titel The curious hard Lefschetz property for character varieties
Referent:in, Affiliation Prof. Dr. Anton Mellit, Universität Vienna
Datum, Zeit 22. Januar 2020, 14:30-15:45
Ort HG G 43
Abstract I will talk about a way to decompose the character variety of a Riemann surface of arbitrary rank with prescribed semisimple generic local monodromies into cells where each cell looks like a product of an affine space and a symplectic torus. This can be thought of as abelianization. As an application, we deduce the curious hard Lefschetz property conjectured by Hausel, Letellier and Rodriguez-Villegas, which claims that the operator of cup product with the class of the holomorphic symplectic form is an isomorphism between complementary degrees of the associated graded with respect to the weight filtration on the cohomology.
The curious hard Lefschetz property for character varietiesread_more
HG G 43
* 22. Januar 2020
16:15-17:30 		Prof. Dr. Georg Oberdieck
Universität Bonn
Details

Algebraic Geometry and Moduli Seminar

Titel Motivic decompositions of the Hilbert scheme of points of a K3 surface
Referent:in, Affiliation Prof. Dr. Georg Oberdieck, Universität Bonn
Datum, Zeit 22. Januar 2020, 16:15-17:30
Ort HG G 43
Abstract I will discuss joint work with Andrei Negut and Qizheng Yin in which we construct an explicit, multiplicative Chow-Kunneth decomposition of any Hilbert scheme of points of a K3 surface. This is parallel to the case of abelian varieties where such a decomposition is induced by the Fourier-Mukai transform. Our approach relies on a lift of the Looijenga-Lunts-Verbitsky Lie algebra to Chow.
Motivic decompositions of the Hilbert scheme of points of a K3 surfaceread_more
HG G 43
* 23. Januar 2020
14:30-15:45 		Tim Bülles
ETH Zürich
Details

Algebraic Geometry and Moduli Seminar

Titel Multiple covers and point counts for K3 surfaces
Referent:in, Affiliation Tim Bülles, ETH Zürich
Datum, Zeit 23. Januar 2020, 14:30-15:45
Ort HG G 43
Abstract The Gromov—Witten theory of K3 surfaces in arbitrary curve classes is conjecturally governed by quasi-modular forms and should satisfy holomorphic anomaly equations. The primitive case is well understood whereas the higher divisibility case is significantly more complicated. We explain the multiple cover conjecture and discuss partial results in divisibility 2. The counts of genus g curves passing through g generic points will serve as a leading example. The talk is based on joint work in progress with Younghan Bae.
Multiple covers and point counts for K3 surfacesread_more
HG G 43
* 23. Januar 2020
16:15-17:30 		Dr. Ulrike Riess
ETH-ITS
Details

Algebraic Geometry and Moduli Seminar

Titel Base loci of big and nef line bundles on irreducible symplectic varieties
Referent:in, Affiliation Dr. Ulrike Riess, ETH-ITS
Datum, Zeit 23. Januar 2020, 16:15-17:30
Ort HG G 43
Abstract In the first part of this talk, I give a complete description of the divisorial part of the base locus of big and nef line bundles on irreducible symplectic varieties (under certain conditions). This is a generalization of well-known results of Mayer and Saint-Donat for K3 surfaces. In the second part, I will present what is currently known on the non-divisorial part, including the results of an ongoing cooperation with Daniele Agostini.
Base loci of big and nef line bundles on irreducible symplectic varietiesread_more
HG G 43
19. Februar 2020
13:30-14:45 		Rosa Schwarz
University of Leiden
Details

Algebraic Geometry and Moduli Seminar

Titel Gromov-Witten invariants for BGm
Referent:in, Affiliation Rosa Schwarz, University of Leiden
Datum, Zeit 19. Februar 2020, 13:30-14:45
Ort HG G 43
Abstract An example of a Gromov-Witten invariant is the number of degree d curves in a projective plane through a number of points. There one considers the moduli space of maps of degree d from stable curves to the projective plane, and represent the passing-through-points conditions as intersecting classes of pullbacks via evaluation maps. Then the numbers are computed by pushing forward to a point. Frenkel, Teleman and Tolland in their article Gromov-Witten Gauge Theory construct similar GW-invariants for the moduli space of stable curves together with line bundles, i.e. stable maps to BGm. To work with this moduli space, they need to make slight modifications and it is not trivial to show the pushforward to a point actually yields numbers. In this talk I will illustrate the construction of and actually compute these Gromov-Witten invariants in the case of genus zero 3-marked curves.
Gromov-Witten invariants for BGmread_more
HG G 43
21. Februar 2020
16:00-17:15 		Dr. Kaloyan Slavov
ETH Zürich
Details

Algebraic Geometry and Moduli Seminar

Titel A refined version of the Bertini irreducibility theorem
Referent:in, Affiliation Dr. Kaloyan Slavov, ETH Zürich
Datum, Zeit 21. Februar 2020, 16:00-17:15
Ort HG G 43
Abstract Consider a map from a geometrically irreducible variety to a projective space and assume that its nonempty fibers are equidimensional. We bound the dimension of the locus of hyperplanes whose pullback is not geometrically irreducible; for example, if the map is dominant, this bad locus is at most one-dimensional. The proof involves a probabilistic counting argument over finite fields. This is joint work with Bjorn Poonen.
A refined version of the Bertini irreducibility theoremread_more
HG G 43
26. Februar 2020
13:30-14:45 		Dr. Longting Wu
ETH Zürich
Details

Algebraic Geometry and Moduli Seminar

Titel Double ramification and Dubrovin-Zhang hierarchies I
Referent:in, Affiliation Dr. Longting Wu, ETH Zürich
Datum, Zeit 26. Februar 2020, 13:30-14:45
Ort HG G 43
Abstract In the lecture series, we focus on the construction of Double ramification (DR) hierarchies and the conjectural relation between DR and Dubrovin-Zhang (DZ) hierarchies. DR hierarchies were constructed by Buryak. Later, a quantization of the DR hierarchies was given by Buryak and Rossi. In the first talk, Longting Wu will give a quick introduction of these constructions and discuss their properties. Various examples will be given. In the second talk, Pierrick Bousseau will talk about the conjectural relation with DZ hierarchies initiated by Buryak and discuss recent progress.
Double ramification and Dubrovin-Zhang hierarchies Iread_more
HG G 43
28. Februar 2020
16:00-17:15 		Yizhen Zhao
Institut de Mathématiques de Jussieu
Details

Algebraic Geometry and Moduli Seminar

Titel Landau-Ginzburg/Calabi-Yau correspondence for a complete intersection via matrix factorizations
Referent:in, Affiliation Yizhen Zhao, Institut de Mathématiques de Jussieu
Datum, Zeit 28. Februar 2020, 16:00-17:15
Ort HG G 43
Abstract In this talk, I will introduce two enumerative theories coming from a variation of GIT stability condition. One of them is the Gromov-Witten theory of a Calabi-Yau complete intersection; the other one is a theory of a family of isolated singularities fibered over a projective line, which is developed by Fan, Jarvis, and Ruan recently. I will show these two theories are equivalent after analytic continuation. For Calabi-Yau complete intersections of two cubics, I will show that this equivalence is directly related - via Chern character - to the equivalences between the derived category of coherent sheaves and that of matrix factorizations of the singularities. This generalizes Chiodo-Iritani-Ruan's theorem matching Orlov's equivalences and quantum LG/CY correspondence for hypersurfaces.
Landau-Ginzburg/Calabi-Yau correspondence for a complete intersection via matrix factorizationsread_more
HG G 43
4. März 2020
13:30-14:45 		Dr. Pierrick Bousseau
ETH-ITS
Details

Algebraic Geometry and Moduli Seminar

Titel Double ramification and Dubrovin-Zhang hierarchies II
Referent:in, Affiliation Dr. Pierrick Bousseau, ETH-ITS
Datum, Zeit 4. März 2020, 13:30-14:45
Ort HG G 43
Abstract In the lecture series, we focus on the construction of Double ramification (DR) hierarchies and the conjectural relation between DR and Dubrovin-Zhang (DZ) hierarchies. DR hierarchies were constructed by Buryak. Later, a quantization of the DR hierarchies was given by Buryak and Rossi. In the first talk, Longting Wu will give a quick introduction of these constructions and discuss their properties. Various examples will be given. In the second talk, Pierrick Bousseau will talk about the conjectural relation with DZ hierarchies initiated by Buryak and discuss recent progress.
Double ramification and Dubrovin-Zhang hierarchies IIread_more
HG G 43
6. März 2020
16:00-17:15 		Dr. Longting Wu
ETH Zürich
Details

Algebraic Geometry and Moduli Seminar

Titel Double ramification and Dubrovin-Zhang hierarchies III
Referent:in, Affiliation Dr. Longting Wu, ETH Zürich
Datum, Zeit 6. März 2020, 16:00-17:15
Ort HG G 43
Abstract In the lecture series, we focus on the construction of Double ramification (DR) hierarchies and the conjectural relation between DR and Dubrovin-Zhang (DZ) hierarchies. DR hierarchies were constructed by Buryak. Later, a quantization of the DR hierarchies was given by Buryak and Rossi. In the first talk, Longting Wu will give a quick introduction of these constructions and discuss their properties. Various examples will be given. In the second talk, Pierrick Bousseau will talk about the conjectural relation with DZ hierarchies initiated by Buryak and discuss recent progress.
Double ramification and Dubrovin-Zhang hierarchies IIIread_more
HG G 43

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