9 03, 2022

Higgs bundles on elliptic surfaces and logarithmic transformations, Kyoung-Seog Lee

2022-03-09T15:59:54+02:00March 9th, 2022|Joint ICMS & IMSA Seminar|

Logarithmic transformation is an important operation introduced by Kodaira in the 1960s. One can obtain an elliptic surface with multiple fibers by performing logarithmic transformations on an elliptic surface without multiple fibers. On the other hand, vector bundles on elliptic surfaces are important objects in many branches of mathematics, e.g., algebraic geometry, gauge theory, mathematical physics, etc. In this talk, I will discuss how certain Higgs bundles on elliptic surfaces are changed via logarithmic transformations. This talk is based on a joint work with Ludmil Katzarkov

17 02, 2022

Birational geometry of sextic double solids with a compound A_n singularity

2022-02-18T09:14:03+02:00February 17th, 2022|Joint ICMS & IMSA Seminar|

Sextic double solids, double covers of ℙ^3 branched along a sextic surface, are the lowest degree Gorenstein Fano 3-folds, hence are expected to behave very rigidly in terms of birational geometry. Smooth sextic double solids, and those which are ℚ-factorial with ordinary double points, are known to be birationally rigid. In this talk, we discuss birational geometry of sextic double solids with an isolated compound A_n singularity. I have shown that n is at most 8, and that rigidity fails for n > 3.

4 02, 2022

Windows and Geometric Invariant Theory

2022-02-04T13:05:05+02:00February 4th, 2022|Joint ICMS & IMSA Seminar|

The theory of windows was introduced relatively recently by both Halpern-Leistner and Ballard, Favero and Katzarkov, and is a great tool to study derived categories of algebraic varieties that appear as GIT constructions, as well as their behaviour at wall crossings as we vary the stability conditions.

20 10, 2021

Homological Mirror Symmetry for Hypersurface Singularities

2021-10-21T12:35:26+03:00October 20th, 2021|Joint ICMS & IMSA Seminar|

In this talk, I will briefly introduce homological mirror symmetry for certain hypersurface singularities. I will introduce basic definitions, Berglund-Hubsch duality of invertible polynomials, and some known results. Then I will discuss several examples in detail.

8 10, 2021

More instances of Steenbrink Spectra in Singularity Categories

2021-10-16T13:48:03+03:00October 8th, 2021|Joint ICMS & IMSA Seminar|

Now that we have set up the Frobenius Manifold of a singularity, we can see 2 more places where the steenbrink spectrum of a singularity appears in its singularity category. Namely, the non-commutative mixed hodge structure of a singularity and the dimensional properties of the category. The former appearance being somehow natural, and the latter somewhat mysterious. In this talk we'll conduct a surface level investigation of these appearances.

8 10, 2021

Windows and the BGMN conjecture

2021-10-26T14:31:25+03:00October 8th, 2021|Joint ICMS & IMSA Seminar|

Let $C$ be a smooth projective curve of genus at least 2, and let $N$ be the moduli space of semistable rank-two vector bundles of odd degree on $C$. We construct a semi-orthogonal decomposition in the derived category of $N$ conjectured by Belmans, Galkin and Mukhopadhyay and by Narasimhan. It has blocks of the form $D(C_d)$ where $C_d$ are $d$-th symmetric powers of $C$, and the semi-orthogonal complement to these blocks is conjecturally trivial. In order to prove our result, we use the moduli spaces of stable pairs over $C$. Such spaces are related to each other via GIT wall crossing, and the method of windows allows us to understand the relationship between the derived categories on either side of a given wall. This is a joint work with J. Tevelev.

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