# Geometri- och topologiseminariet

Wednesdays at 14.15 in room Å64119 (unless specified).

*For more info contact the organiser: Georgios Dimitroglou Rizell.*

**March 4: Olof Bergvall **(Uppsala University).

Title: *Del Pezzo surfaces and rational curves*

Abstract: In this talk we will consider Del Pezzo surfaces, i.e. smooth algebraic surfaces with ample anticanonical bundles, and rational curves on such surfaces. More precisely, I will give very concrete descriptions of certain moduli spaces of Del Pezzo surfaces marked with rational curves and explain how to compute their cohomology (as representations of certain Weyl groups).

**Past seminars 2020:**

**February 26: Jérémy Guéré**(Université Grenoble Alpes).

Title: *Hodge-Gromov-Witten theory*

Abstract: Hodge-Gromov-Witten theory of a smooth projective variety X deals with the cap product of the virtual fundamental cycle on the moduli space of stable maps to X with the Euler class of the Hodge vector bundle. I recently studied its deformation invariance to singular varieties, allowing explicit computations in many cases. An important application of my theorem is a calculation of genus-zero GW invariants for some hypersurfaces in weighted projective spaces which do not satisfy the so-called convexity property. It is a first step towards a mirror symmetry statement for these hypersurfaces.

In a second part of the talk, I will describe my plan towards a calculation of GW invariants of the quintic hypersurface in P^4. It is based on a theorem Costello proved in 2003 expressing genus-g GW invariants of a projective variety X in terms of genus-0 GW invariants of the (g+1)-st symmetric power of X.

**February 19: Mark Lawrence**(Nazarbayev University).

Title: *Polynomial hulls, knots, and holomorphic motions*

Abstract: The study of polynomial hulls of compact sets in C^{n} is too challenging to have a useful general answer. Even for smooth manifolds, there is little known. One restriction is to look at tori or unions of tori in S^{1} × C which fiber over the circle. The knot class of the torus plays a crucial role, both in a positive and negative direction. The main positive result of the author and J. Duval is that a torus modeled on a square root has a polynomial hull which is fibered by varieties. On the negative side, it appears that the knot type of the torus can exclude any hull from appearing over the unit disc, but only preliminary results exist in this direction. Using the theorem of Lawrence and Duval, new types of holomorphic motions

can be constructed, which have a limited amount of branching.

**January 22: Wanmin Liu **(Uppsala University).

Title:

*Contractibility of space of stability conditions on projective plane via global dimension function*

Abstract: The global dimension function $\gldim$ is a continuous function defined on Bridgeland stability manifold, and it maps a stability condition to a non-negative real number. We compute the global dimension function $\gldim$ on the space of stability conditions on projective plane. It has the minimal value $2$, and $\gldim^{-1}(2)$ is contractible. Moreover, $\gldim^{-1}(2)$ is contained in the closure of the subspace consisting of geometric stability conditions. We show that $\gldim^{-1}([2,x)$ contracts to $\gldim^{-1}(2)$ for any real number $x>=2$. The $\gldim$ sheds some lights on why a space of Bridgeland stability conditions should be contractible. This is a joint work with Yu-Wei Fan, Chunyi Li and Yu Qiu.