ICTS

The Donaldson-Segal programme proposes to extend familiar techniques for anti-self-dual connections on 4-manifolds to special, higher-dimensional geometries. This includes the study of Calabi-Yau 3-folds, G2-manifolds, and Spin(7)-holonomy manifolds. The fundamental difficulties are compactness, deformation invariance, and orientations of the moduli space of connections. These questions are, at present, largely open. After introducing some background material, I shall focus on the orientation problem for SU(2)-bundles over G2-manifolds (joint with D. Joyce).

Seshadri constants are invariants of line bundles on projective varieties. Their study is now a very active area of research with connections to several different topics. We will give an overview of the current work on Seshadri constants and discuss some recent results.

I describe foundational aspects in defining the Floer cohomology of a Lagrangian submanifold in a symplectic manifold. An important class of Lagrangian submanifolds comes from toric symplectic manifolds, where they are just fibers of the moment map. I will give many examples of toric Lagrangians with non-trivial Floer homology. These examples will give us ideas on how to find Floer non-trivial Lagrangians in certain situations where a toric structure is only available locally. I will not assume any back ground in Floer theory.

I want to explain why any homogeneous holomorphic foliation on any Kobayashi hyperbolic complex manifold is the set of fibers of an equivariant holomorphic submersion. Along the way, I will demonstrate the connectivity of the fixed point locus of any group of biholomorphisms of any convex hyperbolic domain.

In my talk I will discuss real holomorphic sections of the Deligne-Hitchin moduli space $\mathcal M_{DH}$. This space is defined as the moduli space of $\lambda$-connections on a Riemann surface glued with the moduli space of $\lambda$-connections on the complex conjugate Riemann surface. It can be interpreted as the twistor space of the hyper-Kähler moduli space of solutions of Hitchin’s selfduality equations by the work of C. Simpson. Real holomorphic sections thereof (with respect to certain natural real structures of $\mathcal M_{DH}$) correspond to various types of (equivariant) harmonic maps into symmetric spaces. We construct two types of counter examples to the question of C. Simpson whether all real sections for the real structure corresponding to the antipodal involution are given by twistor lines, i.e., global solutions of the self-duality equations. Sections of the first type consists of harmonic maps into the 2-sphere, while sections of the second type give rise to solutions of the selfduality equations on an open dense subset $U$ of the Riemann surface with a certain asymptotic behavior at the boundary of $U.$ The talk is based on joint work with I. Biswas and M. Roeser and on joint work with L. Heller.

The purpose of this talk is to give a way of constructing real variations of mixed Hodge structures (R-VMHS) over compact Kahler manifolds by using mixed Hodge structures on Sullivan’s 1-minimal models. This construction is very similar to known ideas (e.g. Hain-Zucker 1987, Eyssidieux-Simpson 2011). But this may be essentially different from them, since the obtained R-VMHS depends on Kahler metric.

In this talk we will study the following question: how many genus zero degree d curves are there in P^3, such that their image lies in a plane (and satisfying appropriate constraints). This can be viewed as a "family" version of the classical problem of counting rational curves in P^2. We will also discuss the parallel question of enumerating planar degree d curves in P^3 that have singularities. The content of this talk is joint work with Anantadulal Paul and Rahul Singh.

The well-known determinantal varieties (vanishing of minors) can be generalized in a Jordan theoretic setting, including exceptional symmetric spaces. In joint work with M. Englis (Prague), we prove that these Jordan theoretic determinantal varieties ("Kepler varieties") are normal (using a result of Kempf on homogeneous vector bundles) and we determine the TYZ (Tian-Yau-Zelditch) expansion of reproducing Bergman type kernels associated with Hilbert spaces of holomorphic functions on Kepler varieties. Multi-variable hypergeometric functions and their asymptotic behavior play a crucial role.

In this talk we will explain how torus equivariant principal bundles over a complex affine toric variety with at most factorial singularities admit an equivariant trivialization. We will use this to give a Kaneyama-type description of the isomorphism classes of such principal bundles over a general complex toric variety with at most factorial singularities. This is based on joint work with Indranil Biswas and Arijit Dey.

A large class of moduli spaces of holomorphic vector bundles, compact manifolds and related objects possess a Weil-Petersson form, which reflect the variation of distinguished metrics on the fibers. It is desirable to extend these forms to compactifications of the respective moduli spaces; for degenerating holomorphic families an extension of the Weil-Petersson form as a positive current to the singular locus is being constructed. The resulting current then descends to a compactification of the moduli space. We explain the extension property for moduli of stable vector bundles, for Douady spaces, moduli of canonically polarized manifolds, and for degenerating families of polarized Calabi-Yau manifolds. Applications to the existence of positive holomorphic line bundles are given.

We shall briefly introduce the Kostant - Souriau method of quantization of a symplectic manifold, which is known as geometric quantization. We briefly describe our coadjoint orbit quantization of the finite Toda system (joint work with Dr. Saibal Ganguli). Next we introduce Quillen's determinant line bundle. Then we will describe the quantization of various moduli spaces arising from physics using the Quillen construction. Examples include the Hitchin system and the vortex moduli space. We will also talk about a general theorem which essentially says that the quantum bundle (or a tensor power of the same) of a compact integral K\"{a}hler manifold (or integral symplectic manifold) can be realised as a Quillen determinant bundle (joint work with Professor Mathai Varghese).