Title of Talk: On varieties with Ulrich twisted cotangent, tangent, normal and conormal bundles
Speaker: Debojyoti Bhattacharya, TIFR, Mumbai
Date: November 28, 2023
Time: 16:00:00 Hours
Venue: AG-77
Abstract: Ulrich bundles on a smooth, projective variety can be considered as vector bundles having the simplest possible cohomology (to be more precise, they are vector bundles without intermediate cohomology and with the maximal possible numbers of generators). They appear in the arena of Algebraic Geometry with the foundational paper by Eisenbud and Schreyer in 2003. Since then many efforts have been driven towards finding an answer to the fundamental question: Does every smooth projective variety support an Ulrich bundle (nowadays called the ?Eisenbud -Schreyer conjecture?).
In this talk, we will first recall a brief history of Ulrich bundles followed by some of their importance and major themes of research in this area. We will then move on to discuss the following problem: Let $X$ be a smooth projective variety of dimension ${n\geq1}$ and $H$ be a very ample divisor on $X$ giving the embedding $X \subset \mathbb{P}^N$ . Let ${k \in \mathbb{Z}}$. Then characterize the triples $(X, H, k)$ such that $\Omega_{X}(k)$, $T_{X}(k)$ (and respectively for the other two bundles mentioned in the title) is Ulrich for $(X, H)$.
We will give sketches of the proofs in a few cases where a clear classification is available and statethe results for other cases. This is based on the preprints https://doi.org/10.48550/arXiv.2205.\linebreak06602 by Angelo Felice Lopez,https://doi.org/10.48550/arXiv.2301.03104 by Angelo Felice Lopez and Debaditya Raychaudhury and https://doi.org/10.48550/arXiv.2306.00113 by Vincenzo Antonelli, Gianfranco Casnati, Angelo Felice Lopez and Debaditya Raychaudhury.