A note on a question of Markman

TL;DR

This paper extends the understanding of how certain algebraic invariants of vector bundles and polyvector fields behave under deformations of complex projective varieties, answering a question posed by Markman.

Contribution

It proves that the Hodge type of Mukai vectors remains invariant under broader deformations, not just complex structure deformations, and explores a Lie theoretic analogue.

Findings

Mukai vector remains of Hodge type under all polyvector field deformations.

Answers Markman's question regarding deformation invariance.

Explores Lie algebraic analogue of the deformation invariance.

Abstract

Let $\mathcal{F}$ be a vector bundle on a complex projective algebraic variety $X$. If $\mathcal{F}$ deforms along a first order deformation of $X$, its Mukai vector remains of Hodge type along this deformation. We prove an analogous statement for all polyvector fields, not only for those in $H^1(X,T_X)$ corresponding to deformations of the complex structure. This answers a question of Markman. We also explore a Lie theoretic analogue of the statement above.