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Let $\mathcal{X} \to \Delta^* $ be a family of complex projective varieties over a punctured disc. Then, for any fibre $X$, there is a monodromy action $M: H^* (X) \to H^*(X)$.

Is there a monodromy action on the K-theory of $X$? What about on various derived categories of sheaves?

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    $\begingroup$ What monodromy acts on is the homotopy type, so I'd say it acts on topological K-theory and on categories of derived local systems, but not on algebraic K-theory or on the usual derived categories (coherent or constructible). $\endgroup$
    – Dustin Clausen
    Commented Nov 20, 2010 at 19:34
  • $\begingroup$ @Dustin: I've never thought about the category of derived local systems, what is that thing like? $\endgroup$
    – Vivek Shende
    Commented Nov 20, 2010 at 20:09

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$(1)$ When $X$ is Calabi-Yau, there is a "derived version" of monodromy around a conifold point. It was also investigated in the framework of string theory, see for instance the paper

Paul S. Aspinwall1 and Michael R. Douglas

"D-brane stability and monodromy"

Journal of High Energy Physics (2002).

$(2)$ When $X$ is a $K3$ surface, it is possible to construct a representation of the group of auto-equivalences $\textrm{Aut } D(X)$ on the cohomology of all moduli spaces of stable sheaves (with primitive Mukai vectors) on $X$, and to relate this representation to the monodromy of the Hilbert schemes $X^{[n]}$ of points on the surface. See

Eyal Markman

"On the monodromy of moduli spaces of sheaves on K3 surfaces",

J. Algebraic Geom. 17 (2008), 29-99.

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  • $\begingroup$ The Aspinwall-Douglas paper shows that a monodromy around the conifold point induces an autoequivalence of $D^b(Y)$ on the mirror partner, not on $X$ itself. (Note that the notation of $X$, $Y$, is switched compared to their article, to be consistent with the setup in Vivek's question.) $\endgroup$
    – Arend Bayer
    Commented Nov 21, 2010 at 15:52
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When the fibers $X$ are Calabi-Yau threefolds, the monodromy preserves a symplectic form, and $Y$ is their mirror partner, then one expects a corresponding autoequivalence on $Y$. (The monodromy induces a symplectic auto-equivalence of any fiber, so an auto-equivalence of the Fukaya category, and by Kontsevich's homological mirror symmetry this gives an auto-equivalence of the derived category of coherent sheaves on $Y$.)

In the case of a Dehn twist on the symplectic side, this yields a Seidel-Thomas on the coherent sheaves side, but many other examples have been studied.

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