The Fourier–Mukai orientation conjecture for K3 surfaces

Let $Z_0$ and $X_0$ be smooth projective K3 surfaces, and let

$$\Psi\colon D^b(Z_0)\longrightarrow D^b(X_0)$$

be a Fourier–Mukai equivalence. The induced map on integral cohomology is denoted by

$$\psi\colon H^*(Z_0,\mathbb Z)\longrightarrow H^*(X_0,\mathbb Z).$$

Fix the canonical orientation of the positive definite four-planes in the cohomology of an algebraic K3 surface, obtained from the chosen orientation on the positive part of $H^0\oplus H^4$, the real and imaginary parts of the period, and an ample class. The Fourier–Mukai orientation conjecture. The cohomology action $\psi$ preserves the canonical orientation of positive definite four-planes in cohomology. This is proposed as an analogue for Fourier–Mukai functors of Donaldson’s orientation theorem for orientation-preserving diffeomorphisms of K3 surfaces. A weaker form was proposed independently by Markman, but the source gives no resolution of the stated conjecture.