Tian's theorem for Grassmannian embeddings and degeneracy sets of random sections (Bingxiao LIU - Université de Lille)

In this talk, I will present an extension of Tian’s approximation
theorem to the setting of Grassmannian embeddings and discuss applications to
the equidistribution of degeneracy sets of random holomorphic sections.

Let $(X, \omega)$ be a compact Kähler manifold, $(L, h^L)$ a positive line
bundle, and $(E, h^E)$ a Hermitian holomorphic vector bundle of rank $r \geq
1$ on $X$. We prove that the pullback by the Kodaira embedding associated to
$L^p \otimes E$ of the $k$-th Chern class of the dual of the universal bundle
over the Grassmannian converges as $p \to \infty$ to the $k$-th power of the
Chern form $c_1(L, h^L)$ for $0 \leq k \leq r$. If $c_1(L, h^L) = \omega$, we
also determine the second term in the semiclassical expansion, which involves
$c_1(E, h^E)$. As a consequence, we show that the limit distribution of zeros
of random sequences of holomorphic sections of $L^p \otimes E$ is
equidistributed on $X$ with respect to $c_1(L,h^L)^r$. Furthermore, we compute
the expectation of the currents of integration along degeneracy sets of random
holomorphic sections. This is a joint work with Turgay Bayraktar, Dan Coman,
and George Marinescu.