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Tian's theorem for Grassmannian embeddings and degeneracy sets of random sections

Turgay Bayraktar, Dan Coman, Bingxiao Liu, George Marinescu

TL;DR

The paper proves a Tian-type semiclassical expansion for Grassmannian embeddings associated to high powers of a positive line bundle twisted by a Hermitian vector bundle, giving a precise leading term \(\binom{r}{k} c_1(L,h^L)^k\) and, when \(c_1(L,h^L)=\omega\), a second-order correction involving \(c_1(E,h^E)\). It then connects these geometric asymptotics to probabilistic questions: zeros of random sections of \(L^p\otimes E\) equidistribute with the predicted current and the degeneracy-sets currents have explicit expected values given by pullbacks of higher Chern classes via the Kodaira map, interpreted through meromorphic transforms in the sense of Dinh–Sibony. The framework unifies deterministic asymptotics with stochastic geometry, yielding almost-sure convergence results and precise speed-of-convergence estimates, and includes specializations to simultaneous zeros (recovering classical results) and determinant-based degeneracy currents. Overall, the work extends Tian’s expansion to Grassmannian embeddings, enriches the theory of randomness in holomorphic sections, and provides a robust toolbox for studying zeros and degeneracy sets via Chern-class pullbacks.

Abstract

Let $(X,ω)$ be a compact Kähler manifold, $(L,h^L)$ be a positive line bundle, and $(E,h^E)$ be a Hermitian holomorphic vector bundle of rank $r$ 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 $c_1(L,h^L)=ω$ 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 high powers $L^p\otimes E$ is $c_1(L,h^L)^r$. Furthermore, we compute the expectation of the currents of integration along degeneracy sets of random holomorphic sections.

Tian's theorem for Grassmannian embeddings and degeneracy sets of random sections

TL;DR

The paper proves a Tian-type semiclassical expansion for Grassmannian embeddings associated to high powers of a positive line bundle twisted by a Hermitian vector bundle, giving a precise leading term \(\binom{r}{k} c_1(L,h^L)^k\) and, when \(c_1(L,h^L)=\omega\), a second-order correction involving \(c_1(E,h^E)\). It then connects these geometric asymptotics to probabilistic questions: zeros of random sections of equidistribute with the predicted current and the degeneracy-sets currents have explicit expected values given by pullbacks of higher Chern classes via the Kodaira map, interpreted through meromorphic transforms in the sense of Dinh–Sibony. The framework unifies deterministic asymptotics with stochastic geometry, yielding almost-sure convergence results and precise speed-of-convergence estimates, and includes specializations to simultaneous zeros (recovering classical results) and determinant-based degeneracy currents. Overall, the work extends Tian’s expansion to Grassmannian embeddings, enriches the theory of randomness in holomorphic sections, and provides a robust toolbox for studying zeros and degeneracy sets via Chern-class pullbacks.

Abstract

Let be a compact Kähler manifold, be a positive line bundle, and be a Hermitian holomorphic vector bundle of rank on . We prove that the pullback by the Kodaira embedding associated to of the -th Chern class of the dual of the universal bundle over the Grassmannian converges as to the -th power of the Chern form , for . If we also determine the second term in the semiclassical expansion, which involves . As a consequence we show that the limit distribution of zeros of random sequences of holomorphic sections of high powers is . Furthermore, we compute the expectation of the currents of integration along degeneracy sets of random holomorphic sections.
Paper Structure (15 sections, 21 theorems, 192 equations)

This paper contains 15 sections, 21 theorems, 192 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Chern forms of Hermitian vector bundles
  4. Universal bundle on Grassmannian
  5. Bergman kernel asymptotics
  6. Dimension of the space of holomorphic sections
  7. Tian's theorem for Grassmannian embeddings
  8. Kodaira maps as meromorphic transforms
  9. Expectation currents of degeneracy sets
  10. Cohomology class of expectation currents
  11. Invariance properties under group actions
  12. Formula for the expectation current
  13. Degeneracy set of a tuple of sections
  14. Distribution of zeros of random sections
  15. Simultaneous zero sets of holomorphic sections

Key Result

Theorem 1.1

Let $(X,\omega)$ be a compact Kähler manifold of dimension $n$, $(L,h^L)$ be a positive line bundle on $X$, and $(E,h^E)$ be a Hermitian holomorphic vector bundle on $X$ of rank $r\leq n$. Then for every $0\leq k\leq r$ we have in the $\mathscr{C}^\infty$ topology as $p\to\infty$, Assume further that the bundle $(L,h^L)$ polarizes $(X,\omega)$, that is $\omega=c_1(L,h)$. Then for every $0\leq k\l

Theorems & Definitions (42)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Theorem 2.3: MM07
  • Theorem 3.1
  • ...and 32 more