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On a Class of Singular Complex Manifolds

Alireza Bahraini

TL;DR

This work defines and analyzes a class of singular complex manifolds arising from degenerate complex structures $J'$ on $X\setminus D$, developing a full Kodaira–Hodge theory in this setting. The authors construct a degenerate hyperKähler structure via degenerate Monge–Ampère equations, formulate a precise definition of degenerate complex manifolds, and establish both weak and strong $L^2$ Hodge theory, including a singular $\partial\bar{\partial}$-lemma and Kodaira theory with vanishing and embedding results for complements of a divisor. Key tools include $L^2$-cohomology on spaces with conical degeneracies (à la Cheeger–Dai), Hörmander $L^2$-estimates with plurisubharmonic weights, and solvability results for $\bar{\partial}_{J'}$ near $D$. The framework extends classical Kähler–Kodaira theory to spaces with complex-analytic singularities and interplays with hypoanalytic/involutive structures, offering potential applications to degenerations, mirror symmetry, and the construction of Lagrangian cycles in singular settings.

Abstract

We introduce a new class of singular complex manifolds and we develop a degenerate Kodaira-Hodge theory for this class of singular manifolds.

On a Class of Singular Complex Manifolds

TL;DR

This work defines and analyzes a class of singular complex manifolds arising from degenerate complex structures on , developing a full Kodaira–Hodge theory in this setting. The authors construct a degenerate hyperKähler structure via degenerate Monge–Ampère equations, formulate a precise definition of degenerate complex manifolds, and establish both weak and strong Hodge theory, including a singular -lemma and Kodaira theory with vanishing and embedding results for complements of a divisor. Key tools include -cohomology on spaces with conical degeneracies (à la Cheeger–Dai), Hörmander -estimates with plurisubharmonic weights, and solvability results for near . The framework extends classical Kähler–Kodaira theory to spaces with complex-analytic singularities and interplays with hypoanalytic/involutive structures, offering potential applications to degenerations, mirror symmetry, and the construction of Lagrangian cycles in singular settings.

Abstract

We introduce a new class of singular complex manifolds and we develop a degenerate Kodaira-Hodge theory for this class of singular manifolds.

Paper Structure

This paper contains 19 sections, 22 theorems, 142 equations.

Table of Contents

  1. Introduction
  2. DCMA equations and Degenerate HyperKähler structure
  3. Involutive Structures and Hypoanalytic manifolds following F.Trèvestre
  4. Hodge Theory on Degenerate Complex Manifolds
  5. $L^2$ Hodge Theory on manifolds with non-isolated conical singularities following Cheeger-Dai
  6. Definition of Degenerate Complex Manifolds
  7. Weighted $L^2$ Spaces and $L^2$ Hodge Decomposition
  8. Fundamental Inequality.
  9. Weak Hodge Theorem
  10. Strong Hodge Theory for Degenerate Complex Manifolds
  11. Singular $\partial\bar{\partial}$- Lemma
  12. Kodaira theory
  13. Kodaira Vanishing
  14. Kodaira Embedding Theorem.
  15. Solvability of $\bar{\partial}_{J'}$ in a neighborhood of $D$
  16. ...and 4 more sections

Key Result

Lemma 1

If $A_0 ^{p,q} (X\setminus D)$ denotes the space of smooth $(p,q)$-forms with compact suppport in $X\setminus D$ then $A_0 ^{p,q} (X\setminus D)$ is dense in $W'^{p,q} _l (X)$.

Theorems & Definitions (37)

  • Definition 1
  • Remark 1
  • Definition 2
  • Lemma 1
  • proof
  • Lemma 2
  • Corollary 1
  • proof
  • Theorem 1
  • Lemma 3
  • ...and 27 more