DISEASE CHARACTERISTICS:

• Diagnosis of 1 of the following advanced hematologic malignancies:

  • Acute myeloid leukemia (AML) meeting the following criteria:

    • Not expected to be curable with chemotherapy and meets ≥ 1 of the following criteria:

      • High-risk cytogenetics (-7, -7q, -5, -5q, t[6,9], t[9,11], complex, Philadelphia chromosome positive [Ph+])
      • AML evolved from prior myelodysplasia
      • AML secondary to prior chemotherapy
      • Failed to achieve remission
      • In second or subsequent remission
    • Marrow blasts ≤ 10% (may be achieved using chemotherapy)

  • Myelodysplastic syndromes (MDS) with high-risk features

    • International Prognostic Scoring System (IPSS) score intermediate -2 or high-risk
    • Marrow blasts ≤ 20% (may be achieved using chemotherapy)

  • Acute lymphoblastic leukemia meeting the following criteria:

    • Not expected to be curable with chemotherapy and meets ≥ 1 of the following criteria:

      • High-risk cytogenetics (Ph+, t[4,11], 11q23 abnormalities, and monosomy 7)
      • Required > 1 induction course to achieve remission
      • Failed to achieve remission
      • In second or subsequent remission
    • Marrow blasts ≤ 10% (may be achieved using chemotherapy)

  • Chronic myelogenous leukemia meeting ≥ 1 of the following criteria:

    • Accelerated phase
    • Chronic phase refractory to imatinib mesylate

    • Blastic phase

      • Marrow blasts ≤ 10% (may be achieved using chemotherapy)

  • Multiple myeloma meeting 1 of the following criteria:

    • Stage II or III disease with > first relapse or refractory disease
    • Newly diagnosed disease with chromosome 13 abnormalities

  • Lymphoma meeting the following criteria:

    • One of the following subtypes:

      • Diffuse large cell lymphoma
      • Mantle cell lymphoma
      • Peripheral T-cell lymphoma
      • T-natural killer (NK) cell lymphoma
      • Hodgkin's lymphoma

    • Disease failed to respond to primary therapy, progressed, or recurred after prior therapy

      • Patients who have failed autologous stem cell transplantation are eligible provided it has been > 1 year since transplant
• No rapid progression of malignant disease
• Not eligible for autologous stem cell transplantation

• Available umbilical cord blood (1-3 units) donor matching at ≥ 4 of 6 HLA antigens (A, B, and DR)

  • Patients with an HLA-identical or 1 antigen-mismatched related donor OR a potential HLA-matched unrelated donor matching at > 6/8 (A, B, C, DR) alleles are not eligible

PATIENT CHARACTERISTICS:

• ECOG performance status 0-2
• Creatinine < 2.0 mg/dL
• Creatinine clearance > 40 mL/min
• Bilirubin < 2.0 mg/dL
• AST and alkaline phosphatase < 3 times upper limit of normal
• Hepatitis C and active hepatitis B allowed if patient has ≤ grade 2 inflammation or fibrosis by liver biopsy
• Ejection fraction > 40% by echocardiogram or MUGA
• DLCO > 40% of predicted
• Not pregnant or nursing
• Negative pregnancy test
• No known HIV infection
• No active infection requiring ongoing antibiotic treatment

PRIOR CONCURRENT THERAPY:

• See Disease Characteristics